Transcript for:
Mastering SAT Math Strategies

I've scored 1590 on the sat with perfect 800s on back-to-back SAT Math tests and in this video I show you 216 practical tips tricks and strategies for the digital SAT Math section each demonstrate on an actual question from digital SAT practice tests one through four for more digital SAT prep content make sure to like this video And subscribe to my channel here's how to answer this SCT math question in under 20 seconds question is what is 10 of 470 we can take 10 and express it as a decimal of 0.1 and then multiply it by 470 and that'll give us our answer of 47 so our answer will be answer Choice B here's a trick that'll save you time on the SAT Math section question 2 says which equation has the same solution as the given equation if we take a quick glance at our answer choices we see we have 4X in every single one of our answer choices which means if we look at our equation on top we have 4X plus 6 equals 18 all we have to do is subtract 6 from both sides and we're left with 4X is equal to 12. now we don't actually have to solve for x because we have 4X in every answer choice so our answer is answer Choice C here's a tip to be more efficient on the SAT Math section if you glance the answer choices of question 3 you'll see that all of them are going to be inequalities so as you read through this word problem you want to go ahead and write out the inequality to save you time so if we go and start from the beginning the total cost in dollars to rent a surfboard consists of a 25 service fee so we can go ahead and write down that 25 and that's a one-time charge because the service fee and a 10 per hour rental fee so we have plus 10 H we'll use H to represent hours a person runs a surfboard for T hours so we have to replace H with tidon and intends to spend a maximum of 75 dollars to run the surfboard so that they're going to spend a maximum of 75 so this total has to be less than or equal to 75 we find which inequality represents the situation and as we can see that is going to be answer Choice D so D would be our answer there here's how to solve this sat math question in less than 30 seconds the function G is defined by G of X is equal to x squared plus nine for which value of x is G of x equal to 25. well since we have the G of X equaling 25 we go and plug that in for G of X now we have 25 is equal to x squared plus nine now we're going to subtract 9 from each side so we can start isolating that x squared value now we have 16 is equal to x squared to isolate X we need to take the square root of x squared and the square root of 60 and the square root of 16 is 4 which means 4 will equal x and our answer will be a here's how to solve this probability question on the SAT question 5 States each face of a fair 14 sided die is labeled with the number 1 through 14. with a different number appearing on each face if the die is ruled one time what is the probability of rolling a 2 well only one of the 14 faces has a 2 on it so the answer has to be 1 over 14 which is answer Choice a here's a trick for converting units on the SAT question 6 States a printer produces posters at a constant rate of 42 posters per minute at what rate in posters per hour does the printer produce posters so what we're going to do here is we're going to convert from 42 posters per minute which I'll represent as P per M and we have to convert that into posters per hour so we're going to do is we're going to multiply by 60 minutes per hour so as you can see our minutes which I'm representing as M are going to cancel here and we'll be left with posters P per hour H so 42 times 60 is going to equal 2520 posters per hour so our answer is 2500 and 20. here's how to solve this sat math question in under 40 seconds question 7 States the function f is defined by the equation f of x equals 7x plus 2. what is the value of f of x when x equals four well all we have to do is plug in 4 for X so we do 7 times 4 plus 2. 7 times 4 is going to be 28 28 plus 2 will be 30 30 is going to equal our value of f of x and be our answer here's how to solve this sat math question in under 60 seconds question 8 States a teacher is creating an assignment worth 70 points the assignment will consist of questions worth one point and questions worth three points which equation represents the situation where X represents the number of one point questions and Y represents the number of three-point questions we know our total number of points is going to be 70. so it's gonna have to equal out to 70. now for every X question we get right which is every one point question we get one point so we have X Plus now our three point questions which are represented as y so X Plus 3y must equal 70 and our answer will be D here's how to answer the similar triangles question on the SAT question 9 States right triangles l m n and pqr are similar where l and m correspond to p and Q respectively angle M has a measure of 53 degrees what is the measure of angle Q so what we see here is since we have similar triangles where l and m correspond to p and Q we also know that n must correspond to R okay we see that angle M has a measure of 53 degrees okay and then we're asked for the measure of angle Q we see that M corresponds to angle Q since they're similar triangles so they're going to have the same angle measure okay so that 53 degrees that we are given for the angle M must also be the angle measure of angle Q so our answer will be B here's a trick for the sac math section question 10 states the solution to the given system of equations is X Y what is the value of x whenever you see stacked equations like this on the SAT it's usually a sign that you're going to be able to add subtract or substitute one of the equations with the other in order to get your answer faster in this case we're asked for the value of x and we're given that Y is equal to negative 3x so what we're going to do is we're going to substitute in that negative 3x for y in this below equation so now what we're going to have is we're going to have 4X plus y when we know Y is negative 3x so so 4X minus 3x is equal to 15. now we know that 4X minus 3x is X which means that X must equal 15 and our answer is answer Choice C here's how to answer scatter plot questions on the new digital sat question 11 states which of the following equations is the most appropriate linear model for the data shown in the scatter plot what you're going to want to do here since we're looking for a linear model to represent the scatter plot let's go ahead and draw a line just to give you a rough reference as you can see we have a negative slope which means that answer choices C and D cannot be true since they have positive slopes the next thing we'll look at is the differences between s choices A and B which are their y-intercept we see that we have a positive y-intercept of right around 10. so if we look at answer Choice a it would have a y intercept of negative 10 which cannot be correct and our answer Choice must be answer Choice B here's how to answer this polynomials question on the SAT question 12 States the graph of y equals f of x is shown where the function f is defined by f of x equals ax cubed plus BX squared plus CX plus d where a b c and d are constants for how many values of X is f of x equals zero well all we have to do is find where f of x equals zero and count the number of points we see we have one here one here and one here so our answer will be three which is Ann's Choice C here's a tip to be more efficient on the SAT Math section if you see a word problem like this you're going to want to start writing out the equation as you read it so watch How I do this problem question 13 states Vivian bought party hats and cupcakes for 71 dollars so 71 is going to be the total amount she spends each package of party hats costs three dollars so three dollars times our number of packages of party hats which I'll represent as p and each cupcake costs one dollar so you could make it plus 1C but since a one in front of C is redundant we'll just get rid of the one so now we have our equation well if you bought 10 packages of party hats that means 10 is equal to P so we can go and substitute that in so now I'll go ahead and erase that P and I can put in 10 so we know that's the amount that she had bought of those party hats we have three times ten we know that's going to equal 30. so we're gonna have 71 is equal to 30 plus C now we want to isolate C to solve for it to do that we're going to subtract 30 from each side 71 minus 30 is going to leave us with 41. so now we have 41 is equal to C and our answer will be 41. here's an example of when you should use your graphing calculator on the SAT Math section if you take a look at question four 14 it states what is one of the solutions to the given equation now you could solve this without a graphing calculator by factoring but it's really a smarter choice just to go ahead and use the graphing calculator now as you can see here this is Desmos and you're going to have essentially a version of Desmos on the Blue Book sat on the new digital sat now what I recommend doing is you just go ahead and put in this equation into the graphing calculator now keep in mind you want to substitute out Z or any other variable they put in there for x and once you do that you see that you have your equation graphed out and then since it's set equal to zero all you have to do is find where it equals zero you can see that's at negative 12 and at 2. so you have two potential answers here one is negative 12 and that would be correct and the other answer is 2 and that would also be correct so you could put in one of either of those answers but I want to quickly touch on why I think it's smart to use the graphing calculator here instead of actually Factor this out yourself if you're factoring this out yourself the chances of you making a mistake go up versus if you use a graphing calculator now keep in mind using the graphing calculator that's built into the digital sat only takes a couple of seconds right and factoring May save you a couple of seconds but in my opinion it's not worth the risk of you actually making a mistake doing multiplication or doing addition or doing subtraction anything along those lines in my opinion with questions like this it's just much smarter to go ahead and use the graphing calculator you avoid making many many many errors that can be made when factoring so with questions like these just use the graphing calculator here's how to solve this sat math question about exponential growth question 15 states that bacteria are growing liquid growth medium there were 300 000 cells per milliliter during an initial observation the number of cells per milliliter doubles every three hours that's important how many cells per milliliter will there be 15 hours after the initial observation so our starting amount is 300 000 so let's go ahead and write that down now we're going to be using the equation for exponential growth keep in mind we are doubling every three hours since we're doubling a two is what's going to go in these parentheses now we are going to raise it to the power of 15 over 3. and I'm going to explain why here and it's very important you pay attention because this equation is commonly tested on the SAT Math section so the reason we are raising it to the power of 15 over 3 and keep in mind we're raising this 2 to the power of 15 over 3 three is because we are doubling every three hours since we're doubling every three hours that gives us that three in that denominator of that exponent now we're asked how many there's going to be after 15 hours which is why we have that 15 in the numerator okay so now as you can see we have our equation we put this in our calculator and we get our answer of D 9.6 million here's how to solve this question about simplifying exponents on the SAT question 16 states which expression is equivalent to 6X to the power of 8 times y squared plus 12x squared y squared what you're going to do here is you're going to look for common factors that you can pull out of each of these so as we can see we have y squared in both of them we also have 6 and 12 which we can pull out a 6 from 12. we also have x to the power of 8 and x squared okay so the way that we're going to do this is we're going to pull 6 out we're going to pull x squared out and we're going to pull y squared out now we still have to multiply get this equation still okay so what do we have to multiply by here well we're going to need x to the power of 6 still to get to that x to the power of 8. keep in mind that this x squared times this x to the 6 is what's going to give us x to the power of 8. we're still going to have that y squared we have that six out front so now we have our first term now we know that we also have to have another term so we have six here but we have 12 up here okay so to get to that 12 there we're gonna have to add 2 and we still have x squared and Y squared here as well so we see that we're ultimately going to end up with 6X squared y squared times x to the power of 6 plus 2 which we see is going to be answer Choice C here's how to better interpret word problems on the SAT Math section question 17 States a neighborhood consists of a two hectare Park and a 35 hectare residential area the total number of trees in the neighborhood is 3934. the equation 2x plus 35y is equal to 3934 represents the situation which of the following is the best interpretation of X in this context well what we have to look at here is what does this total represent well it represents the total amount of trees now we also have to look at what gets us to that total okay we have 35 times y well 35 is our area but we have to multiply by the number of trees per unit of area now keep in mind we're asked for to solve for x here so we're looking at the amount of trees per hectare in the park okay so X must be the average number of trees per hectare in the Park which is answer Choice a here's a tip to solve SAT Math Questions when your answer choices aren't in point-slope form if we take a look at this question it says the graph shows the relationship between the number of shares of stock from company a which is represented as X and the number of shares of stock from Company B which is represented as y that Simone can purchase which equation can represent this relationship and we see we have company B and Company a and obviously we have the correct correspondence of x to a and Y to B so that's good to know now in this case I see that A and C both are in point slope form so I'm going to go ahead and just check if they're correct since that's the easiest thing to do so here I see that both of these have y-intercepts that are not correct I have 12 and 8 but the actual Y intercept is 40. so I can go ahead and get rid of a and I can get rid of C now you could go ahead and convert B and D into point-slope form and there's not an issue with that but it does take longer than what I'm about to show you instead what you could do is you could know that the slope which will represent as m is equal to negative A over B when we have our equation set up like this where we have a coefficient in front of X plus a coefficient in front of y equals some constant now the next thing I'm going to do is I'm going to check what my slopes are for each one okay so for answer Choice B here we see that our slope would be negative 8 since 8 is our a value over 12. okay and then for our D value we see that we would have negative 12 over 8. now I'm going to see what my actual slope is okay my actual slope is I go down 40 and I go over 60. so that'd be negative 40 over 60 which we know we can simplify to negative 4 over 6. Okay negative 4 over 6 is equal to negative 8 over 12 when we simplify negative 8 over 12. so our answer there is going to be answer Choice B here's how to solve this sat math question in under 60 seconds question 19 States Circle A has a radius of 3 and in Circle B has a radius of 129 and where N is a positive constant the area of circle B is how many times the area of circle a well let's start by finding the area of circle B then so Circle B has a radius of 129 n now keep in mind the formula for the area of a circle is pi r squared we know our radius is 129 n and all of that is going to get squared now we have to divide that then by the area of circle a to find out how many times bigger it is so we know that the radius of circle a is 3n squared now keep in mind that n here is going to end up being squared in both the top and the bottom okay so first thing it's very obvious that those Pi's cancels most people will get that now the next thing we need to understand is that 129 squared times N squared is equal to what we have here of 129 times N All To The Power of 2. Now on the bottom then we end up with 3 squared times N squared as well and we see that those n squareds are going to cancel okay that's really important to know because now the question becomes very easy we can just plug it into our calculator and when we plug into our calculator 129 squared divided by 3 squared which we can also just know is 9 then you're going to end up with answer Choice D here's how to avoid this tricky question on the SAT question 20 States the frequency table summarizes the 57 data values in a data set what is the maximum data value in the data set now most people when they look at this might think that the answer is 11 because it has the highest frequency and they think that that's the maximum since it's the highest value in the frequency section of the table but that's incorrect the actual answer here is 14. now the reason why is because they're asking for the maximum data value they're not asking for the highest frequency value okay that would be 11 because the data value there is 11 but they are asking for the maximum data value which means our answer has to be 14. here's how to answer this question about circles on the SAT question 21 says a circle in the X Y plane as a diameter within points 2 4 and 214 an equation of the circle is x minus 2 squared plus y minus 9 Square equals r squared where R is a positive constant what is the value of R okay what you need to understand here is that this is a question dealing with the circle equation so it's important you know the circle equation going into the SAT because it is tested fairly often now as far as how you answer this it's actually pretty easy as long as you can recognize this simple trick when you're given a circle question you're given endpoints oftentimes they're endpoints that are specifically designed to make the question easier so you want to look at them pretty closely and here what we notice is that we have an x coordinate that is the same in both of these sometimes it'll be a y coordinate that is the same in both but either way try to find if you can see either the x coordinates are the same or the Y coordinates are the same because it'll make the question a lot easier in this case since our x coordinates are the same we're going to look at how far apart are our Y coordinates and we see that from 4 to 14 as a difference of 10. now keep in mind that the circle has a diameter with these endpoints okay so if the circles like this and this is our diameter between our points and it's 10 well that means our radius must be 5 because our diameter is double our radius okay so the value of R and keep in mind that this is the equation of a circle okay where R is representing our radius and we're asked for the value of R not r squared the value of R then must be five okay so our answer is 5. here's how to solve questions that ask you to convert from radians to degrees on the SAT question 22 States the measure of angle R is 2 pi over 3 radians the measure of angle T is five pi over 12 radians greater than the measure of angle R what is the measure of angle T in degrees so we have to start by finding the measure of angle T in radians and then after that we'll convert to degrees so we know that the angle R is 2 pi over 3 radians now we have to add on to that the 5 pi over 12 radians since 5 pi over 12 is the amount of radians that we are greater than the angle measure of R to get to angle T from here we need to have a common denominator in order to add these fractions so to do that I'm just going to go I see 3 and 12 I'm just going to go ahead and make it 12. I know that you could sort of simplify it but I'm not going to so I'm going to multiply 3 by 4 to get to 12 and I'm going to multiply with 2 by 4 which will give me 8. so now I have 8 pi over 12. now I can go and add these together and I'm going to see that's going to get me 13 pi over 12. now from here I need to go ahead and convert 13 pi over 12 in 2 degrees now this is really easy but some people can get tripped up on if they don't know this trick in order to convert from radians to degrees you need to multiply by 180 degrees over Pi radians now keep in mind that we have radians as our units in here with this 13 pi over 12. so to get rid of those radians we have to have radians in the denominator right here which is why we have Pi on the bottom okay so when we go ahead and do this we're going to end up getting rid of our PI on each one sorry I did not mean to erase that let me go and put that back okay we're going to end up canceling out those pies and we're going to end up with 13 times 180 over 12 which we're going to go ahead and put into our calculator and once we do that we're going to see that that turns out to be 195 degrees which is answer Choice C here's how to answer questions about converting units on the SAT question 23 States a certain town has an area of 4.36 square miles what is the area in square yards of this town now when you're converting area as your unit you have to keep in mind that you up squaring it right since we have square miles and square yards so when we make this conversion of one mile equaling 1 760 yards what we have to actually do is we have to take our first our 4.36 square miles and then we have to actually multiply that by 1760 squared since we're converting from square miles to square yards we have to square the unit that we are converting okay and when we do that we put it in our calculator and we end up with an answer of answer Choice D 13 million five hundred and five thousand five hundred and thirty six here's how to solve this difficult sat math question question 24 States for line H the table shows three values of X and their corresponding values of Y Line K is the result of translating line H down 5 units in the X Y plane what is the x-intercept of line K well in order to get to line K we need to first know the equation for line H so let's go and find that so let's start by finding our slope we see that we go over 5 and we go up 30. so we have 30 over 5 as our slope so let's go ahead and write that down we're going to get 30 over 5 which we know is equal to 6 as you equal to our slope which we'll call M now in order to solve for our full equation we're going to go ahead and put in some of these points to get our y-intercept okay so we see that X is 18 when Y is 1. so Y is 130 so we're going to have 130 is equal to 6 times 18 plus some constant B now let's go ahead and solve for this so 6 times 18 is going to give us 108 so let's go ahead and subtract 108 from each side here 130 minus 108 is going to leave us with 22 is equal to B okay so now we have our equation so let's go ahead and write out our equation all right so we have y is equal to 6 X Plus 22. now we don't use this to find our answer reacts we have to keep in mind that line H is translated down 5 units so if we're going down 5 units instead of plus 22 we would have plus 17. okay so let's go ahead and put in that plus 17. okay so now we have our equation for line H now we can go and find our x-intercept well where's our x-intercept that is where Y is equal to zero so let's go ahead and take out y we'll put in 0. and now I'll be able to track 17 from both sides okay when we do that we're going to end up with negative 17 is equal to 6X now we're going to go ahead and divide both sides by 6 and we're going to get that X is equal to negative 17 over 6 which we see is going to be answer Choice D here's a great example of an SAT question that is made way easier by using the graphing calculator that is available on the digital sat question 25 says in the X Y plane the graph of the equation y equals negative x squared plus 9x minus 100 intersects the line Y equals c at exactly one point what is the value of C now the key thing here is that it intersects at exactly one point which is what is going to allow us to do this super easily using Desmos so as you can see what I did here is I went ahead and I plugged in this equation into Desmos now as you can see from this negative x squared and also from the graph it's going to open down so in order for us to to intersect at exactly one point with the equation y equals c that's going to have to happen right at this maximum value so if we take a look at the graph all we got to do is find that point we see that point is negative 79.75 we just to match that up with one of our answer choices now now I see with our answer choices they're all negative but this is obviously going to be way too small value negative 100 we know is not equal to negative 79.75 negative 481 over 4 would be something below negative 100 so it can't be correct and our answer would have to be C here's how to make this really difficult sat math question much easier if you take a look at question 26 States for each real number R which of the following points lies on the graph of each equation in the X Y plane for the given system now if you see stacked equations like this oftentimes you'll be looking to sort of add or subtract to get to your answer but in this case since we're asked for which of the following points lies on the graph of each equation what we're actually going to do is check if if something lies in the first equation if it also will will be a solution to the second equation so the way we're going to do this is in this case we have 2X plus 3y equals 7 we have 10x plus 15y equals 35. I'm going to look for proportions or ratios here I see that to get from 7 to 35 I'd be multiplying by five I see to get from 2 to 10 I'd multiply by 5. I see to get from 3 to 15 I would multiply by 5. so in this case I know that since all of those numbers if they're multi by 5 will get me that next equation I know that if if a solution set it works on the first equation it will also be true on the second equation right so now in this case I only have to actually check on one of the equations and if it is true then it means that it will work for the other equation as well so that's going to save me half the time now the next important thing to help you here is the order at which you're going to check these because the reason this question is so difficult is because it's time consuming it's not necessarily that is is extremely complex but the real only way to check these is substitution which is obviously not ideal at all substitution takes a very long time but thankfully we already gotten rid of half the work let's let's be smart about the way we go about this so what I'm going to look for here is which of these answer choices I would start with because if you start with the wrong answer choices oftentimes it's going to be a little bit more lengthy to figure out that it's actually wrong and the reason why is because typically the correct answer choices will just fit nicely it's just kind of the way it works usually most of the time not always but most of the time so the way that I'm going to choose which answer Choice actually substitute in first here is I'm going to look for for things that will cancel out okay so the first thing I'm going to do is since I see three out of my four answer choices are dealing with sort of denominators of two or three I'm going to be working with this first equation okay and I would usually pick that one anyway just because it's smaller numbers than sort of 10 15 and 35 so that's where I'm going to be checking these possible solutions now the next thing I'm going to look at is which ones will cancel out nicely right and the way that I look at that is I'm going to look for X values that are divided by 2 since I have this 2 and look for y values that are divided by 3 because I have that 3. okay so I see here my X values are over two okay so that is one that I would probably start with and if I look at b or I'm sorry if I look at number c i see that in this y value I have ones that are over three so that's the one that I would check next and then I have DNA and I'd probably just check D third and then a fourth I'm checking a fourth because it's the most complex I'm checking D third because it doesn't have something that nicely cancels out okay so from here this is the order that I would check these in on a substitution question now from this point I would just go ahead and start substituting so let's go ahead and do that so we're going to have 2 multiplied by negative three R over two plus seven over two and then we are also going to have that plus 3 times y we know that Y is going to be R so plus 3r is going to equal 7. now let's find out if this is true so we can go ahead and cancel this 2 out with this 2 and that two out as well with this two because we know that we would distribute this okay so once we do that we end up with negative three r plus seven and then plus 3r is equal to 7. now we have a negative three R and A plus three R which will cancel out to zero so we're going to end up with 7 is equal to seven okay we end up with 7 is equal to seven now is that true yes so is that a solution yes so is our answer B yes here's a really difficult sat math question made super easy question 27 States the perimeter of an equilateral triangle is 624 centimeters the height of the triangle is K times root 3 centimeters where K is a constant what is the value of K well if we go ahead and draw this out it's going to make us a lot easier to visualize so we have our equilateral triangle we have our height which is going to form a right angle there since it's equilateral we know that we have side lengths 60 and 60. and we also know that since we have a 90 degree angle and a 60 degree angle this angle has to be 30 degrees the next thing we know is that our perimeter is 624 okay and that's perimeter since we have an equilateral triangle if we divide it by 3 we're going to get our side lengths so that's going to be 208 as our side length now the next thing to note is that we have our height K times root 3. now this is important because when we have a 60 30 90 triangle we know that the value here must actually be K and the reason why if I go ahead and draw this out very quickly okay is that that in a 60 30 90 triangle which I'll go ahead and just put those numbers there so you can recognize them and you should try to memorize this for the SAT if you have side length x there you're gonna have side length x times root 3 there and you're going to have 2X here so in order to actually solve for the value of K I just have to divide this 208 by 2 to get the value of K right here okay so 208 over 2 is going to leave me with 104 as my answer and I'll just go ahead and put 104 as the value of K here's how to solve this oddly worded sat math question question 1 states that Tilly earns P dollars for every W hours of work which expression represents the amount of money in dollars till he earns for 39 W hours of work well we know that she's earning P dollars per every W hours of work and we know that we can set that proportion equal to the amount that she's earning per 39 W hours of work and now on our numerator we'll just have that labeled as X and that's what we need to solve for here so to isolate X we'll multiply both sides by 39w to isolate X so now we have 39 W time P all over W is equal to X now we see our W's are going to cancel and we're going to be left with 39p keep in mind that that is p so I'm gonna put it parentheses just to make it clear so 39 times p is equal to X and we see that that is going to be answer Choice a here's how to solve this seemingly simple SAT Math problem question 2 States for a training program one rides his bike at an average rate of 5.7 minutes per mile which function m models the number of minutes it will take one to write X miles at this rate now the reason this question can be Troublesome for some people taking the SAT is because it really does deal in details so let's go ahead and focus on the details that we need to pay attention to in this case we are asked for the model of the number of minutes it will take to ride X miles at this rate now we have to keep in mind that our units were given for our rate is 5.7 minutes per mile so this is actually very easy but you have to pay attention to the details like I said now we're modeling the amount of time it takes him to ride x amount of miles so we're already given that ratio and our answer is d m of x equals 5.7 X you have to make sure here that you're paying attention to the details because some people will would have fall for the misconception that you have to convert minutes per mile into miles per minute or miles per hour and you do not need to do that in this case your answer is here's an SAT Math tip that can save you a ton of time question 3 states the solution to the given system of equations is X Y what is the value of y since we want to solve for y and we see that we have stacked equations we're going to look for a way to get rid of the X to do that in this case I see we can add these two equations together because 3x plus negative 3x is going to give us 0x so we're going to end up with then is y is equal to 12 plus negative 6 which is the same as 12 minus 6 which is 6. so our answer there will be B for the value of y here's how to solve this sat math question in under 60 seconds question 4 States the equation gives a speed s in miles per hour of a certain car T seconds after it began to accelerate what is the speed in miles per hour of the car five seconds after it began to accelerate well given our equation we want to know how fast is it going five seconds after it begins to accelerate we know that the amount of time after it begins to accelerate is represented as T so we plug in 5 for T and we're going to have our speed is equal to 40 plus three times five three times five is going to be fifteen fifteen plus forty is going to equal 55 and our answer will be answer Choice D here's how to solve this question that deals with the Pythagorean theorem on the SAT question 5 States for the right triangle shown a equals 4 and b equals 5 which expression represents the value of C well when you're given a triangle question on the SAT it's often helpful to go ahead and label your size so as you can see we go ahead and do that we're asked for the value of C we're going to use the Pythagorean theorem which means that we're going to take a squared plus b squared and make it equal to c squared and when you do that you're going to get your value of C as the square root of 4 squared plus 5 squared which we see is going to be answer Choice D here's how to solve this sat math question in under 60 seconds the question Six States what is the solution to the given equation we have 4X plus 5 equals 165 we want to isolate X so we're going to subtract 5 from both sides we're going to end up with 160 is equal to 4X then we'll divide both sides by 4 to isolate X when we do that we have 160 over 4 which is going to end up giving us 40 as our value of x right our answer there is 40. here's how to solve this sat math question in under 30 seconds the question States the x-intercept of the graph is X 0 what is the value of x in this case since we're asking for value of x and zero we know that our y coordinate is 0 at y being zero we have our x value of 7 so the value of x is 7. here's how to solve this sat math question that deals with slope intercept form the function f is defined by f of x equals 1 10 x minus two what is the y-intercept of the graph of y equals f of x in the X Y plane well all that this means is you have an equation y equals 110 x minus 2. all you got to do is find the y-intercept your y-intercept is where X is zero if you substitute in 0 for X you're going to end up with Y is equal to negative 2. so as you can see your answer has to be where x is 0 and where Y is negative 2 and your answer has to be answer Choice B here's how to solve this sat math question that deals with Transformations question nines is the function f is defined by f of x equals seven X cubed and the X Y plane the graph of y equals g of X is the result of Shifting the graph of y equals f of x down two units which equation defines the function G well the function G of X is the same as f of x except we are moving it down two units which means that g of X will equal seven X cubed minus two we can see that that is answer Choice D here's how to solve this system of equations question really easily the question states which ordered pair X y's assume solution to the given system of equations now as we can see we have one equation x plus seven equals 10 we know that in that case we can subtract 7 from both sides and we're going to get that X is equal to 3. now that we have the value of x we can go ahead and plug that in on our second equation we will end up with three plus seven squared which is equal to 10 squared which is equal to 100 is equal to Y so we know that it will have to be 3 and 100 which is answer Choice a here's how to solve this sat math question that deals with adding and subtracting variables with exponents the question states which expression is equivalent to 7x cubed plus 7x in parenthesis minus 6X cubed minus 3x what we're going to do here is distribute this minus sign to both of that 6X cubed and that minus 3x once we do that we're going to go ahead and combine like terms so we have 7x minus negative 3x which is the same as plus 3x and then we also have a 7x cubed minus 6X cubed now keep in mind that these are going to get some together I just didn't have room on one line to write them but that is ultimately going to give us 10x plus our X cubed value for a total of X cubed plus 10x which we see as answer Choice a so a is our answer there here's how to solve this sat math question in under 60 seconds question 12 States the function p is defined by P of n is equal to 7 n cubed what is the value of n when P of n is equal to 56 well when P of n is equal to 56 means we just substitute in that 56 for p of n in our equation and we get 56 is equal to 7 n cubed next thing we're going to do is try to isolate n because that's what we need to solve for is the value of n now we got 56 over 7 that's going to leave us with 8 is equal to n cubed now from here you should hopefully be able to recognize that if we take the third root of each side we'll be able to isolate the value of N and we will also have the third root of 8 which is 2. so 2 is equal to the value of N and our answer will be a here's how to solve SAT Math questions that deal with intersecting lines this question states in the figure shown line C intersects parallel lines s and t What is the value of x well since we have parallel lines s and t with something intersecting them we know that this angle next to X also has to be 110. the next thing that we can notice is that since we have this straight line here that has to sum to 180 so we can go and do 180 minus 110 and that will give us our answer for the value of x as 70 degrees so our answer will be 70. here's how to solve questions that ask you to find the mean of a data set on the SAT question 14 States a list of 10 data values is shown what is the mean of these data since the SAT has gone digital you will be able to use a calculator on each part of the test as of the filming of this video so in that case if you're asked to find the mean I will 100 recommend that you use a calculator exclusively because you can and it makes much more sense to find the mean of a data set using a calculator rather than doing it by handwritten math because you're less likely to make an error so what you would do is you would take all of these values and you would sum them together and then you you would take that sum and you would divide it by the number of values and that will get you your mean so that is how you would solve this question and once you do that you will end up with an answer of 10. here's how to answer this question that asks about the exponential growth formula on the SAT question 15 states the equation E of T is equal to 5 times 1.8 to the power of T gives the estimated number of employees at a restaurant where T is the number of years since the restaurant opened which of the following is the best interpretation of the number five in this context the number five is our initial value it is our starting value it is the amount that when T is zero we will have because any number raised to the power of zero is one so this is our initial amount and that is a very important thing to know for the SAT Math section let's go and run through our answer choices we have a the estimated number of employees when the restaurant opened that is the correct answer here's how to solve this sat math question super quickly question 16 States what is the minimum value of the given function we have G of x equals x squared plus 55. now since we have an x squared value we don't have any sort of minus five x value or anything like that we know that any number that is squared is going to be positive and that's just being added to 50 5 so our minimum value will be 55. now if you come across a question like this and it's a little bit more complicated for example say that the question was something like x squared minus 12x plus 55 then in that case what you should do is you should go ahead and graph it use on the digital sat you pretty much have access to Desmos within sort of the testing environment and you should use that and then just find the minimum on the graph but in this case this is a trick you can use since you know that x squared will always be positive you're not subtracting anything as X changes so in this case this is a trick you can use to move faster here's something you need to know about exponential and linear functions for the SAT question 17 States each year the value of an investment increases by 0.49 of its value the previous year which of the fine functions best models how the value of the investment changes over time now since it is dependent on the value of the investment the previous year since we are going up by 0.49 percent each year that is exponential growth so that is increasing exponential now I want to cover these other three options as well so you know them and you can answer them correctly on the SAT if they come up on a different question so let's go over it increasing exponential in this case since we're going up by 0.49 of our previous year's value the number that would be within our exponential growth equation for the growth rate would be 1.0049 now what if that number instead of being above one what if it was below one right let's say that our rate instead of being that our rate was 0.09 well 0.09 is a decreasing exponential function so in that case it would be a now what if instead of being dependent on the previous year's value it was independent of the previous year's value so in that case let's say that we just went up by fifty dollars so we went up by 50 each day right it's a dollar sign we're going up by 50 each day well that would be increasing linear and what if we're going down by 50 each day that would be decreasing linear if we are are not dependent on the previous year's value then it's likely going to be then it will be linear now if we are dependent on the previous year's value then it will be exponential and then you can use the rate of growth to determine if it's increasing or decreasing here's how to solve this sat math question about percentages question 18 states the population of Greenville increased by seven percent from 2015 to 2016. if the 2016 population is K times the 2015 population what is the value of K well if we're increasing by seven percent then the way we would represent this is the 2015 population which we're just going to label as P multiplied by K which is how many times we are bigger than the 2015 population will give us our new population now we have to solve for K and we know we're increasing by seven percent and since we're increasing by seven percent we would represent that as 1.07 so our answer will be answer Choice C here's how to answer this tricky sat math question about exponents the question states which expression is equivalent to a to the power of 11 over 12 where a is positive now the first thing I would immediately check for is if we have a to the 11 over 12 just Rewritten in the same same format which would be the 12th root of a to the 11th power but we see that that's not the case so instead what we're going to do is we're going to go ahead and multiply by 12 over 12 to this exponent up here and the reason why we're doing that is because I see that all of these have 132 on the power on the you know what a is being raised to the power as and I know to get to 132 I ultimately have to multiply by 12. so we're going to go ahead and multiply by 12 over 12. now why is this okay well it's okay because we're multiplying by once we're taking that 11 12 we're multiplying by 1 so it would stay the same we're just making it a larger number so when we do that 11 times 12 we know is going to equal 132 and we know in our denominator 12 times 12 will equal 144. so now we're going to find our answer Choice it will be answer Choice B which is 144th root of a to the power of 132. here's how to solve this sat math question that deals with an inequality the question States an event planner is planning a party it costs the event planner a one-time fee of 35 dollars now keep in mind when you're dealing with a word problem like this you want to start writing out equations as you come across the number so in this case we have a one-time fee of 35 dollars that's not reoccurring so I can just go ahead and add that to rent the venue and to rent the venue and 10.25 per attendee so it's gonna be our cost will be plus 10.25 per attendee which I'm just going to call a for now unless I'm given a variable it needs to be the event planner has a budget of 200 what is the greatest number of attendees possible without exceeding the budget so since we cannot exceed the budget it has to be less than or equal to 200. now from here I want to go ahead and solve for the maximum amount of attendees I can have without going over so I need to solve for a so I'm going to subtract 35 from both sides that will cancel to zero subtract 35 here okay so now I'm going to end up with 10.25 a and then 200 minus 35 is going to end up giving us 165. from here I will divide both sides by 10.25 okay so I divide both sides by 10.25 now 165 divided by 10.25 is going to give me 16.09 now since I know I can't have you know a partial amount of a person and I can't go over my budget since that gives me 16.09 I know that a must equal 16 okay and I got that by using my calculator on the new digital sat you can use your calculator on the section so our answer will be a is equal to 16 so our answer would just be 16. here's how to solve this sat math question that deals with absolute value question 21 states the absolute value of 4X minus 4 is equal to 112 what is the positive value of x minus 1. now keep in mind here that when you have an absolute value question like this you can have two answers for the value of x but in this case we are asked for the positive value of x minus 1. so there is only one right answer here but there are two answers for the value of x so let's go ahead and to talk about why that is and how you can approach this question so the first thing I would do is I would do 4X minus 4 is equal to 112 then I will add 4 to both sides add forward both sides this will cancel to zero 112 plus 4 will give us 116 so I can just go ahead and write this as 4X is equal to 116 and then from there I divide both sides by 4 and we're going to get that X is equal to see what that is in my calculator that'll be 29 okay so X is equal to 29. now we see that if we plug in 29 for X we're going to get 29 minus 1 that's positive that's 28. so 28 will be our answer okay so 28 is our answer now I want to show you what would happen if we came across the negative number first okay because the other way to get X because there are two values that X can be here is you would do negative and I'm just going to actually erase this so you can see how you do this right so you saw me get one of the values of X there is another value of x however because when you have absolute value like this you also have to do this to find the other solution okay so negative 4X minus 4 you distribute your negative sign okay and notice how I'm Distributing that negative sign to everything that was within that absolute value and that gets us negative 4 4X plus 4 is equal to 112 we will subtract 4 and we will get 108 okay so now we have negative 4X is equal to 108 I'm just going to go ahead and write that neater negative 4X is equal to 108 we divide both sides by negative 4 divide both sides by negative 4. okay and we see that X is going to equal 108 divided by negative 4 which if we go ahead and put that in our calculator 108 divided by negative 4 that's going to give us negative 27 okay and then in that case you would see negative 27 minus 1 is negative 28 that is not a positive solution which is why our answer there has to be 28 okay so it's important that you understand absolute value on the SAT Math section which is why I wanted to go over this even though we got to the answer sooner without having to get this other value for x here's how to solve this difficult sat math question that deals with volume question 22 states that a cube has an edge length of 68 inches a solid sphere with a radius of 34 inches is inside the cube such that the sphere touches the center of each face of the cube to the nearest cubic inch what is the volume of the space in the cube that is not taken up by the sphere and just a quick note anytime you have something that is underlined on the SAT pay close attention to it because they are underlining it for a reason because in a sense it's almost trying to trick you they're telling you hey this is important it sticks out from the rest because it is different and you need to pay attention to it so anytime you see that pay close attention to it now since we're asked to find the volume of the space in the cube not taken up by the sphere we have to calculate first the volume of the cube then we have to subtract the volume of the sphere so let's start with the volume of the cube we know that the cube has an edge length of 68 inches so what we're going to do is we're going to have 68 inches now the volume of a cube okay all the side lengths are the same so that's going to be 68 to the power of 3 is the volume of the cube now let's subtracting the volume of the sphere so the volume of a sphere is four thirds times pi times the radius cubed we know our radius is 34 inches so I'll go ahead and put 34 in parentheses cubed now all you have to do is put this into your calculator and you will ultimately get your answer of 149 700 196 so our answer there will be answer Choice a here's how to solve this question on the SAT about circles super easily question 23 States what is the diameter of the circle in the X Y plane with this equation now if you know the circle equation you know that this value of 16 is equal to the radius squared now if 16 is equal to the radius squared all you have to do is take the square root of the radius squared to isolate the radius the square root of 16 we know is four so we know that 4 is equal to the radius to get the diameter we know that the diameter which we'll just call d is equal to two times the radius so that's going to be 2 times 4 which will equal 8 and our answer there will be answer Choice B here's a trick to make this difficult Sac math question super simple question 24 States for the exponential function f the value of f of 1 is K where K is a constant which of the following equivalent forms the function f shows the value of K as the coefficient or the base well in this case all of our ant's choices are in the exponential growth formula or the exponential growth equation so our coefficients are going to be 50 80 128 and 204.8 now in order for the value of f of 1 to be that in a situation where we have exponential growth like this we need to have the exponent be zero because if the exponent is zero and we're multiplying it by 1.6 that's going to give us one which means that our value will just be that of the coefficient so f of x will equal the value of the coefficient when X is equal to 1 which is what we need so all we got to do is is do find the answer choice where we are raising it to the power of zero so one plus one would be two so we can go ahead and get rid of a if we just have one that's not zero if we have one minus one we get that equal to zero as the exponent so our answer there has to be answer Choice C and I will just quickly show you how this works just to prove it to you in case you are in disbelief 128 times 1.6 raised to the power of zero well 1.6 to the power of 0 is going to be one so that will equal 128 when we have F of 1. we know that we have 128 as our coefficient therefore our answer has to be C so there's just some quick proof in case you were doubting me here's a really difficult sat math question made super simple just by knowing the exponential growth formula so question 25 States a model estimates that at the end of each year from 2015 to 2020 the number of squirrels in population was one hundred and fifty percent more than the number of squirrels in the population at the end of the previous year the model estimates that at the end of 2016 now the reason that I'm underlining this is because I noticed that here we had 2015 to 2020 now we have 2016. so we are moving one year in to this growth there were 180 squirrels in the population which of the following equations represents this model where n is the estimated number of squirrels in the population T years after the end of 2015 and T must be less than or equal to five key thing here is they're kind of trying to throw you off with ants choices C and D because we are modeling it after the end of 2015 and that 180 number that they gave us is at the end of 2016. so that's one year in so our initial value is not 180 our initial value is 72. now as far as the rate of increase they're also trying to throw you off with answer Choice a because they know that you're going to see that 150 percent and think oh that's just 1.5 but it's not you have to keep in mind that it's 150 more than the number of squirrels in the population at the end of the previous year right so if we're multiplying by let's say we're increasing by 50 then it would be 1.5 but we're not increasing by 50 we're increasing by 150 which means we have to multiply that by 2.5 so our answer has to be B so if you know the exponential growth formula this question is really easy if you don't don't you're probably going to get it wrong here's one of the most difficult SAT Math Questions made super simple question 26 states that in the given pair of equations A and B are constants the graph of the pair of equations the X Y plane is a pair of perpendicular lines which of the following pairs of equations also represents a pair of perpendicular lines so the key thing here is that you understand what perpendicular lines mean it means that these slopes are negative reciprocals of each other so for example if we look at our equations that we are given we have 5x plus 7y equals one ax plus b y equals one now the slope of this first one I'm going to call M our slope so m is going to equal negative A over B now the form negative A over B works when you have the form of a coefficient in front of X plus a coefficient in front of Y is equal to some constant which we have in both of these equations so we can use it okay so that means that our slope is going to equal let me go ahead and get that it's going to equal negative 5 over 7. all right now our slope for this bottom equation is going to be m is going to equal well that's going to have to equal seven over five and how do we know that well we need the opposite sign and the reciprocal okay so that's going to have to be 7 over 5. let me go and write that a little bit bigger so you can see it better 7 over 5. okay and I'll just draw the arrow that that's there all right well if that's going to be 7 over 5 we know that our a value then well that must be negative 7. okay and the reason we don't ask me negative seven is because we know that the formula for our slope is negative A over B in this form okay and we know our B value then has to be five okay so I'm just going to go ahead and write that down that our B value is going to equal 5. because we need to know that in order to solve this now the other thing that I think is important to notice here and that will ultimately help you solve this extremely quickly once you know this trick is that if you look at our equations up top here we have negative 7x plus 5y is equal to one now what I want you to notice here is that when you're dealing with perpendicular lines in this format of you know the the format where you have them set equal to a constant right notice how we have 5x and 5x and then notice how we have 7y and 7y obviously they are changed places right that that coefficient of seven has now changed from being attached to the Y to being attached to the X and then we have One Singular change in sign so that change in sign it can be out in front of this a value or it could also be in front of the B value but either way we have that one change in sign so now that we've noticed that we can pretty much solve these pretty quickly right so if if we look at answer Choice a if we were to plug in for a we know our value of a is negative seven so we can go ahead and do that so negative seven we know that we have the sign change right we're switching from that 70 in front of the Y being in front of X that part's good but then we have minus 2 times B we know that our B value is five right so we're going to end up getting minus 10y now the problem with that is that we change both sides we can only change one of the signs so a is going to be wrong and then we look at B we've got 10x plus 7y and then down here we know that our a value is negative seven so once again we're changing the sign that looks good and then we got plus 2 times B we know B's fives that's going to end up giving us plus 10 y right and now we see that both of the the you know the coefficient in front of X is now in front of Y coefficient that was in front of Y is now in front of X and we only changed one of the signs so that means that B has to be our correct answer okay so I know it took a little while for me to explain that but I wanted to make sure that you guys understand how this works and why this works because once you understand this trick you can then apply it and it becomes very very quick to find the right answer and also I will just note that this is one of the more difficult questions on the SAT so being able save time by understanding this trick is going to be very very beneficial especially for students who are looking to score high on the math section in particular above 700. here's what you need to know about calculating the number of real solutions on the SAT Math section along with a problem to help you apply it question 27 says in the equation above C is a constant the equation has no real solutions if C is greater than n what is the least possible value of n well let's start by discussing how you know the number of real solutions the way you know that for a quadratic equation like this is by using the value of B squared minus 4AC now one thing I want to have you make note of is that in the quadratic equation it is set equal to zero that's very important okay so B squared minus 4AC if that value is greater than zero then that means that you're going to have two real solutions which I'll represent as RS now if B squared minus 4AC is equal to zero then that means you're going to have one real solution now if you can kind of figure out where I'm going with this B squared minus 4AC if that is less than zero then you're going to have no real solutions okay so now that we know this and we know that in our question we are asked to find the equation having no real solutions of C is greater than n we know that we can go ahead and set up this equation so we know our a value is one we know our B value is going to be negative 34 and we know our C value is just going to be represented as C in this case so we have B squared we know B is negative 34. so we'll go ahead and take that and square it we're going to subtract 4 times a a is one and then we'll multiply it by C the next thing we have to do is we have to set that as less than zero so negative 34 squared is going to become 11 I believe let me just quick check on the calculator 1156 1156 okay so 1156 minus 4C is going to be less than zero I'm going to go ahead and add 4C to each side okay so I'll add 4C to each side and then next thing I'm going to do is I'm going to divide by 4 in order order to isolate C so we can get it all alone so let's go ahead and do that we'll divide both sides by four we'll get that c must be greater than 11 56 over 4 which is going to be 289. all right so now that we have this we know that c has to be greater than n okay and we know that n if we go ahead and let me just draw this right this inequality is going to equal this other inequality of C being greater than n well in that case n is going to have to equal 289 okay so our answer for the least possible value of n is 289 here's how to answer this sat math question in under 30 seconds the question States the line graph shows the percent of cars for sale at a used car lot on a given day by model year for what model years the percent of cars for sale the smallest well we see on our Y axis we have the percent of cars for sale and we just have to find where that's the smallest we see that that's in the year 2014 so our answer is going to be answer Choice C here's how to answer this probability question on the SAT question two states for a particular machine that produces beads 29 out of every 100 beads it produces have a defect a b produced by the machine will be selected at random them what is the probability of selecting a beat that has a defect in this case we're looking for those that have a defect which we know is 29 out of every 100 so we know that our probability of that must be in Choice C 29 out of 100. here's how to answer this question about intersecting lines on the SAT question 3 states in the figure line M is parallel to line n and line T intersects both lines what is the value of x well since M and N are parallel 19 is intersecting them we know that X is going to be the same as the angle X I just drew next to that Angle 33. now we also know that a semicircle has 180 degrees in it so all we have to do is 180 degrees minus 33 degrees and that will give us our answer of 147 degrees as the angle X so our answer is D here's how to find the y-intercept of this graph on the SAT the question States what is the Y intercept of the graph shown all you got to do is find where we're crossing that y-axis where X is zero and we see that that is at eight so our answer there is going to be 0 8. here's how to solve this sat math question in under 60 seconds the question States the total cost f of x in dollars to lease a car for 36 months from a particular car dealership is given by f of x equals 36x plus 1000 where X is the monthly payment in dollars what is the total cost to lease a car when the monthly payments 400 we know the monthly payments are represented by X so we can go and substitute 400 in for x and we will have 36 times 400 plus 1000 and that will end up giving us our answer of fifteen thousand and four hundred dollars which is answer Choice C here's how to solve this question about perimeters on the SAT question 6 States each side of the square has a length of 45 what is the perimeter of a square well we know Square sides are all the same so the perimeter will just be four times the side length which we know is 45 we can go and put 4 times 45 in our calculator or you can just do it in your head and you're going to get 180 for your answer as the perimeter of the square here's how to solve this sat math question that deals with factor in question seven states what is the positive solution to the given equation in order to solve this we need to ultimately get it set equal to zero so let's go ahead and do that by multiplying both sides by X plus six once we do that we have x times X plus six we'll also go ahead and subtract 55 from each side so we can set it equal to zero and now we have our equation so let's go ahead and destroy attribute this x to this other X to get x squared and then also plus 6X so now we have x squared plus 6X minus 55 is equal to 0. now let's go ahead and Factor so I'm looking for factors of negative 55 that will sum to have positive 6. so when I do that I can think we've got 11 and 5. I need to keep it positive so I'm going to have to have X plus 11 and I would have to have x minus 5. now we Factor we have to keep in mind that we can't just pick either one we need to have the positive solution to the given equation so we know that X plus 11 will ultimately mean that X could equal negative 11 but we can't choose that because that's negative we need to have x equals 5. so our answer there would be 5. here's how to translate this word problem into an equation and solve it on the SAT question eight states and object travels at a constant speed of 12 centimeters per second at the speed what is the time in seconds that it will take for the object to travel 108 centimeters well this is pretty easy the distance that you're traveling is 108 centimeters you want to know how long it's going to take you you are traveling at 12 centimeters per second okay you're multiplying by your time in seconds okay we'll have t represented as time in order to solve for T then all you got to do is divide 108 by 12 to get your answer when you do that you're going to end up with your answer of 9 which is Choice a here's how to solve this question about comparing the mean of two data sets on the SAT question 9 States the list gives the values in data sets X and Y which statement correctly compares the mean of data set x with the mean of data set y well let's go and find the means of both we'll start with data set X so the mean is going to be the sum of these numbers divided by 4 so 5 plus 9 plus 9 plus 13. now I see I have these same numbers in data set y now I also have a 27 add line to data set y now since we've got the same four numbers in data set X as in y and then we are just adding in 27 we know 27 has to be greater than whatever the mean is in data set X because the maximum value in data X is 13 and 27 is higher than that so it's going to have to bring that mean up therefore we know that data set y has to have a larger mean than data set X without even having to calculate the means just by looking at this so we know our answer there has to be B if you want to calculate the means for this you can but you could also just recognize that 27 is larger than the maximum data set X other than that they're the same therefore data set y has to be a larger mean here's how to solve this sat math question in under 60 seconds question 10 states a rocket contained 467 000 kilograms of propellant before launch exactly 21 seconds after launch 360 105 kilograms of the propellant remained on average approximately how much propellant in kilograms did the rocket burn each second after launch while our starting amount of propellant is 467 000 we then have to subtract how much propellant remained and then we have to divide all of that by the amount of seconds that this took place which is 21 seconds once we put that into our calculator we're going to end up with answer Choice a and that'll give us exactly or approximately how much repellent in kilograms the rocket burned each second after launch here's how to use ratios to move faster on the SAT Math section question 11 states if 4X plus 2 equals 12 what is the value of 16x Plus 8 whenever I see a question like this I'm always looking for proportions I see 4X plus 2 If I multiply that by 4 I get 16x plus eight so all I've got to do is multiply 12 by 4 12 times 4 is going to equal 48 So my answer is 48 my answer there's B this way I don't have to solve for x so I'll get it done faster here's how to interpret a common situation on the SAT Math section question 12 States an object is kicked from a platform the equation H equals negative 4.9 t squared plus 7t plus 9 represents the situation where H is the height of the object above the ground in meters T seconds after it is kicked which number represents the height in meters from which the object was kicked okay a common a question on the SAT deals with something falling from some certain height whether it's thrown off kicked off launched off things like that and the way that it's usually represented is in an equation like this in this case we're asked which number represents the height from which the object was kicked well the height at which it is kicked is when it time is at zero right it's only kicked once the time is zero so that means T would be zero which makes this zero which makes that zero which means that nine is the height when it is first kicked so our answer has to be D here's the type of question where you should use Desmos on the digital sat question 13 states that f of x is equal to 4x squared minus 50x plus 126 the given equation defines the function f for what value of x does f of x reach its minimum if I saw a question on like like this on the new digital sat I would use Desmos for it 100 I would just check where the minimum is so I'll go and show you how you would do that all you got to do pull up Desmos you can do this on the digital sat it's built into the college board's Blue Book so you put in the equation you look for where the minimum is the minimum is at 6.25 for the x value of the minimum so your answer there would be 6.25 so this is a great example of the type of question where it really does make sense to use Desmos on the new digital sat here's how to set up and solve this system of inequalities question on the SAT question 14 States a small business owner budgets twenty two hundred dollars to purchase candles the owner must purchase a minimum of 200 candles to maintain the discounted pricing if the owner pays 4.90 per candle to purchase small candles and 1160 per candle to purchase large candles okay I'm using S for small candles I'll use l for large can handles what is the maximum number of large candles the owner can purchase to stay within the budget and maintain the discount pricing so we only have two candle sizes so we can go ahead and determine the total number of our candles will be L plus S I want to quickly point something out you saw as I read through this I was writing out the system of inequalities as I go if you encounter a question on the SAT Math section that is three lines or longer it is most likely going to end up being a question that involves either system of inequalities two equations that you need to set up possibly something to do with converting units something along those lines but either way my general recommendation is if a question is three lines or longer on the SAT Math section you try to write down units variables and obviously like the coefficients in front of certain units any sort of constants things like that basically you want to start writing your equation as you read through the word problem because otherwise if I hadn't written down these two inequalities I would have to go back and do a second read through so this is a strategy that I use and that I recommend you use to save time that being said let's go ahead and solve for the number of large candles one thing I want to point out here you'll notice I'm solving for L right away not s that's because I don't give about S I don't sure about s at all right we don't need s we just need l so that's all we're going to focus on solving okay so I'm going to go ahead and subtract uh I'm sorry we're going to subtract L from each side because we want to isolate s and the reason why is we want to substitute in for S something with L okay so let's go ahead and get S by subtracting L from both sides we get 200 minus L is going to be less than or equal to S now we can go ahead and substitute in okay so let's go ahead and substitute that in for S Okay so we've got 4.9 times s which we know we can represent as 200 minus L okay and keep in mind that we still have a plus 11.6 L right here so let's go and distribute the 4.9 to the negative L into the 200 4.9 times 200 is going to give us 980 okay so equals 980 and then we're going to have minus 4.9 L and plus 11.6 L which will give us plus 6.7 l in total okay because 11.6 L minus 4.9 L gives us that plus 6.7 L now that all is still going to be 2200 over there we'll subtract 980 from both sides minus 980 that's going to end up leaving us with 12 20 20 is going to be greater than or equal to 6.7 l okay from there we would divide both sides by 6.7 so we'll go ahead and do that divide both sides by 6.7 okay 1220 divided by 6.7 go ahead and put that in my calculator that's going to be 182.09 okay 182 point I'll just call it 0.1 okay it's 0.09 all right now we have to see that L has to be less than or equal to that okay so in order to be less than 182.1 we would go to 182 we can't have 0.1 of something okay so our answer there in this case we can't because we have 0.1 of the candle right so our answer there would be 182 so we can go ahead and write that down and we'll Circle that so 182 there's our answer here's how to solve this sat math question in under 30 seconds question 15 states in a linear function f f of 0 is equal to 8 and F of 1 is equal to 12. which equation defines F immediately I see I have that F of zero is equal to 8 which means that my y-intercept is 8. I see all of these are in slope intercept form and I see The Intercept of c and b is not 8 so I can get rid of them next thing I see is the difference between a and d is the slope if I look at my slope I'm going over 1 and I'm going up by four so my slope is four So my answer there has to be answer Choice D here's how to interpret this SCT math problem question 16 states that the function f of w is equal to 6w squared gives the area in the area of a rectangle in square feet if its width is W feet and its length is six times its width which of the following is the best interpretation of f of 14 is equal to 1 176. all right well ultimately this function is giving us our area we know that W represents our width so F of 14 means that our width is 14 feet and then that 1176 means that when our width is 14 feet our area of our rectangle is 1176 feet that's answer Choice a so a is our correct answer there here's how to answer this SAT Math problem about Arc Length by using ratios and proportions question 17 states that the circle shown has Center o circumference 144 pi and diameters PR and Qs the length of Arc PS is twice the length of Arc p q so we'll call Arc p q X and the length of arcps we know is twice that so we'll call that 2X I'll go ahead and Mark those links here okay the next thing we are asked is what is the length of Arc QR well Arc QR is going to be the same length as arcps so I'm going to go ahead and focus on solving for PS just because it's easier for me to show on this video so if we go ahead and draw this we see that it's X plus 2X so that's going to be 3x we also know our circumference is 144 Pi if our circumference is 144 Pi our circumference is that whole line around the circle right it's all of this that line right there okay so our circumference is 144 Pi then that semi-circle right that semi-arc semi-arc is going to be 72 Pi okay so then we know that 72 Pi is equal to 3x right because it's the sum of that 2X and that X therefore you would know that PS which is the same as QR okay is two-thirds two-thirds of that okay so it'd just be 72 Pi just be 72 pi times two-thirds okay and when we put that in your calculator or you can just do it in your head you're gonna end up with 48 pi as the value you for Q to R you could have also done it by doing that 72 Pi is equal to 3x okay and then solving for x and then taking it in and multiplying it by two but easier ways just to recognize that hey this is a ratio right it's two-thirds two-thirds times 72 Pi gets you 48 Pi it's a lot easier than going through and solving for x so I wanted to put the X down there just so you guys could see and understand sort of how it breaks down but ultimately it's a lot better if you can just recognize hey this is a ratio we've got half of this whole circumference 72 Pi we are taking two thirds of that to get Arc of QR so answer there is B here's a couple tips for the SAT Math section to help you be more efficient I'm going to illustrate them with question 18. question 18 states a company that provides will watching tours takes groups of 21 people at a time the company's revenue is eighty dollars per adult so I'm going to represent that as 80 a and sixty dollars per child which I'll represent as 60 times C if the company's revenue for one group consisting of adults and children was one thousand four hundred and forty dollars okay so that's gonna be the sum of of these the value of the number of adults times the cost per adult plus number of kids times cost per kid how many people in the group were children well we know that it consists of adults and children so we can represent that as April c as you can see I did one read through and I had both my equations so that's tip number one here as you read through an SAT Math problem that is three or more lines as you can see in this case it was six that's why I knew that I wanted to use this sort of this sort of tip or trick anytime you've got an SAT Math problem longer than three lines try to write out the equation or the system of equations as you read through that way you save time by not having to re do a re-read through right right afterwards to find your variables and things like that okay that being said let's go ahead and keep talking because there's more tips to come here now keep in mind we are asked to solve for how many people in the group were children so you could go through and you could solve for a first and then get to C but you shouldn't do that you should go ahead and solve for C first you want the most direct path to your answer on the SAT Math section because it's a timed exam and you want to be efficient so we're going straight for C we're not we don't want to get a we don't need it so we won't we just don't want to even see it all right so here's what we're going to do we're going to substitute in a value with C for this a value there the way we do that is by subtracting C from both sides to get 21 minus C is equal to a now we go ahead and sub substitute in okay so it will be 80 times 21 minus C plus 60 C is equal to 14. we go ahead and distribute our 80 to our 21 into our C 80 times 21 is going to be 1680 and then we are also going to have minus 80c plus 60c is going to equal 14. now we want to start isolating our C so minus 80c and plus 60 C is going to leave us with negative 20c okay and we're also going to go ahead and subtract 1680 from both sides okay okay subtract 1680 from both sides this is going to leave us with 240 or I guess negative 240 is equal to negative 20 C now from here oh we're going to go ahead and divide both sides by negative 20. negative 240 divided by negative 20 is going to leave you with your answer of C is 12. so our answer there will be answer Choice C because we have 12 children and as you can see by solving for the number of children first we didn't have to go back through and solve for the number of adults we never have to know the number of adults if you wanted to go back through and see obviously you could just put in the number 12 there for for C but it's not necessary you don't need to know the number of adults so don't care about it don't need to focus on it just get the number you need to get to and move on to the next question here's how to find x and y intercepts quickly when you have an equation in slope intercept form question 19 States the function H is defined by h of X is equal to 4X plus 28. the graph of y equals h of X and X Y plane has an x-intercept at a and zero and a y intercept at zero B where A and B are constants what is the value of a plus b well in this case there's no way to just get to a plus b normally if I see something like that that's what I'd like to do in this case you have to actually find both A and B so in this case let's go and start by finding a we see that a is where our y value is zero well where our y value is zero we have four X Plus 28 is equal to zero we subtract 28 from both sides we get negative 28 is equal to 4X we divide both sides by four negative 28 over 4 is going to equal negative 7 as the value of x now let's go ahead and solve for our B value and keep in mind that X there would equal that a value okay so a is negative 7. now let's go ahead and solve for what B is equal to okay B is where our x value is zero so we just put in 0 there and we see that we ultimately are going to get 28 okay so we're asked for the value of a plus b so we just do 28 plus negative 7 which is same as 28 Minus 7 that's going to leave us with 21 so our answer there is a here's an example of an SAT math question that's gotten way easier now that the SAT has gone digital question 20 States one of the factors of 2x cubed plus 42x squared plus 208x is X plus b where B is a positive constant what is the smallest possible value of B now keep in mind you need the smallest possible value of B and that's important here because there's going to be two possible answers but I want to show you why this question is so much easier now that you have Desmos built into Blue Book on the the digital sat before you would have to factor this out and that could be a bit difficult and there's a lot of room for error when you're doing that but now that you have Desmos and the blue book app when you're taking the digital sat you can just put it into Desmos now you have to also understand how to work the problem with Desmos so in this case you know that you need the value X plus b now what that means is that you're looking specifically in this case at the negative X values or the negative X values in regards to the x-intercept right so let me go ahead and zoom in here the reason that we're going to the negative sections because it can't be a positive number because these are ultimately our zeros they're not our values for B okay our zeros here and as you can see the first negative or the first zero there is that negative eight okay now that does not mean negative 8 is our answer because it's not okay because you keep in mind that we're looking for the value of B and X plus b and X plus b since it's one of our factors has to be set equal to zero which means that X is actually going to equal negative B now we know our value for x okay our value for x is negative 8 which means that B has to equal positive eight okay so our answer there will be eight now you really need to make sure that if you are using Desmos which I do recommend you do you do take close account to the Sines of values and you understand how to interpret these graphs because if you don't understand how to interpret these graphs but you try to use them it's not going to help you so as you saw when I looked here I had to understand that I'm looking for the value of B okay to get to that I need to understand how a factor works right the factor has to be set equal to zero and then once I find this value of x I have to remember to get that back to B so make sure that when you do use Desmos on the digital sat which I do recommend you do because it helps a ton and makes you faster less prone to error that you do at least understand how to interpret these graphs here's what you need to know about questions that ask you about the number of real solutions to a quadratic equation on the SAT I'm going to illustrate read this with question 21 question 21 states in the given system of equations a is a positive constant the system has exactly one distinct real solution what is the value of a whenever you have a question asking you about two real solutions one real solutions no real solutions or the number of real solutions it's basically testing do you know this crucial Rule and that rule is B squared minus 4AC okay now this works when you have an equation a quadratic equation that is set equal to zero okay so a quadratic equation looks like this and it's got to equal out to zero now I'll show you how we get to there in regards to this question in a second but first I want to explain this rule so B squared minus 4AC if this value is greater than zero then that means that there are going to be two real solutions now what if this value is equal to zero in that case there is one real solution and what if this value is less than zero well if B squared minus 4AC is less than zero then that means that there are going to be no real solutions okay okay so zero real solutions okay I guess I should actually Express that as zero because that's the way it is commonly shown on the SAT is zero real solutions okay with this being said I want to go ahead and go over how we apply this in this question okay so in order to get our B squared minus 4AC value like I said we have to set it equal to zero in this case it is not set to equal to zero yet we have to substitute in this minus one and a half four y first so then we would end up with minus one and a half is equal to x squared plus eight X plus a okay from here I'm gonna go ahead and add one and a half to each side so that we can get it set equal to zero so we can use this this rule of B squared minus 4AC okay so then we have that much plus a plus one and a half now what I'm going to do is I'm actually going to pick essentially a random variable which I'm just going to call uh we'll use Jet and I'm trying to pick one that doesn't sound like a or look like a number let's use a we're gonna use Z okay and I'll try to make it very differentiated okay I'm gonna let Z equal a plus 1.5 all right so now I'm going to go ahead and substitute in Z for a plus 1.5 and you'll see why I'm doing this in a moment here okay so now we have our quadratic equation it's set equal to zero now we can do B squared minus 4AC now you may be wondering what's B what's a and what's C I'll go ahead and show you okay our a value is whatever coefficient is in front of this x squared since there's no coefficient in front of it it's one okay so our a value is one our B value is going to be eight it's whatever is in front of that X okay and our C value is the constant which in this case we're representing a z all right so B is eight so that's going to be squared 8 squared is 64. okay so B squared minus 4 times AR a value is 1 okay so we don't need to show that okay and then times C which we're representing as Z to represent that a plus 1.5 all right now we know that there has to be exactly one real solution which means we have to set it equal to zero now it's pretty easy to solve we're gonna go ahead and add 4z to each side so we'll have 4z is equal to 64. we'll divide both sides by 4 to isolate Z so now we have Z all alone we know that Z is equal to 64 over 4 which is going to equal 16 okay so now Z is 16. well now that we have Z is 16 we can go back up here and we have 16 is equal to a plus one and a half I'm going to erase this so you can kind of see what I'm doing back up here 16 is equal to a plus one and a half I subtract one and a half from each side and I get 14.5 is equal to a okay so there's what you need to know about the number of real solutions on the SAT Math section along with this question as an example of how you can apply it here's how to quickly answer this sat math question by understanding zeros question 22 States f of x is equal to X plus six times X plus 5 times x minus four the function f is given which table of values represents Y is equal to f of x minus three very important thing to hear is to recognize is that f of x is this equation we have to do that minus 3 there okay so what we can actually call this is I'm just going to scratch this out I'll call it Y and then I'll just add that minus 3 at the end it's the same thing all right so now we have our new equation Y the next thing that we see is that we Define the table values that represents it the key thing that I'm looking for here is I want to find where I have a 0 in parentheses which we're going to have when X is sorry let me write that where you can see it we're going to have that where X is at negative 6 because negative six plus six is zero we're going to have that where X is negative 5 because negative 5 plus 5 is 0 we'll have that where X is 4 4 minus 5 4-0 I'll look for those X values in my table here I see all of them are in here okay so I know they're all there anytime I have one of these I'm going to have a y value of negative 3 because 0 times anything is going to end up being zero so then I just end up with zero minus three okay as we can see a has to then be incorrect B has to be correct because as all of these values of our zeros and this x side of the table and as negative three for all of them so B is our correct answer here's how to quickly solve this sat math question by understanding the equation for exponential growth question 23 states that for the function Q the value of Q of X decreases by 45 percent for every increase in the value of x by 1. if Q of 0 is equal to 14 which equation defines Q well Q of 0 being 14 means that our y-intercept where x is 0 is going to be 14 which means our initial value is 14. so we can get rid of A and B because their initial values are not 14. next thing we need to recognize is our growth rate okay our growth factor we have q of X decreasing by 45 for every increase in the value of x by 1. if we're d decreasing by 45 percent that means our growth factor is 0.55 okay we're not getting bigger ants Choice D would mean that we are increasing we're decreasing so that growth factor has to be less than one so our answer there is C here's a trap to avoid on the SAT Math section question 24 States the graph of y equals f of x plus 14 is shown which equation defines the function f many people will pick answer Choice C on this question because they'll see the y-intercept is 2 and pick answer Choice C but that's not correct and the reason why is because you have to pay close attention to the details on the SAT Math section this question States the graph of y equals f of x plus 14 is shown and we're asked to find the function f not the function y not y equals f of x plus 14. so the function f has to be down it has to have a y-intercept that is 14 below where it is on this graph so if we go down 14 from 2 we have to be at negative 12 which means our y-intercept has to be negative 12 and our answer has to be answer Choice a because that's the only answer Choice with negative 12 as the y-intercept so on the SAT maths section pay very close attention to details with what you're asked to answer with and any equations that you're given any equations or functions that you're given in the context of the question here's how to solve this question about tangents here's how to solve this question about similar triangles and tangents on the SAT Math section question 25 States the side lengths of right triangle rst are given triangle rst is similar to triangle uvw where s corresponds to V and T corresponds to w What is the value of tan of w well the first thing I'm going to do is since I have rst as the triangle and they're similar the tangent values are also going to be the same for the corresponding angle so I'm going to go ahead and find that W what is it equal to an rst okay so if I've got V and T correspond let's see s corresponds to V all right so s corresponds to V okay I see that's the middle one that's the middle one and then T corresponds to W okay I see that that's the last one that's the last one that means that R will correspond to UK but we've got tan of w okay so that's going to be T all right so we have to find tan of T okay I can just write that as then tan of T okay it's the same thing because these are similar triangles they're going to have the same tan values for the same angle all right so now what is the value of it well let's go ahead and draw out this triangle okay because that's just going to make it easier to visualize we have a right triangle we have we've got our St so let's go ahead and just Mark that our St we know T to R is 52. we know s to T is 48 okay and keep in mind this is obviously not drawn to scale okay it's just a quick sketch because we're only really worried about the actual values not really the the angle measures all right so now we want to know the value of tan of w or tan of T all right well we know that the tan is the opposite over our adjacent okay so our opposite from T would be 20 and our adjacent would be 48 so that's going to equal 20 over 48 which we can simplify down we see we have to because there are 20 48 it's not an answer choice so if we're simplifying this down we go ahead and divide them both into four and that's going to give us 5 over 12 as our answer so that'd be answer Choice B here's how to solve this sat math question that deals with rearranging variables question 26 States one gallon of paint will cover 220 square feet of a surface a room has a total wall area of w square feet which equation represents the total amount of paint p in gallons needed to paint the walls of the room twice all right so here's how I ended up with this equation one quick note you saw I started writing down variables and sort of the coefficients in front of those variables as I read through that's a strategy I recommend that you do anytime that you have an SAT Math word problem that is three lines or longer okay just because it helps you avoid having to reread as much so just want to make note of that that being said let's go and talk about how I got there all right so we have one gallon of paint is going to cover 220 square feet of the surface okay so we have 220 let's see amount of square feet one gallon p is going to cover okay so this gets us the amount of coverage that we're getting now we know that has to equal the area of the wall times two because we're going to do it twice right we're going to paint the walls twice okay so now we have our equation now we see that we need to solve for p so we just have to find what P equals now all of this question basically hinges on you setting up this equation correctly initially if you don't set it up correctly initially you're not going to get to the right answer so you have to make sure that you understand how how the units work right because as you can see here we are ultimately getting to an area by taking the amount of coverage we get per gallon of paint multiplying by the number of gallons now we get our area and we're setting that equal to an area as well so checking if your units match is a good way to check your check your work as well so I want to point that out okay we're ultimately getting area equals area all right that being said let's go ahead and solve for p now we know that P is going to equal W Times 2 all over 220 because we divide both sides by 220. next thing we'll do is we'll go ahead and simplify okay we can divide 2 by 2 and 220 by 2 that's ultimately going to give us if we go and cross both those out we'll end up with 110 in that denominator so p is equal to W over 110 are in so there will be answer Choice a here's how to solve this difficult sat math question question 27 States the number a is 110 greater than the number B well that means that a is going to equal 2.1 times B because it's 110 greater than the number B okay keep in mind pay attention to that that greater there that means it's going to be 2.1 times B all right the number B is 90 less than 4.7 well another way to say that is the number B is 10 of 47 okay so that means that b is going to equal 0.1 times 47 which is going to equal 4.7 now we're asked what is the value of a well this is pretty easy to solve for now okay in order to to get a we need to know the value of B which we now know is 4.7 so we just do 2.1 times 4.7 if we put that into our calculator we will get 9.87 as the value of a so our answer is 9.87 here's how to answer statistical inference questions on the SAT question one states that there are 55 students in Spanish Club a sample of Spanish club students was selected at random that's important whenever you're doing statistical sampling it must be random sampling in order to provide an accurate statistical inference for you to use later on and ask whether they intend to enroll in a new study program of those surveyed 20 responded they intend to enroll always pay attention to if it's they intend or they not to intend because sometimes they'll put in something that has only two options and they will give you one of those being they do or the other being they do not and you have to pay attention to what you're given in this case we're given that they do intend to enroll in the study program based on the survey which of the following is the best estimate of the total number of Spanish club students who intended to enroll in the study program in this case this question's super straightforward forward we have 55 total students we're told that of those who responded which was a random sample 20 said they would intend to enroll were asked for those who would intend to enroll so all we have to do is 55 times 0.2 and that'll give us 11 so our answer would be a here's a tip to help you be more efficient on the SAT question two states Jay walks at a speed of 3 miles per hour and runs at a speed of 5 miles per hour he walks for w hours and runs for our hours for a combined total of 14 miles which equation represents this situation okay well as you can see I've already got my answer without having to read through a second time and the reason why is because I wrote down my variables as I read through as you can see it's a little bit messy because my focus is on reading but I don't have to read through again and the reason you want to do this is because if you were to read through this word problem without writing anything down and then you get to the end and it says which equation represents the situation you don't have to go back three if you look for your numbers you have to look for your variables so what I do when I encounter a word problem on the SAT and you can usually tell pretty much right off the bat if it's a word problem because it'll be usually at least three lines or longer in this case it was four lines you want to write down any numbers they give you and any variables they give you as you go through and you want to try to sort of get it in the equation form that you're expecting and this comes with a little bit of practice but as you can see once I saw we've got walking at three miles per hour right running at five miles per hour I'm guessing it's going to be a summation to get us distance so I went ahead and I put the three here the five here and then once I got my variables went ahead and attached them to the approach appropriate number that I got earlier in the question and then got the equation not having to read through twice so this is a great way to save time on the SAT and I highly recommend doing it here's how I recommend approaching questions about Scatter Plots on the SAT question 3 states that scatter plot shows the relationship between two variables X and Y a line of best fit is also shown which of the following equations best represents the line of best fit shown what I do in this situation is I take a quick glance at the ends choices to see what format they're in in this case I see they're all in slope intercept form which is going to make this question really easy so let's start by finding our intercept we see our intercept is positive right around three which means we can go and get rid of c and d because they both have negative y-intercept the next thing I'll take a look at is the slope in this case I see my slope is positive if I look at n's choices A and B I see only one of them has a positive slope So my answer has to be against Choice a here's how to answer this question that deals with y-intercepts really quickly on the SAT the question States the graph of y equals f of x is shown in the X Y plane what is the value of f of zero well the value of f of zero is just the y-intercept so if we take a look at our graph we see our answer is going to be 3. so our answer is answer Choice D here's how to answer questions about adding exponents on the SAT question five states which expression is equivalent to and then we have this multiplication question where m q and z are positive what you need to understand in order to answer this question correctly is how to add exponents when you have the same base so to illustrate this let's look at m m to the power 4 times m is equal to m to the power of 5 because it's the same base we add their exponents when we multiply them together same thing with Q we have q to the power 4 times Q to the power five that'll give us Q to the power of nine same thing with Z in this case we have Z to the power of negative 1 times Z to the power of three in this case what we have is is z to the power of 2 because it's 3 minus 1 or in other words 3 plus negative 1. and as we can see our answer then has to be answer Choice B here's how to find the median on the SAT question 6 States what is the median of the data shown we're going to start by getting rid of our minimum value then our maximum value then our next minimum value then our next maximum value then our next minimum value then our next maximum value then our next minimum value then our next maximum value to find our answer of 79. now what if we had two values in the middle and we didn't have an odd number total of values we had an even number well in that case let's say for example that we had 79 plus 81 as or in this case I guess we'll use 82 since that's the one that's left of it but it's all arbitrary we've already answered this question but I just want to show you what we do in that case in that case we had 79 plus 82 we'd divide by 2 we take the average and we use that as the median here's how to solve this sat math question in under 30 seconds the question States what value of x is the solution to the given equation well we got to isolate X then so let's go ahead and subtract 40 from both sides 95 minus 40 is going to leave us with 55 is equal to X so the value of x that is solution to the given equation is 55. here's a tip to be way more efficient on the SAT question 8 States the solution to the given system of equations is X Y what is the value of x plus Y what you always want to do on the SAT is look for what you're asked to answer with in this case that's the value of x plus y so does that mean we need to solve for x and y not necessarily and I'll show you why here if I see I have stacked equations like this I'm almost always going to look to add or subtract the bottom equation from the top equation and sometimes that will also mean I need to multiply by some number but in this case I don't need to multiply by a number and what I see immediately is I have 5x and 4x and I have to get X Plus y so what I'm going to do here is I'm actually just going to add these equations together and that's going to give me X Plus y because 5x plus negative 4X is going to leave me with X and then I have plus Y is going to equal 15 plus negative 2 which is equal to 15 minus 2 which is equal to 13. so our answer there is C and we don't have to solve for x and y because we were able to get the value of x plus y here's an example of how to interpret a word problem on the SAT Math section question 9 States a given function G models the number of gallons of gasoline that remains from a full gas tank in a car after driving miles according to the model about how many gallons of gasoline are used to drive each mile well we know that g of M is how much gas remains in the tank okay presumably since before we have driven any miles this number here is going to end up being zero since M would be zero we would start with 12.1 gallons of gas which means that for every mile M that we drive we are losing 0.05 gallons of gas our answer would have to be answer Choice a here's an example of how to solve questions that ask you to rearrange variables on the SAT Math section in this case we're asked the given equation relates the positive numbers B X and Y which equation correctly expresses X in terms of B and Y well in order to get our answer we have to isolate X let's go ahead and do that by multiplying both sides by y to start and then to get 11x to just X we have to divide both sides by 11 and now we have isolated X so we have y over 11 times 7 B which as we can see 11 times 7 is going to give us 77 and we would have y over 77b here's how to solve this sat math question in two different ways question 11 states the graph of the given equations in the X Y plane intersect at the point x y what does it possible value of x now many people would go through and solve this by hand so let's go ahead and do that and I'll talk about the other way you can solve this one so you have y equals 76 and y equals x squared minus 5. in this case we can go ahead and plug in 76 for y so we have 76 is equal to x squared minus 5. next thing we'll do is we'll isolate that x squared value so we'll add 5 to each side and we'll end up with 81 is equal to x squared from here you should be able to recognize that your answer is going to be positive and negative 9 for the value of x but I'll go ahead and show you how we do that we can take the square root of both sides which is going to give us plus or minus 9 is equal to X the reason why it's plus or minus is because x squared if you put in a negative number it will still be composite if you put in a positive number it would composite because it's being squared so our answer there would be B now here's what I want to show you about this question that's the way I just showed you is one way to do it and it's perfectly fine but there is another way to do it there's questions from the digital sat which means you'll have access to Desmos while you are taking the SAT and this is another perfectly valid way to do it you just put in both equations and Desmos and you find where they intersect which in this case we can see here is at negative 9. so both ways are valid the way I would recommend approaching it is with a question as simple as this you can do it by hand if you want to and if you are very confident for example if you're someone who's consistently scoring over 1400 on SAT practice tests in particular over 700 on the math section and you come across this question you think it's fast for you to do it by hand that's fine you can go ahead and do it if you're someone who's scoring less than 700 on the SAT Math section on practice tests I'd use Desmos in my opinion now that you can use Desmos on the digital sat you should be using it fairly often for questions that involve graphing because it helps you avoid human error so even on a question as simple as this if someone wanted to use Desmos I wouldn't fault them at all and I wouldn't tell them not to Desmos is a great tool for the digital sat and you shouldn't be afraid to use it here's how to solve this sat math question about the solution to a system of inequalities the question States the point x and 53 is a solution to the system of qualities in the X Y plane which of the following could be the value of x so in this case we need to solve for x and we already know that this point is a solution so what we're going to do is we're going to take this 53y and we're going to as which is a y value I'm going to plug it in for y so pretty simple on how we're going to set this up it's going to end up being 40 I'm sorry 4X plus 53 must be less than 18. okay now that we've got this set up let's go ahead and isolate X and solve for it so we'll subtract 53 from each side that'll end up leaving us with negative 35 must be greater than 4X from there we want to isolate X as I said so we divide both sides by 4. now we see that X must be less than negative 35 over 4. now what you should be able to recognize here is that 35 over 4 pretty close to 36 over 4 or in other words 9 okay so we know that X has to be less than negative 35 over 4 which means X cannot be negative 5 5 or 9 because they do not pass that inequality the only number that does here is answer Choice a here's how to efficiently answer this sat math question about percentages as well as one thing you want to look out for on questions that are simple like this one question 13 states that out of 300 seeds that were planted 80 sprouted how many of these seeds sprouted so it's pretty easy to get to our answer that's going to be 0.8 to represent that 80 we're going to multiply it by 300 and that is going to give us our answer of 240. now what you want to watch out for on questions that are as simple as this one as whether or not the SAT is going to put something like this how many of these seeds did not Sprout okay and they would probably underline the word not okay because that would change the math right if you're calculating for the percent or the amount of seeds that didn't Sprout then you would end up with 60. so that is something you want to watch out for on questions on the SAT that are simple like this is whether or not there's something that could trip you up like that here's how to efficiently answer this sat math question question 14 States the function f is defined by f of x equals 4X for what value of x does f of x equal 8 well since we're asked to find the value where f of x equals eight we go and plug in 8 for f of x which leaves us with 8 is equal to 4X from there we just want to isolate X by dividing both sides by 4 8 over 4 is equal to 2 which means that the value of x would be 2 and 2 is our answer here's how to efficiently answer this sat math question that deals with division and factors question 15 states which expression is equivalent to 8X times x minus 7 minus 3 times x minus 7 all over 2x minus 14 where X is greater than 7. immediately what I'm seeing is I have x minus 7 here and here so I'm going to see is if I have x minus 7 in my denominator which in this case I see I do because I have 2X minus 14 and I can change that into 2 times x minus 7. so from here what I'm going to do is I'm going to rewrite this division problem so you can see what I'm looking for a little bit more neatly the way we rewrite this is 8x times x minus 7 all over 2 times x minus 7 minus 3 times x minus 7 all over 2 times x minus 7. and now that we've written this it's very easy to see that these x minus 7s are going to cancel out which makes this equation very very simple okay ultimately we have 8X minus 3 all over 2 which once again we can rewrite this as 8X minus 3 all over 2 which we see as answer Choice B so to answer this question what you really need to understand is one how to factor so you can get that x minus 7 out and then second how to sort of change between these similar sort of forms right of division right you can write 8X minus 3 over 2 like this and you can also also write it like this so that's really what you need to understand for this question here's how to find the slope of equations in standard linear form and use that slope to answer questions very quickly question 16 States line p is defined by 2y plus 18x equals nine line r is perpendicular to line p in the X Y plane what is the slope of line r in order to find the slope of line r we need to first know the slope of line P now the first thing you're going to notice is that we have 2y plus 18x equals 9. this is called standard linear form it's when we have our Y and our X on the same side being added together and they are set equal to some constant now because it's in standard linear form we can use the equation m equals negative A over B where a is the value or the coefficient in front of X and B is the coefficient in front of Y so in this case that would be m equals negative a in this case our a values 18 so negative 18 over 2 which is going to equal negative 9. now in order to get a perpendicular slope to negative nine you have to do the negative reciprocal so negative 9 is the same as negative nine over one so to do the negative reciprocal we have to change the sign from negative the positive and then we have to flip the fraction to one over nine so one over nine will be our new slope of line r so we can go slope of R which means that our answer has to be answer Choice C here's an example of why you need to know the exponential growth formula for the SAT question 17 States the given function f models the number of coupons a company sent to their customers at the end of each year where T represents the number of years since the end of 1998 and 0 or and T is between 0 and 5 inclusive if Y is equal to F of T is graphed in the t y plane which of the following is the best interpretation of the y-intercept of the graph in this context your y-intercept when you have an exponential growth formula like this is going to be where T is equal to zero where that exponent is equal to zero and when that exponent is equal to zero zero point six five to the power of 0 is just one and you end up with this Y intercept of 8 000. now in this case we know that that is at the end of 1998 since that's our start year that is where T is zero so from that it's very easy to find the answer so let's go and read through a through D and let's find the answer a the minimum estimated number of coupons a company sent to their customers during the five years was 1400 and no will be the minimum estimated number of coupons companies sent to their customers during five years was eight thousand no this is just the value at the beginning of the time so this is the number of coupons that they sent at the end of 1998 and then see that speed number of coupons the company sent to their customers was 1428 no we know it's eight thousand eight thousand is our initial value D the estimated number of coupons the company sent to their customers at the end of 1998 was 8 000 yes a d is our answer here's something you need to know about similar triangles for the SAT Math section question 18 states that triangle XYZ is similar to Triangle rst such that X Y and Z correspond to r s and t respectively the measure of angle Z is 20 degrees and 2xy is equal to RS what is the measure of angle T the answer is C 20 degrees now how do you know that right away well this piece of information is completely irrelevant and you don't need to know it in order to know the answer is 20 degrees and here is why you are told that these are similar triangles you are also told that Z corresponds to angle T these two angles correspond to one another which means that they must have the same angle measure sure so this piece of information does not matter okay all you need is that angle Z is 20 angle T must also be 20 and your answer is C here's what you need to know about systems of linear equations that have no solution on the SAT question 19 States one of the equations in a system of two linear equations is given the system has no solution which equation could be the second equation in the system first thing we need to know is our slope which we see is going to be six okay we see our y-intercept is going to be 18. now the reason we need to know these things is because in order for a system of linear equations to have no solution they need to have the same slope but a different y-intercept next thing I notice is answer choices a through D are all in standard linear form which means that we can use the formula m equals negative A over B where m is representing the slope now if we do that for answer Choice a we see that answer Choice a would be negative negative 6 which would give us positive six over one since one is the coefficient in front of Y so answer choices A and B I see are both going to have a slope of six now they have the right slope about which one of them has the correct y-intercept and by correct y-intercept it means the Y intercept that is different from the one that we are given because if they have the same y-intercept there is infinitely many solutions and as we can see s Choice a has y being equal to 6X plus 18 if we were able to convert it to slope intercept form and in that case it's the same equation as is up here so a is incorrect because that would be the same equation and there are infinitely infinitely many solutions to that now B has a different y-intercept it's y-intercept here is going to be 22. so our answer is going to be as Choice b as a different Y intercept but the same slope so there is no Solutions possible here's how to quickly solve this question about area on the SAT Math section question 20 States what is the area in square centimeters of a rectangle with a length of 34 centimeters and a width of 29 centimeters this question is pretty easy all you need to know is the area formula for a rectangle and that's going to be the length times the width these are both the same unit they're both centimeters that's one thing you do want to watch out for on the SAT is if you have different units for one side than the other then you'd have to convert units in this case we do not have to do that and it is very simple it is just 29 times 34. we are able to use the calculator on the section this is the digital sat so you're going to be able to use your calculator on both of the sections for the math so you just put in your calculator you're going to end up with 986 as your answer here's a bunch of tips that I can illustrate with this question to help you Ace the SAT question 21 states the solution to the given system of equations is X Y what is the value of x minus y anytime that you're taking the SAT Math section you always want to pay very very close attention to what you're asked to answer with that is the key to moving faster and being as efficient as possible know what you want to answer in this case they're asking for the value of x minus y so what I would do is if I have a system of equations like this I would look for if there's a way that I can do an addition or subtraction of this bottom equation from the top one in order to get the value of x minus y right away and sometimes that may involve multiplying one of these equations by some number now in this case I don't see any way that I can quickly get to x minus y doing this so it's not applicable here but we still have the the plus minus in order to get the value of x quickly and I'll show you how we're going to do that okay like I said sometimes you have to multiply in this case I can see that I have y and 4y so I'll just multiply this top one by four to get rid of my y so I'll do subtraction here okay so I distribute this 4 to the Y I get 4y minus 4y now my y's are gone okay so I'm going to end up with 0 here and then I'm going to have 4 times 4X and 16x minus 15x will leave me with just X then 4 times 1 I'm going to have 4 minus negative 8 which is same as 4 plus 8 that's going to be plus 12. so I can see here that if I subtract 12 from both sides I'm going to end up with X is equal to negative 12. okay so now that solve for x I'll go ahead and solve for y so if x is negative 12 I'll go ahead and erase up here so we can do this quickly X is negative 12 I'll substitute that in here so we have negative 12 and the reason I'm choosing this top equation is because we already have y isolated with no coefficients in front of it so keep in mind that's strategic now that's since I need y now I'm not going to pick the Y as a coefficient in front of it that doesn't make sense to do we want to just go ahead and get y right away so we plug in that negative 12 with the multiply it by 4 we get negative 48 negative 48 plus 1 is going to equal negative 47 okay so now we want the value of x minus y so that's going to be negative 12 minus negative 47 okay so keep in mind subtracting a negative it's the same as adding so we're going to have negative 12 plus 47 and that is going to give us our answer of positive 35. here's what you need to know about calculating the number of real solutions for quadratic equations on the SAT Math section question 22 States how many distinct real solutions does a given equation have well in order to solve this we're going to use B squared minus 4AC because this is a quadratic equation now if the value of B squared minus 4z is positive then that means that there are going to be two real solutions now if the value of B squared minus 4AC is equal to 0 then that means that there are going to be one real solution and I should have said there is going to be one real solution don't at me in the comments okay and then B squared minus 4AC if that is less than zero then that means that you're going to have no real solutions okay now that we've gone ahead and gone over the value of B squared minus 4AC and we that means for the number of real solutions let's go and calculate it so your B value is what's in front of the X your a value is what's in front of your x squared and your C value is your constant keep in mind it must be set equal to 0 for this to work so B squared minus 4AC our B value is 10 10 squared minus 4 times 5 times 16 what does that equal out to that is going to equal out to a negative 220 we know is less than zero therefore we must have no real solutions so our answer is going to be D 0 there are zero real solutions here's how to convert units on the SAT question 23 States a certain part has an area of 11 million 863 808 square yards what is the area in square miles of this park and one thing I want to point out here is anytime you have something underlied on the SAT it means you need to pay very close attention to it in this case it's because we're converting units from Square UI to square miles that being said we are given what one mile is equal to in yards and that's 1 760. so the key to this question is that you are dealing with square yards not just yards square yards not cubic yards square yards that's the key to this whole thing this this makes it super easy as long as you understand that what you're going to do here is you're going to take your 11 million 863 808 and you're going to divide it by 1760 squared and the reason why you have to square that 1760 is because you are dealing in units of square yards two square miles so you have to square that conversion unit as well once you do this you just put it in your calculator and you're going to end up with 3.83 so your answer there is going to be answer Choice B here's how to use the circle equation to answer this question on the SAT question 24 states which of the following equations represents a circle in the X Y plane that intersects the y-axis at exactly one point well first I'm going to quickly teach the circle equation to anyone who doesn't know it so the circle equation we can write as this x minus H squared plus y minus K squared is equal to our radius squared where R is our radius okay now in this situation we would have a center for the circle of h k so as you can see you want to pay attention to the fact that we have those minus signs but our Center is still at H and K so now that we've gone over that I'm going to quickly show you how I would approach this the first thing I would take a look at is what my radius value is for all of these now keep in mind 16 is the radius squared which means that the actual radius here your radius is actually going to equal 4. okay and that's important because we're only intersecting the y-axis at one point and if you're ever kind of in doubt on a question dealing with sort of circles and graphs go ahead and draw it out it's very useful so I'll go ahead and draw that out to illustrate what I'm looking for here so if I know my radius is 4. if my x value is anything greater than 4 and I'll put 4 here for example I see that my x value for the center right is going to be 8 for A and B there's no way that I can reach that y-intercept right no matter where my Y is there's just no way I can get there because my radius is only four so I can get rid of A and B already okay I have to find somewhere where I'm either at 4 or I'm at negative 4. and the only place that that happens and keep in mind I'm specifically referring to the x coordinate for your Center here and we see the only place that happens is C and as Choice D we see that X is Center is at zero now if we had a center at zero we'd end up with a circle that looks like this right and we're crossing that y-intercept twice so our answer haspian's Choice C here's a term that you need to know for the SAT Math section question 25 States and triangles a b c and Def angles B and E each have measure 27 we see that b and er here okay so they match up in that triangle angle C and F each have a measure of 41 so we see C and F they also match up in this triangle which additional piece of information is sufficient to determine whether triangle ABC is congruent to Triangle d e f the key thing to understand here and pay attention to is the fact that we are asked about congruency which means that they are the same triangles we are not asked about similarity two triangles that are similar can have different side lengths but two triangles that are congruent must have the same side lengths and the same angles now we already know that these triangles are similar because if two out of these three angles are the same that means the third angle is the same in them as well so what we need to know then is if the side lengths are all the same and the way that we can get that we can't get that from just one side lane because that's only from one of the triangles okay we can't get that from another angle measure we already know they're all the same we do need more information because we need to know whether or not the side lengths are the same and to do that we need to get the side lengths of two corresponding sides which we see we get here in BC Larry if we go up here BC right here these last two letters in EF okay so our answer has to be C and once again make sure you understand the difference between congruency which is when triangles are the exact same and similarity which is just when they have the same angle measures this is one of the most difficult SAT Math questions and it has less to do with the actual math involved and more to do with the actual logic that's required to set up the problem so with that being said let's go ahead and go over it and let's simplify it down so question 26 states that the two data sets of 23 integers each are summarized in the histograms shown for each of the histograms the first interval represents the frequency of integers greater than or equal to 10 but less than 20. so the key thing that we need to understand here is that this line right here right this 30 number is included inside of here right so it's really going whatever is sort of inside of a box keep in mind it's it's sort of the zero through nine it doesn't include that next zero okay so that being said let's keep reading through the second interval represents the frequency of integers greater than or equal to 20 but less than 30 and so on what is the smallest possible difference between the mean of data set a and the mean of data set B so as you can see pretty much sort of a logical sort of question as for the setup of this right so you get the smallest possible difference between these two data sets you want to keep a couple things in mind first of all if you ever have histograms on the SAT you want to pay attention to the shape of them now if you have two of them like you do in this case you want to pay attention to the shape of each of them and as we can see the shape of them is the same there's just a shift right so the actual shape is the same as you can see this one right here goes across into nine this goes to seven this goes to four this goes to three same thing over here we have three four seven and nine right we have the same number of bars obviously so these are the same shape but you want to also pay attention to your x-axis as well and as we can see data set B is shifted to the left 10 for each of these right so now in order to set up the smallest difference in means we would actually have to calculate the the mean of each right but we don't just calculate the mean because we can't right it's a histogram we know these numbers fall in between these so the key thing we need to recognize is in order to find the difference we need to know which one's larger and which one is smaller now this data set a it has to be larger because all of these are shifted to the right okay so they can't all fall on the same number because as I pointed out right this 30 and this is true for all of these you know these these uh rectangles or these bars on the graph it's not inclusive of you know this for example right it's 20 to 29 that doesn't include that 30 right that 30 is its own and that's what makes it so there has to be a difference here okay so we already know that the answer can't be zero because like I said there has to be a difference um you know because we are dealing with the fact that it's 20 to 29 30 to 39 and so on so if we are shifting one of them you know to the in this case to the right right if you're doing from B to a it would be to the right if you're doing a to B it would be to the left there has to be some difference there okay so with that being said let's go and calculate that difference well if we're going to calculate what the smallest difference is we need the minimum value for data set a and we would need the maximum value for data set B because we are subtracting B from a okay so let's start with getting the minimum value of data set a okay so minimum value of data set a let me go ahead and go up here okay so minimum value of data set a what we would do to set this up is we would start by getting the lowest value from our first bar right so that's going to be 20. okay so we're getting our minimum value of a so that'd be 20 times 3 okay and then we'll have to add on to that 30 times 4 and we'll have to add on to that 40 times 7 and then we'll have to add on to that 50 times 9. okay and then once we get that we're gonna have to divide by our total which is going to be 3 plus 4 which is seven seven plus seven which is 14 and then 14 plus 9 which is 23. okay so we're going to take all that we will divide it by 23. okay now keep in mind that we're going to also have that division by 23 over in data set b as well all right now we have to calculate our number for data set B and I'm gonna have to scroll down a little bit just so you can see all of this getting written out so go ahead and try to scroll down Okay so we're gonna have to set up a pretty similar thing right so I'm going to put in our subtraction sign and I know it's not super great formatted but that's okay all right so we're going to have minus now we have to do kind of the opposite thing that we just did in a so in this case we're taking our maximum values now keep in mind that our maximum values are not inclusive right so this is a range of 10 to 19 okay so we have to take 19 there so that would be minus 19 times 3 plus 29 times 4 plus 39 times 7 plus 49 times 9 okay and this is all also over 23 okay and then when we put this into our calculator which is what you'd do okay I wrote this out you would just be typing this in your calculator okay this is also going to get you your answer of one so your answer there is going to be B and that's how you solve this tricky SAT Math problem here's something you need to know for the SAT and that's how to extract numbers from underneath the square root question 27 States a triangle has legs with lengths of 24 centimeters and 21 centimeters if the length of the triangle is hypotenuse and centimeters can be written in the form form three times the square root of D where D is an integer what is the value of D well in order to get in this form of 3 times root D we need to First actually know what the hypotenuse length is so let's go ahead and solve for that so we have a right triangle we know we have lengths of 24 and 21 as our side lengths we need to solve our hypotenuse which we're going to represent as c squared we're going to use Pythagorean theorem so we're going to go ahead and do 24 squared plus 21 squared that's going to equal c squared okay we know that this is going to equal let's put it in our calculator and we're going to get 10 17 okay so we'll take the square root of each side to isolate C okay so we know that c is going to equal that square root of 10 17. okay so now that we have that go ahead and write this out a little bit more neatly we know C is equal to root of 10 17. C is our hypotenuse that's what we're trying to solve for now we need to ultimately rewrite this in a form where 3 is extracted okay well underneath the square root in order to extract the value of 3 we would have to have the value of 9 underneath that square root and the way we get that's just pretty simple right it's by 3 squared okay since we have square root here we'd be squaring three and we would get nine all right so now what we have to do is we have to do 10 17 divided by 9 and that's going to give us 113. now what we do is we set this equal to 9 times 113 which as you can see still going to equal 10 17 since we had 10 17 over 9 equal 113. so because of that we now have 9 underneath the square root which means that we can extract 3 because and I'll just write this over here okay because the root 9 times 113 is the same as root nine times root 113 and what is root nine equal to well root nine is equal to 3. okay so now we've extracted that 3 and we see we have 3 root 113 which is the same format as we got in this question so we know the value of D has to be 113 as well so our answer is 113. here's how to solve this sat math question in under 30 seconds question one States what is the solution to the given equation all we have to do here is isolate K by subtracting 12 from both sides 336 minus 12 is 324 so we know that K will equal 324 and our answer is B here's how to efficiently solve this sat math question question two states the function f is defined by f of x is equal to X cubed plus 15. what is the value of f of 2. in this case all we have to do is plug in 2 for x and we're going to get 2 cubed which is equal to 8 and we have to add 15 to that so 8 plus 15 will leave us with 23 so our answer is C here's how to interpret this sat word problem question 3 states that Sean runs a tent at a cost of 11 per day plus a one-time Insurance fee of ten dollars which equation represents the total cost C in dollars to run the time with insurance for D days we know our total cost is going to equal our one-time Insurance fee of ten dollars plus our eleven dollars per day giving us our answer as answer Choice C here's how to quickly solve this sat math question about intersecting parallel lines question four states in the figure show and line M is parallel the line and what is the value of x well we know that this angle of 26 is also going to be the same angle that is right here therefore we know that we have this semicircle of X and angle that is 26 degrees so we can do 180 degrees for that angle that semicircle minus the 26 degrees and then we're going to get the value of x in degrees which we know will equal 154. so our answer is answer Choice D so I'll go and scroll down so you can see that our answer will be D here's how to be more efficient on the SAT Math section question 5 states that John paid a total of 165 dollars for a microscope by making a down payment of 37 plus P monthly payments of 16 each which of the following equations represents the situation and as you can see as I read through the problem I wrote down the equation that I would eventually come up with so we see that our answer there has to be C now I want to talk about why this is a good strategy and when to use it what I do is when I see an SAT math question with three lines or more then as I read through the question I'll start writing down any coefficients variables or key units that I will need to solve the question so as you saw when I saw that I had a total of 165 I knew I was probably going to sum up something else to get to that total of 165 so I wrote down 165 first then as I kept reading I saw I had a down payment of 37 a down payment is a one-time payment so there's not going to be a variable next to it so I went ahead and wrote down 37 okay and keep in mind I'm trying to add up to a total here so I could probably insinuate that I'm going to have a summation after that then we have plus P monthly payments so plus indicating I need to add plus P monthly payments of 16 each in this case p is our variable representing the monthly payments and our coefficient in front of that is 16 each which is how I was able to write out this equation before I finish the question okay so this is a key strategy I recommend you use on the SAT when you have a word problem that is three lines or longer here's how to solve this sat math question in under 60 seconds question 6 states that if Y is equal to 5x plus 10 what is the value of y when X is equal to 8 well all we got to do here is plug in a for X we're gonna have 5 times 8 which is going to give us 40. 40 plus 10 is going to give us 50 and our answer will be 50. here's how to interpret bar graphs on the SAT Math section question 7 States the bar graph shows the distribution of 419 cans collected by 10 different groups for a food drive how many cans were collected by group six well let's first take a look at our y-axis we see we have their number of cans we need to know how many cans were collected by group six so we find group six we see that they collected 40 cans they the answer there is 40. here's how to efficiently solve this sat math question about probability question 8 States the table gives a distribution of votes for a new school mascot and grade level for 80 students if one of these students is selected at random what is the probability of selecting a student who's a vote for the new mascot was a lion so a couple key things to look for on probability questions number one is look for the pool that you are selecting from in this case we are selecting one of the students at random keep in mind sometimes they may ask if we are selecting a 7th grader and eighth grade or sixth grader if we are selecting one of the students who voted for the badger mascot so you always want to pay attention to the pool that you are selecting from in this case it's just one of the students so that means we're going to be pulling from this total right here okay now the key thing here is we are looking for the probability that student voter for a mascot that was a line case which we see is going to be this value here so our total probability then is going to be 20 which is the part that we are selecting which is these those students who voted for a lion over the total that we are selecting from which is all of the students so 20 over 80 is going to equal one over four so if we go ahead and scroll down we're going to see that that'll be answer choice C so c will be our answer there here's how to interpret slope in context for the SAT Math section question 9 States the graph represents the total charge in dollars by an electrician for X hours of work the electrician charges a one-time fee plus an hourly rate what is the best interpretation of the slope of the graph well we know that he has a one-time fee which is already represented by our y-intercept and then the slope is representative of his hourly rate okay so we have option A the electrician's hourly rate and that'll be the correct answer there here's how to solve this sat math question that is really a proportions question but disguised as a perimeter's question question 10 states that square X is a side length of 12 centimeters the perimeter of square Y is two times the perimeter of square x what is the length in centimeters of one side of square y well a couple things to keep in mind here one these are both squares number two the perimeter is going to be the same equation for both of them just with a different value for the side length so if we want to actually double the perimeter all we have to do is actually double the side length and because we know that we know the answer is D having to do very little math here keep in mind we didn't actually have to calculate the perimeter for either Square X or Square y because we're able to recognize that the equation for the calculation of the perimeter is the same the only difference is the actual side light so if we wanted to double the output of the perimeter or double the perimeter value in this equation all we have to do is double that side length because the equations are set up the same here's how to quickly solve this sat math question by understanding slope and y-intercepts question 11 states what is the equation of a line that passes through the point zero five well the .05 means that we have a y-intercept of 5 because that x coordinate is zero so that is where we are crossing the y axis and it's parallel to the graph of y is equal to 7x plus 4 in the X Y plane well if we're parallel to the graph of y equals seven X plus four that graph has a slope of 7 and since we're parallel to it we also must have a slope of seven so we know that our slope must be 7x okay so slope equals 7x we know our Y intercept must be five okay so our y-intercept must be five therefore we know that our equation has to be b y is equal to 7x plus 5. here's how to efficiently solve this sat math question by understanding how to calculate slope and y-intercept quickly question 12 States in linear function h h of 0 is 41 we know that when we have H of 0 f of zero g of 0 any function where we have that zero as our x coordinate then we are getting our y-intercept so in this case our Y intercept is 41. then we have H of 1 is 40 immediately all going to finish reading the question first which equation defines H well immediately what I see here is that we are moving over one okay so if we go over one we see that we are going down by one from 41 to 40. okay so negative 1 over 1 is going to be our slope okay so a negative one coefficient for our slope so now we have our Y intercept of 41 we have our slope of negative one it's very easy to find Define this equation all right so we're going to have that Negative X as our slope so our B is incorrect because it doesn't have that 41 as our Y intercept okay so a is our correct answer there we've got Negative X is the slope plus 41 y-intercept answer there has to be answer Choice a here's how to quickly solve this sat math question by understanding the equation for exponential growth question 13 states the function f of T is equal to 60 000 times 2 to the power of T over 410 gives the number of bacteria in a population T minutes after an initial observation how much time in minutes does it take for the number of bacteria in the population to double well we know that we are doubling when we have this 2 to the power of one this two is our growth factor the 60 000 is our initial amount and this T over 410 is representing how often we are doing this doubling okay and we will double every 410 minutes okay because at 410 minutes we'll get our first double and then if we go up another 410 then we'll get our second double okay in this case we are asked for the amount of time it takes for the population to double only one time so T would be 410 and our answer will indeed be 410. here's how to translate graphs up on the SAT Math section question 14 States the function f is defined by f of x is equal to x minus 6 times x minus 2 times X plus 6 and the X Y plane the graph of y equals g of X is the result translating the graph of y equals f of x up four units now keep in mind our f of x graph is already defined up here what is the value of G of 0 in other words that's the y-intercept of G of X okay now keep in mind that g of X is f of x translated up four units so we have to start by finding what the y-intercept is of f of x and then add another four units on top of that since G of X is a translation up four units of it all right so let's go ahead and solve for first the Y intercept of f of x so we know that the y-intercept is where X is zero so let's go ahead and plug in 0 for X so we'd have 0 times 0 minus six and then zero minus two and then 0 plus 6. okay and this is all going to equal out to negative 6 times negative 2 which will give us positive 12 positive 12 times 6 which will give us 72. okay so now we have our y-intercept for our F of zero right now G of 0 is what we need to solve for here and we know that's going to be up four units so that means that g of 0 will equal F of zero which we know is 72 plus 4 which means that g of 0 will equal 76 and our final answer will be 76. here's how to efficiently solve this sat math question question 15 states a candle is made of 17 ounces of wax when the candle is burning the amount of wax in the candle decreases by one ounce every four hours if six ounces of Wax remain in this candle how for how many hours has it been burning okay so we know that we start out with 17 ounces we know that we are going down by one quarter ounce every hour so in this case I'm having X represent hours and the way that I got this minus one quarter ax is by the fact that we're losing one ounce every four hours that means every hour we are losing one quarter of an ounce all right so from here the way I'm going to set up my equation is actually setting it equal to how many ounces are remaining okay so the amount of ounces that are remaining I'm going to call f of x okay and that's going to equal our initial amount minus how much we're losing every hour and keep in mind hour I'm representing is X here all right so now from here we want to find if six ounces remain okay so that would mean that we have six remaining and that's going to equal our starting amount minus one quarter X now we want to solve for what time for how many hours it's been burning so we want to solve for x now that we have this equation we'll go ahead and subtract 17 from both sides that's going to giving us negative 11 is equal to negative one quarter X from there we will multiply both sides by negative four in order to get rid of this minus one quarter order in case we multiply both sides by negative four negative 4 times negative 11 is going to give us positive 44 is equal to X so our answer will be 44. here's how to rearrange variables on the SAT Math section question 16 States the given equation relates the numbers J K and M which equation correctly expresses K in terms of J and M all right so we have to solve for what K is equal to so to do that we're going to isolate it by subtracting 14 J from both sides so we'll subtract 14 J from both sides okay so now this is gone next thing we have to do is divide both sides by 5 to isolate k all right so now we're going to end up with K is equal to M minus 14 J all over 5. okay and keep in mind that we divided that by five all right so let's find which equation matches up with r's we see that's going to be answer Choice a so a will be our answer there here's something you need to know about similar triangles for the SAT Math section question 17 States triangle fgh is similar to triangle jkl where angle F corresponds to angle J and angles G and K are right angles if sine of f is equal to 308 over 317 what is the value of sine of J okay well we know that F and J are referring to the same angle these are similar triangles they have all the same angle measures because they have the same angle measures the value for their signs are going to be the same when it's the same angle that is inputted in these parentheses which we see it is okay J and F are the corresponding angles Within These similar triangles okay as you can see up here it says F corresponds to angle J therefore they're going to have the same sine values and our answer for the sine of J will also be 308 over 317. here's how to efficiently set up and solve this sat math question question 18 states the product of two positive integers is 546. if the first integer is 11 greater than this then twice the second integer what is the smaller of the two integers okay so you want to solve for the smaller of the two integers well in this case since we have greater and smaller I'm going to use G and S as my variables okay so G and S we know 546 is going to equal the product of these two variables G and S we also know that the first integer which we're representing as or actually I guess let's use F and S for the first and the second okay so we're going to use F and S all right so we know that the first integer is equal to 11 plus twice the second integer okay that's 2s now we want to solve for the smaller so we want to solve for S here so to solve for S I'm going to go ahead and plug in F into this top equation okay we know f is also represented as 11 plus 2s we're going to end up with 546 is equal to 11 plus 2s times s we'll go ahead and distribute our s and then we're going to end up with 11s plus 2s squared okay from here we want to set this all equal to zero so we can solve for S so we're going to subtract 546 from both sides to get 0 is equal to 2. s squared plus 11s minus 546. now this question is from the digital sat which means that you have access to Desmos within the testing environment okay it's through blue Buck which is the sort of the software that they're using to test um for you to take the SAT the digital sat so at this point what I would do is I would actually plug this into my graphing calculator so I've got to show you what I did so I put it in my graphing calculator and now it's very easy to actually solve for where it equals zero because it equals zero where it crosses the x axis and as you can see there's two points but we only are able to use one of them and the reason why is because we have to remember that the product is of two positive integers so yes there are two values where s will equal where this equation here will equal zero But ultimately there's only one that is a positive value and that is going to be at 14 okay so 14 is going to have to be your answer for the value of s okay so our answer is 14 which is answer Choice B here's an example of an SAT math question where I would use substitution to solve the question quickly question 19 states which point XY is a solution to the given system of inequalities in the X Y plane so I've got negative 14 for my x value I'll go ahead and plug that in up here okay negative 14 plus 7 is ultimately going to give me a value of negative 7 my y value 0 is 0 less than or equal to negative seven no so a is wrong B I go ahead and put in a 0 for the value of x this time and a negative 14 for the value of y is negative 14 less than or equal to 7 yes it's true let me check my next Point okay I'd have negative 14 again for the value of y I've got 0 here all right is negative 1 less than or equal to negative 14 no B's wrong next one I jump up onto C I plug in 0 for X again okay so I'm going to end up with 7 I'm going to end up with negative one and then this time I have 14 for my value of y is 14 less than or equal to 7 no it is not so C is wrong our answer there would have to be D I will check it just to make sure that I did this correctly and as you should if you get to a question like this and if you're using substitution on a question I actually do recommend that unless you're in a massive time crunch you do actually check the last answer just to make sure you don't mess up because as you can see I just did like Mental Math pretty much six times so odds of making a mistake for me aren't high but for the average test taker like you could have made a mistake even I could have made a mistake uh I definitely just came across really arrogant but I'm just gonna keep going anyways I would plug in 14 for the value of x so 14 plus 7 that's gonna be 21 uh and then right here for the value of y 0 is 0 less than or equal to 21 yes uh plugging in zero again plugging in 14 here I'm going to end up with negative 28 Minus one uh so that'd be negative 29 okay as we can see negative 29 is less than or equal to zero so our answer would be D so I guess the quick lesson here is if you're doing substitution try to move quickly once you do eliminate three answer choices you should still check the fourth one because there's a chance you made a mistake so that's something I do on the SAT Math section even as someone who scored perfectly on back to back SAT Math sections I would still check that answer D just to make sure I didn't make a mistake on the mental math because as I said I did six basically six sort of situations of mental math before getting to answer twist D so there is a chance that you know anyone can make a mistake there so I wanted to quickly go over that question on substitution and with that being said let's move on here's how to solve this difficult sat math question efficiently question 20 States what is the smallest solution to the given equation to start let me get rid of these square roots by going ahead and squaring both sides from there I'm going to end up getting x minus 2 squared which is going to end up equaling 30X plus 34. now from here I want to ultimately get this set equal to zero so I can solve for x so to do that I'm going to expand x minus 2 squared to do that I use the foil method so I'm going to end up with x squared minus 4X plus 4 is equal to 3x plus 34. now like I said I want to get this set equal to zero so that I can ultimately solve for it okay so setting equal to zero I'm going to get x squared minus 4X but I also have to subtract 3x from both sides get it set equal to zero so that's going to be minus 7x total and then I have that 4 but I have to subtract 34 so that's going to be minus 30. now from here this is from the digital sat so you could plug this into Desmos and then solve for x but in this case I want to just go ahead and show you how you could also solve for this by hand since that would be a little bit more difficult way to to do it but it could possibly be a little bit quicker depending on whether you're skilled at factoring like this if you're not Skillet factoring you can go and use Desmos like I said perfectly valid way of doing it but in this case I'll talk to you about how to factor this so 30. you want to think about what multiplies together to get Negative 30 that adds to get negative 7. so immediately I would think negative 10 and positive 3. so I would end up with x minus 10 and then X Plus 3. okay so that's set equal to zero now from there I've got two possible solutions one would be X is equal to 10 and one would be X is equal to negative 3. now keep in mind that we have to have the smallest solution to the given equations now the smallest solution here is negative 3 because it's obviously negative and other value is positive so our answer here would be negative three now if you were to graph this out you also still need to pay attention to the fact that you need the smallest solution of the given equation so make sure on the SAT that you always pay really close attention to what you're asked to answer with because if you don't you're probably going to miss points here's how to efficiently solve this multi-step sat math question question 21 states that the regular price of a shirt at a store is 11.70 events the sale price of a shirt or the shirt is 80 less than the regular price and the sale price is 30 greater than the store's cost for the shirt what was the store's cost in dollars for sure okay so you want to solve for the stores cost well we know that the price the regular price is 11.7 we know it's on sale for 80 so let's go ahead and solve for the sale price because in order to get the stores cost we need to first know the sale price so the sale price is 11.7 times 0.2 because the shirt is 80 off all right so now we have the sale price so that's going to equal our sale price so I'm just going to write that as sale well we also know that the sale price is equal to well it's 30 greater than the stores cost we want to sell it for the stores cost so the stores cost times 1.3 is going to give us the sale price well we already know the sale price and we want to solve for cost we know that the sale price divided by 1.3 will equal our cost we have our sale price up here so we can go ahead and go like this 11.2 or I'm sorry 11.7 divided by 0.2 or I'm sorry 11.7 times 0.2 that is our sale price we then divide divide that by 1.3 and we get our cost okay so now we can go ahead and plug this into our calculator and let's see what we get so we got 11.7 times 0.2 divided by 1.3 and that'll give us our answer of 1.8 is equal to the cost so 1.8 is our answer there here's how to efficiently solve this sat math question about density question 22 states that the sample of Oak has a density of 807 kilograms per cubic meter the sample is in the shape of a cube where each Edge has a length of 0.9 meters to the nearest whole number what is the mass in kilograms of this sample all right so we know our equation for density is mass over volume is equal to our density in this case we told our density is 807 kilograms per cubic meter make sure we're paying close attention to units in this case it's kilograms per cubic meter we also see we're dealing in meters and we're asked for answering kilograms we're not going to be dealing with any unit conversions so that makes this a lot easier now we know that our mass is what we're trying to solve for okay so we know our mass then is going to equal 807 times our volume now we have to solve for our volume we know that is from a cube which has each length of 0.9 meters now keep in mind our equation for volume of a cube would be the Cube's Edge length cubed okay because all these all these side lengths are the same in a cube all right so ultimately that's going to be 0.9 cubed all right so now we've got our equation let's go ahead and put it into our calculator and we will get our answer so we're going to do 807 multiplied by 0.9 cubed and that'll give us our answer of 588 for the value of our Mass so our answer is answer Choice a here's how to avoid a common misconception on the SAT question 23 States for X is positive the function f is defined as follows f of x equals 201 of X which of the following can describe this function many people would be tempted to pick increasing exponential here because they see this 201 but that's not correct the correct answer here is increasing linear and the reason why is because if f of x is equaling 201 of X it's going to scale linearly with X and here's why if you were to write this out as an equation it would be f of x is equal to 2 1 times x because think about it if x is 1 and f of x is 200 and I'll just draw this as a table actually really quick okay we'll do X is one and X is 2 and you'll kind of be able to see how this works okay so our y value when X is 1 in this case would be two well what's our y value when X is 2 then well that would be 4.02 now what about when it's 3 well that'd be six point and as you can see each time we are increasing by 2.01 so this is linear increasing okay so if you see a question like this on the SAT don't fall for the misconception of it being exponential when it's actually linear and if you want something to help you figure out if it is exponential or linear try making a table with three values and that should help you figure it out here's how to solve this difficult sat math question that deals with Transformations left and right of graphs question 24 States the rational function f is defined by an equation in the form f of x equals a over X plus b where A and B are constants the partial graph of y equals f of x is shown if G of X is equal to f of x plus 4 which equation could Define G the key thing to have you solve this quickly is to understand that when G of X is equal to f of x plus 4 that is a transformation of the graph of f of x to the left by four units okay now if it was x minus four it'd be a transformation to the right four units but since we know it's a transformation to the left four units this is going to make this question a lot easier now as you can see I'm making a DOT up here at the point negative 10 and negative 1. the reason why I'm doing that is because if you take a quick glance at your answer choices it's going to help you see that getting to the point of negative one for your y value or your G of x value should be pretty easy because all of your numerators are six so in order to find that value you have to find the answer choice that will get you to a g of x value of negative one okay because that's that y value when X is negative 14. and you may be wondering how I get to that negative 14 number but I'll quickly show you okay if we're transforming this by going to the left four okay I'll go ahead and draw a negative 11 negative 12. negative 13 negative 14. okay then this dot has to shift over to that negative 14. okay so the new point will need to be at negative 14 and negative 1. so as you can see what I did is I just put those points in and now I'm just going to check my answer choices to find which one has that correct I see an option A that'd be 6 over negative 14 which does not equal negative 1 I see and B I'd have 6 over negative 10 which is also not equal to negative 1. I see in C that I would have 6 over negative 14 plus 8 negative 14 plus 8 is a negative 6 that'd be 6 over negative 6 which is equal to negative one so I see C would be my answer I can quickly check d d i c is actually just going to be G of X is equal to 6 so that could never be the correct answer okay so my answer would have to be answer Choice C here's how to solve questions about adding fractions on the SAT Math section question 25 states which expression is equivalent to and then you've got kind of this mess of fractions that you're adding together with a bunch of variables but I'm going to simplify this down for you and show you how you can do this pretty easily what you ultimately need to first get is a common denominator of these fractions I see I have x minus 8 here which means I'm probably going to be looking at x minus 8 as one of the factors in this right here so let's go ahead and see if we do have that as a factor okay we see that x squared y minus 8xy can also be Rewritten as let's see x minus 8 times x y all right so that's going to be the same so from there we see that this right here we don't really need to do anything to but what we do have to do is with this y plus 12 over x minus 8. okay we ultimately need to multiply this by X Y over X Y okay so now we have to go ahead and distribute our XY to 12 and also to Y once we do that we're going to end up with X Y squared plus 12 x y okay and since we have the same denominators here we can go ahead and add this y times x minus 8. well we know that y times x is going to give us another y x so plus another x y and then we'll also have a y times minus 8 which will give us a minus 8y okay and keep in mind that this is still all over that denominator of x minus 8 times x y well let's go into simplify this down that's going to end up being X Y squared let me go ahead and write that a little bit neater X Y squared plus 13 x y since we are combining these two terms okay and then we also have minus 8y and keep in mind this is all still over that denominator of x squared y minus eight all right from here we go and find which answer Choice matches this we see all the answer Choice C so c will be our answer there here's why you need to pay attention to details on the SAT Math section question 26 states that the table shows the results of a pole a total of 803 voters selected at random were asked if the candidate they would vote for in the upcoming were asked which candidate they would vote for in the upcoming election according to the poll if 6424 people would vote in the election by how many votes would Angel Cruz be expected to win the key thing with the SAT Math section in general is you need to know what you need to answer with in this case it is by how many votes Angel Cruz would be expected to win so it's not how many votes Angel Crews would get it's not how many votes his competitor would get it's about how many votes he would beat his competitor by so it's the difference between the two so because it's the difference between the two what we ultimately want to do is we want to take this poll and we want to find the difference between those two so we're going to do 483 minus 320. we have to put that all over the number of people who are in the poll which is 803 okay and the next thing we have to do is we have to multiply that number by the amount of people who are actually going to be voting in the election because we're wanting to write how many votes they are expecting him to win by right the difference in the expectations of what will happen with the actual number of people which is 6424. okay so I'll go ahead and multiply that by six four two four all right now we'll put this into our calculator and get the answer so the answer is 1 304 which we see is going to be answer Choice B here's how to use Desmos to be more efficient on the digital sat question 27 states that the graph of x squared plus X Plus y squared plus Y is equal to 199 over 2 in the X Y plane is a circle what is the length of the circle's radius with a question like this there's a lot of room for error if you try to do it by hand so I highly recommend using the graphing calculator that's available on the digital sat to do this so the graphing calculator on the digital sat pretty much works just like desmo so I'll go ahead and give you a look at how I would set this up it's really simple all I would do is put in this equation into the graphing calculator and since I'm looking for the circle's radius I would first find two points that are really easy to calculate so I see I have one here at the x coordinate negative 0.5 and the y coordinate 9.5 and then I just find the opposite coordinate right so I just go straight down and I see I have that negative 0.5 so the same x coordinate my y coordinate is negative 10 and a half so that difference is from nine and a half down to negative 10 and a half so that means now that I know my diameter is 20. go ahead and write that we know that our diameter is 20. so 20 is equal to D we know that our radius then has to equal 10 and we're asked for to answer with our radius so that gives us our answer right there the answer is 10. here's how to quickly solve this SAT Math problem question one States Isabel grows potatoes in our garden this year she harvested 760 potatoes and saved 10 of them to plant next year how many of the harvested potatoes it is about saved plant next year all you got to do is take your total amount of potatoes multiply that by the percentage that she saved which represent as a decimal 0.1 you're going to get your answer of 76 which is answer Choice B here's how to find the y-intercept of this graph on the SAT question two states what is the y-intercept of the graph shown we find where it crosses that y-axis and we see that's at two so our answer there will be 0 2 which is answer Choice B here's how to convert units on the SAT Math section question three states what length in centimeters is equivalent to a length of 51 meters where one meter is equal to 100 centimeters well we know that we have 51 meters which I'm going to represent with meters as M for that unit it's not a variable but it's the unit okay now we want to get rid of meters so what we're going to do is we're going to multiply by the fact that we have one meter per 100 centimeters now notice how my number of meters is going to cancel out and the unit that I'm going to be left with is centimeters so as we can see 51 times 100 is going to leave us with 5100 centimeters so our answer is answer Choice C here's how to quickly solve this sat math question about distance question four states a bus is traveling at a constant speed along a straight portion of a road the equation D is equal to 30 T gives the distance D and feet from a road marker that the bus will be T seconds after pass passing the marker how many feet from the marker will the bus be two seconds after passing the marker well all we have to do is plug in two for T since T represents the number of seconds after passing the marker and that'll give us our distance D that we are from the marker so D will equal 30 times 2 in this situation since we are doing the distance two seconds after passing the marker and that will give us 60 so our answer there will be answer Choice C here's how to simplify on the SAT Math section by Distributing a negative sign question five states which expression is equivalent to 20 W minus four W plus 3w what you'll do here is you'll distribute that negative sign to that 4w and to that 3w and that'll leave you with 20 W minus 4w minus 3w that's ultimately going to end up being the same as 20w minus 7w which is 13w so our answer there is answer Choice B here's how to use proportions to solve this sat math question faster the question is if 6 Plus x equals 9 what is the value of 18 plus 3x immediately I recognize that 6 Plus X if we multiply it by 3 we get 18 plus 3x all we got to do is take this 9 and multiply it by 3 which will give us 27 as our answer so if you see a question like this on the SAT see if you can multiply it by something to get what's right here and that will help you get to your answer faster without having to solve for your variable X here's how to quickly find the x coordinate of a vertex of a quadratic if you take a look at this question it states the given equation relates the variables X and Y for what value of x does the value of y reach its minimum well in this case we see we have a quadratic x squared minus 14x plus 22. now what you have to do then is you have to take the value of negative B over 2A and that will give you the x coordinate of your vertex now in this case we're asked for the value of x where y reaches its minimum now y will reach its minimum at the x coordinate of the vertex so all we got to do is do negative B over 2A well our B value is negative 14. so we have negative negative 14 which will give us positive 14 over 2 times a we see in this case our a values just one so 14 over 2 will give us a value of 7. so our answer there has to be 7. here's how to find an equivalent expression for a question like number eight which expression is equivalent to 9x squared plus 5x in this situation I see I can pull out an X by factoring so so if I do that I'm going to end up with x times 9x plus 5 okay let's see if we have that anywhere we see we do have that as option A so that'd be our answer I'll quickly show you why BC and D are wrong as well okay obviously 5 times 9 gives us 45 that's not found anywhere 9 times 5 again okay that's not found anywhere and then right here we'd end up with an X cubed value okay so as you can see our answer there would have to be D okay you can tell that just by factoring but I want to go and show you why BC and Dr wrong as well here's how to quickly solve this question about finding angle measures in a triangle on the SAT question 9 States and triangle ABC the measure of angle B is 52 the measure of angle C is 17. what is the measure of angle a well we know that the sum of all the angles in a triangle must add to 180 so all we have to do here is do 180 degrees minus 17 degrees minus 52 degrees and that'll give us the value of angle a and 50 plus 17 is going to give us 69 180 minus 69 will give us 111 so our answer there would be answer Choice D here's how to quickly solve this sat math question question 10 states the graphs of the equations in the given systems of equations intersect at the point x y and the X Y plane what is the value of y i it's pretty easy to solve for y here all you got to do is go ahead and substitute x equals 8 and for X we know 8 squared is 64. so we're going to end up with Y is equal to 64 plus 8 which we know is going to equal 72 so our answer there is D here's how to answer scatter plot questions on the SAT Math section question 11 states the scatter plot shows the relationship between two variables X and Y a line of best fit is also shown which of the following equations best represents the line of us if it's shown so immediately what I'm going to look for is my y-intercept which I see is right around 13 and a half then I'm also going to look at my slope which I see is negative so I see that I have positive slopes and answer choices C and answer choices a so I can get rid of them I see I have the wrong y-intercept and asteroids D So my answer there has to be answer Choice B here's a trick to make your life easier on the SAT when dealing with square roots question 12 States the function f is defined by f of x equals eight times the square root of x for what value of x does f of x equal 48 well let's go and start by writing out our equation so we're going to have 48 is equal to 8 root X now you could go ahead and square both sides but I don't recommend doing that because you're going to end up with a really large number instead what I recommend it doing is dividing both sides by 8 to start 48 over 8 is going to leave you with 6 is equal to the square root of x from here now is a great time to go ahead and square both sides because you don't have to square the value of 48 now you end up with 36 is equal to X so your answer there is going to be C here's how to efficiently solve this geometry problem on the SAT question 13 states a circle has Center o and points R and S lie on the circle and triangle ORS the measure of angle Ros is 88 and what is the measure of RSO in degrees all right let's draw this out because it's going to make it a lot easier to visualize okay we've got this Center o and we've got Ros which is 88 degrees so Ros do you think we need to understand here is that the side length from R to O and O to S are going to be the same length now that's important because we can go and put in this 88 degrees for this angle measure now we want to know what the measure of angle RSO is so let's go ahead and draw this triangle okay now as you can see we have an isosceles triangle now an isosceles triangle means that these two sides are the same which also means that the angle or s and the angle RSO are the same so now since we know all of the degrees in a triangle will sum or I'm sorry all of the angles in a triangle will sum to 180 degrees we knew 180 degrees we'll subtract the 88 degrees that we have in this angle here and then we will divide it by 2 since we know that we have two of these angles that will have the same measure okay and once we do that we'll get that angle RSO so 180 minus 88 is going to leave us with 92 92 divided by 2 is going to leave us with 46 as our answer for angle RSO here's how to efficiently solve this sat math question question 14 States what is the sum of the solutions to give an equation well let's go ahead and distribute r x ultimately we're going to want to set this equal to zero so let's go ahead and just try to do that right so we'll distribute this x we're going to end up with x squared plus x minus 56 is equal to 4x squared so we'll distribute that 4X and then we will have minus 28x all right let's go ahead and set it equal to zero to do that I'm actually going to subtract everything on my left side because I don't want to deal with subtracting 4x squared because then I have a negative coefficient in front of my x squared value and just a piece of advice I would generally try to avoid having a negative coefficient from your x squared value just something that that I would usually recommend right and it's also easier to subtract you know just a coefficient of one than it is one of four so anyways that's why I'm choosing to do it this way so we'll go ahead and do this we'll have to add 56 okay so we're going to end up getting a 4x squared minus x squared it's going to leave us with 3x squared we have minus 28x which we also have to subtract another X that's going to end up leaving us with a minus 29x and then we also have this plus 56 so plus 56. okay now from here you have a couple of options you could try to factor this out or you could just use Desmos a situation like this honestly I would just use Desmos you're dealing with 56 29 3. you might be able to factor something out there but if there's really not necessary so let's go ahead and use Desmos to solve this so as you can see in the graph we've got two solutions and keep in mind we have to answer with the sum of the solutions so the first solution that we have is at 2.667 so in other words if we convert that into thirds that would be eight over three okay so we have eight over three and then we also have seven okay if we convert seven into thirds that'd be twenty one over three so eight plus twenty one is going to give us 29 over 3 as our final answer there okay so 29 over 3 is the sum of the solutions now once again as always on the SAT Math section you have to be very pay very close attention to what you're asked to answer with in this case it is the sum of the solutions so that's how we get to that 29 over 3. so always make sure you're paying close attention to what you're asked to answer with on the SAT Math section in this case the answer 29 over 3. here's a trick you can use to save time on the SAT Math section question 15 it says the solution to the given system of equations is X Y what is the value of 5x on the SAT you always want to pay attention to what you're asked to answer with in this case it's the value of 5x we don't necessarily have to actually solve for x to get our answer we just need to know the value of 5x so the reason that's important is because if you see stacked equations like this on the SAT you're typically going to look to add subtract or substitute something in in order to get to the value asked or answer with because like I said you don't always actually have to solve for x and you don't always have to actually solve for x and y so in this case what I'm actually going to be looking to do is substitution because I see I have y is equal to 3x and I see I have a 2X here I know 2x plus 3x is going to give me 5x so when I substitute in that 3x for y I end up with 5x is equal to 12 and I know my answer has to be C and I never had to actually find the value of x here's how to solve this question about volume on the SAT question 16 States a cube has an edge length of 41 inches what is the volume in cubic inches of the cube one thing you want to pay attention to on the SAT Math section is units in this case you see they're both Inches so it's very easy okay all you have to do is take your 41 inches for your Edge length and raise it to the power of 3. okay so 41 raised to the power of 3. you put it in your calculator and you'll get answer choice of D 68 921 the reason we're raising it to the power of 3 is because a cube has a length width and height that are all the same so the volume is just 41 cubed here's how to answer this question about population growth on the SAT Math section question 17 States the given function P models the population of LOL T years after a census which of the following functions best models the population of LOL M months after the consensus the key thing to understand here is that this equation up here P of T is representing the population T years after consensus but we have to convert this into months after the consensus so to do that we need this T to turn into that t needs to turn into an M over 12 that way at 12 months we have 12 over 12 which is one year which would be the same as T being 1 1 year so we can go ahead and get rid of A and B next thing we can look at is the difference between C and D now we have to keep in mind we aren't needing to divide our growth factor at all because think about it if we divide 1.06 over 12 we now have a decreasing population because that's going to result in a value that's less than one and that's not the case we know our population is increasing every year okay our growth factor Remains the Same all we're changing is the unit of time so we have M over 12 to account for the fact we are changing from years to months our growth factor Remains the Same and our initial population Remains the Same and our answer is D here's a trick to use on the SAT Math section to be way more efficient if you take a look at question 18 it says the solution to the given system of equations is X Y what is the value of y on the SAT Math section especially whenever you see stacked equations like this always pay really close attention to what you're asked to answer with in this case it's the value of y now whenever I see stacked equations like this what I'm usually going to look to do is either substitute if I have something like y equals and then something that doesn't include another variable but in this case they both have variables there's an X and Y in each of these equations so instead of substitution I am looking for addition and subtraction of adding or subtracting this bottom equation from the top one now sometimes it'll require me to multiply the bottom of the top equation by some number in this case it does I'm going to multiply this bottom equation by 3 and the reason why is because once I do that and I distribute this 3 to all these numbers I will end up with 6X and then I can subtract my bottom equation for my top and I'll end up with 6X minus 6X I'll be able to get rid of my x's I'll have 7y minus 6y and that will leave me with my value of y so let's go ahead and do that we're going to end up with 6X minus 6X X's are gone 7y minus 6y which means that we have one y left and then we have 28 Minus s30 which leaves us with negative 2 and our value of y is negative 2 and our answer there's a so there's some advice on how you can deal with stacked equations on the SAT and work them to your advantage to be more efficient here's how to solve this sat math question under 60 seconds question 19 States the minimum value of x is 12 less than 6 times another number n which inequality shows possible values of X well we know that the minimum value of x which means that X must be greater than or equal to this number okay and it's 12 less so it's going to be minus 12 6 times another number n so 6 times n and as we can see this perfectly matches up with answer Choice B so B will be our correct answer there here's how to efficiently solve this sat math question question 20 States data set a consists of the height of 75 buildings and has a mean of 32 meters data set B consists of a height of 50 buildings and has a mean of 62 meters data set C consists of the heights of 125 buildings from data sets A and B what is the mean in meters of data set C okay so ultimate for this question we need to calculate the mean of data set C what we know that that is a composite of the data set a and b together so what we need to do is we need to get the total amount of height in meters from data sets A and B combined so to do that we're going to take the 75 buildings that are in data set a multiply it by the mean of 32 and that'll give us that total height from data set a we then have to add on to that the total height from data set B which is going to be those 50 buildings multiplied by those 62 meters of those 50 buildings so from there we then have to divide by that 125 buildings total so here we're getting our total height in our numerator and the number of buildings in the denominator and that will ultimately get us to our answer once we put it in our calculator let's see what we get of 44. so 44 is our answer here's three different strategies that you can use to solve question 21 and stick around to the end because at the end I'm going to discuss how to know which strategies you should be using on certain questions when there's different ways that you can get to the right answer all right so let's go and read through the question the graph of 9x minus 10y equals 19 is translated down 4 units in the X Y plane what is the x coordinate of the x-intercept of the results and graphs so a couple things that jump out right away if we're looking for the x-intercept we know that's going to be where Y is 0 we need to find the x coordinate of that x-intercept we're going down four units and we also want to make note that this is in standard linear form not slope intercept form okay so let's start with the first and probably the most commonly used approach that would be converting this into slope intercept form and then translating it down those four units so let's go ahead and do that if We're translating this into if we're converting this into slope intercept form the way that I would approach it is we're going to have 9x minus 10y is equal to 19. I would isolate y in this case I would actually add 10 y to the right and left sides and I'd subtract 19. so I'd end up with 9x minus 19 is equal to 10y I divide both sides by 10 and I'd end up with 9 over 10 x minus 19 over 10. now keep in mind that we are ultimately going to translate this graph down four units in the X Y plane so if We're translating it down four units we would subtract by 40 over 10 okay is equal to Y now we want to find where our x-intercept is well our x-intercept is going to be where Y is equal to zero so we can now set this equal to zero now from here I'm going to combine this negative 19 over 10 and this negative 40 over 10 into negative 59 over 10. then I'll add it to both sides and I'll end up with 59 over 10 is equal to 9 over 10 x okay now from here all we got to do is multiply both sides by 10 and we get 59 is equal to 9x from there to isolate X we see that X would equal 59 over 9. okay so 59 over 9 is our right answer but that's just one of the ways to get there let's talk about the other two different ways to get there okay so one of them is pretty similar to the way that we just did but probably a couple less steps okay and this is kind of for people who are a little bit more advanced in math who maybe are able to sort of pick up on something like this a little bit quicker okay let's say that we have 9x minus 10 y equals 19. all right now if we have We're translating this down four units okay then that's the same as doing 9x minus 10 y plus 4 in parenthesis equals 19. okay and this just kind of saves a couple steps right we distribute that 10 over we're going to end up with 9x minus 10y and then minus 40 is equal to 19. all right so from here once again like I said before our x-intercepts where Y is equal to zero so this will ultimately and I'm just going to kind of bring this up here okay we put in 0 for y we end up with 9x minus 40. it's equal to 19. we add 40 to both sides we get 59 is equal to 9x we divide both sides by nine and once again we get that same answer of 59 over 9. okay so now there is also a third way to solve this okay I want to make a point at the end that will help you with other questions on the sit too so the Third Way is kind of a halfway point between this one and the other one but it's more so like the the first step that we did so let me go ahead and pull out the Desmos because this is for the digital sat so you do have access to Desmos in your testing environment so the way that I would do this using Desmos is I would pretty much convert it into slope intercept form and then I would put it into Desmos and use that to get to the rest of the answer so as you can see if you do that you look for where it equals zero and that's at 6.556 which is 59 over 9. now the disadvantage of using does most is it's not going to give you a perfect fraction but most of the time it's pretty easy to figure it out this is probably one of the more difficult cases where it's something being divided over nine and it's a little bit more difficult to discern what the fraction is but most of the time it's usually something over three or over five or over four and it's pretty easy to do but now that we've talked about the three different methods that you can use for solving this question I want to talk about how you know which method you should use just generally on certain questions because there are often times certain shortcuts you can use and many times maybe not many times but some people may not know those shortcuts exist so are they worth learning when should you use them how much focus should you put into using shortcuts versus using Desmos I think it's something we really do need to talk about because with the digital sat everything is very different and I think it really comes down to how comfortable you are using sort of shortcuts like the one I showed you in the second way of solving this where you know that it's translated y plus four right if you don't know those shortcuts you know rely more on Desmos convert things into slope intercept form there's so many different maybe not so many but there's sometimes a couple different ways to get to an answer and you ultimately need to use what you're most comfortable with now is it good to still learn shortcuts yes because it can save you time so especially if you're someone who struggles with time it's probably worth learning them but if you're someone who can move really fast by converting things into slope intercept form and using Desmos then do it and maybe don't worry as much about learning these shortcuts my job is to give you as many tools as you can to be successful But ultimately you do need to pay attention to what you're most comfortable with there are certain times where the method that I use I'll show you two different methods and you might choose one that I don't necessarily prefer to use and that's okay but you ultimately should be focused on what you're most comfortable with what you're fastest with what your fastest and most accurate with so you kind of got to make a judgment calling for yourself my job is to give you as many strategies as you can to be successful if you find that some of them are a little bit too advanced for you that's okay notice how you could get to this answer using slope intercept form and Desmos right so it's possible to still do highly well extremely well on the SAT Math section and not know every shortcut but like I said if you do struggle with time it is worth knowing the shortcuts so I guess to wrap all this up and to bring it to a conclusion I'm going to give you the tools and you know as many different ways to solve things as I can I'll give you what I recommend I'll try to give you strategies but you also need to take into account your personal situation how comfortable you are using certain strategies and that'll help you do the best that you possibly can on the SAT so there's my little rant for this question here's how to answer this question about the exponential growth formula on the SAT question 22 States two variables X and Y are related such that for each increase of 1 and the value of x the value of y increases by a factor of four when x equals zero y equals 200 which equation represents the relationship well this piece of information right here is giving us our initial value when x is 0 Y is 200 so we need to have that y equals 200 so we can go ahead and get rid of A and B because they both don't have the correct initial value now as for our growth factor we know our growth factor is four it's stated pretty explicitly right here for every increase in the value of x by 1 we increase our value into y by a factor of four now we have to keep in mind we are going to raise this to the power of X because this is exponential growth every time we increase the value of x Pi 1 we increase y by value by a factor of 4. that is exponential growth down to the definition so we know our answer there has to be answer choice of D and there's a quick review on exponential growth for you here's how to solve one of the most difficult SAT Math questions in a standard way and a way to finesse the question if you're not sure how to do you the standard way all right so question 23 States one solution to give an equation can be written as one plus root K where K is a constant what is the value of K so the key part to this is you seeing that you have one plus the square root of K and recognizing that you're probably going to be dealing with some sort of squares there all right so the next thing you really need to understand is how to ultimately get this into the point where you're completing a square right and if this sounds complicated it kind of is but at the end I'll show you a way that you can finesse this so just just hang with me here all right so you're going to start by adding 9 to both sides you'll end up with x squared minus 2x is equal to positive 9. now what you ultimately want to get and what your goal is here is since you saw you have this you want to get to a point where you have X something squared set equal to some constant okay and the reason why is because I guess I'll just get to that kind of as I go through this but you'll see why you want to get there and that's what I'm trying to get to all right so we have x squared minus 2x is equal to 9. now like I said I want to get to X something squared now x squared minus 2x I know x minus one squared isn't going to quite get me exactly x squared minus 2x it would give me x squared minus 2x plus 1. so what do I have to do here well I've got to add 1 to both sides so if I add 1 to both sides I'm going to end up with x squared minus 2x plus 1 is equal to 9 plus 1 which is 10. well now from here I can set that equal to that x minus 1 squared so now I'm where I want to be okay because from here I'm going to take the square root of both sides and now I have root 10 and now keep in mind that root 10 is plus or minus okay because we are taking the square root of it so be plus or minus root 10 is going to equal our x minus 1. okay now from here we go ahead and add 1 to both sides we get 1 plus or minus root 10 is equal to X okay now if we look at the format for the solutions right we need to be writing it as 1 plus root K well we see that one of our Solutions here would be 1 plus root 10. okay so we know our answer for the value of K is 10. okay so our answer would be B now let's say you don't know how to do this which honestly like it's one of the most difficult SAT Math questions so it wouldn't be that surprising okay if you don't know how to do this and you get to the SAT and maybe you are able to do it in practice but on the SAT you just kind of Panic or you're running out of time here's an alternative method that you can kind of finesse to get to the right answer all right so I'm going to use Desmos here and keep in mind the reason I'm using this is because on the new digital sat you pretty much have access to Desmos within Bluebook the testing platform so you'll be able to use this so what I'm going to do here is I plug in my equation now keep in mind this is set equal to zero right I'll go ahead and circle that so you see what I'm referring to right we're set equal to zero so we want to find where we are equaling zero okay so where we are crossing our x-axis now if we take a look we see that we cross our x-axis at 4.162 now what else do we know and I'm going to go ahead and erase this since we have that the solution must be one plus root K and we know that our solution is equal to 4.162 because we see that on Desmos we can do this we've got 4.162 we subtract 1 from that we get 3.162 and we know that's got to equal the square root of some number K now since this is multiple choice we can actually just plug in these numbers and get to our right answer but I want to talk about what if this was free response you can still finesse it even if it was free response and here's how the way I would look at this is I know that the square root of 9 is 3. and I see that we need to get a little bit higher than that so then what I would do is I would put in the square root of 10. and keep in mind this might not work if you're dealing with numbers that are like the square root of 500 but that question typically wouldn't be asked on the SAT because that would be extremely complex and if you already have a question as complex as this you're probably going to be dealing with with a K value that's not you know extreme right so I see 3.162 I know that 9 if I take square root of that's three I go up one I take square root of 10 and I get 3.162 so that's how you can finesse this even if it's not multiple choice obviously since it's multiple choice you could just go ahead and plug in um the 8 the 10 the 20 and see which one works and you'd get 10. so I want to go and show you this because it's a great example of how you you can use the digital SATs desmo is basically being built in to your advantage and if you have a graphing calculator it's the same thing so hopefully this was helpful and hopefully you understand better how you can finesse the SAT if you do forget how to do things you know the normal way quote unquote here's something you need to know about standard deviation for the SAT Math section I'm going to illustrate this through question 24. the dot plots represent the distributions of values and data sets A and B which of the following statements must be true then we have two statements here one being the median of data set a is equal to median and data sub B and then two being the standard deviation of data set a is equal to the standard deviation of data set B now whenever you have anything like a a histogram or Scatter Plots or two data sets you always want to compare the shape of them to each other okay or the values of them to each other if it's the case that they are just listed out in numerical form but in this case we have the shape so we're going to compare the shapes of these two together and as you can see they're the same except for these two dots in data set a you see that there's three dots here for 13 but in data set B there's only one okay so we're looking at this as the only difference now as far as the medians we see that since everything else is the same right we would cross out these and we would end up getting a median of 13 and then over here in data set B we'd also get a median of 13s their medians are the same okay so we know that one is true let's see if we can answer it just based on that we know that one must be true uh it doesn't look like that will get us there on its own so let's go and look at number two standard deviation of data set a is equal to standard deviation of data set B well we know that that can't be true and here's why okay even though these things have the sh the mostly the same shape this difference of two more values being centered around the median is going to make it so that the standard deviation of data set a is lower than that of data set B okay they can't be the same in order for the standard deviations to be the same they have to be the same right they have to be the same data sets for the most part generally on the SAT that's that's pretty much the case Okay because they don't really get too in depth in the actual equation for standard of deviation they pretty much just want you to be able to visually tell in terms of standard deviation so yeah what you need to know here is that since we're adding more concentration around the median around the mean value okay that means that our standard deviation is going to be lower in data set a okay so the answer here is that one is true number two is false okay so that would be which of the following must be true the answer is a one only here's how to solve this really difficult SAT Math Problem the first way being how you can finesse it since it's a multiple choice and the second way I'll show you how you can solve it on the off case that you come up with a question like this that is a free response answer so that being said let's go and get into it question 25 States and isosceles right triangle has a perimeter of 94 plus 94 root 2 inches what is the length in inches of one leg of the triangle okay so ultimately the length of one leg is pretty easy to find and the reason why is that the isosceles right to Triangle so isosceles right triangle and I'm going to draw this out as I recommend you do with questions about triangles and circles on the SAT we've got these two side lengths will be the same because like I said isosceles now that means it's gonna be a 45 45 90 triangle okay therefore we know our hotness will be X root 2 where X is the value of one of the legs okay from here let's go and set up our equation we know our perimeter is 94 plus 94 root 2. we know that's going to equal 2x plus X root 2. okay we get that by taking X Plus X Plus X root 2. okay from here I'm looking to solve for x right so I'm going to factor out X for my right side I'm going to get x times 2 plus root 2. okay now to ultimately solve for x then I just divide both sides by this 2. plus root 2. okay so now I have isolated my X this is over 2 plus root 2. now since this is multiple choice what I would do at this point is I would just go ahead and put this right here in my calculator I'd get whatever it is as a decimal and then I would just check a through D and find the answer that is the fastest way to do it since it's multiple choice so that's what I would 100 recommend and I would without a doubt with no question that's what I'm doing if it's multiple choice which it is in this case so that's a strategy you should use on multiple choice however this question or one like it could come up as a free response question and the way that they would do that since it's dealing with a square root is they would probably say they would do something like this they would put like some variable underneath the square root and say What's the value of D and the answer um so I am going to show you how you solve this in case you come across one of these that is free response so let's continue this question okay so we know that X is equal to this right here now we need to ultimately get this so it's actually a number right we need to get it so that it's something we can put in on free response now to do that we have to multiply by two my minus the square root of 2. okay so we multiply by 2 minus square root of 2. and the reason that we have to do this is because we have 2 plus root 2 here and by taking the difference of these squares it's going to help us get rid of the square root that's in the denominator all right so let's go ahead and do this okay we're going to have this 2 times 2 and I'll do the denominator first so we'd have this 2 times 2 that's going to give us four okay we're gonna have 4 times let's see we're going to have this plus root 2 times 2 and we're gonna have this minus root 2 times 2. so keep in mind those are going to cancel that's why we're doing this we're doing the difference of these squares okay so that square root is now going to get canceled out okay one thing we will still have is this root 2 times this negative root 2 which will give us a minus 2 right so we're going to end up with a 4 minus 2 minus 2. okay now in our denominator I'm sorry our numerator we also have to do this right so we have to take the same thing we have to foil it out all right so we got 94 times 2. we know that's going to be 188 so we got 188 we've also got 94 root 2 times 2 so that's going to give us plus a one root two we've also got a 94 times negative root 2 which will give us 8 minus 94 root two okay and I know I'm running out of room there so I'm actually going to erase this and bring it up okay so as you can see I folded out I brought it up here so you can see it a little bit better and it's a little bit bigger now from here we can go ahead and see that we have 188 minus 188 okay so it's going to cancel so I'm just going to go ahead and erase those okay we still have this plus 188 square root 2 minus 94 square root 2. that's going to leave us with a positive 94 square root 2. so now we have 94 square root 2 over 2. 94 over 2 will leave us with 47 and now we are left with our answer of 47 root 2. so that's how you solve it if it is free response okay the key thing being right here down in your denominator you have to multiply by flipping the sign within that parentheses okay in order to get that difference of squares okay so that's the key to solving this if it is free response remember though if it's multiple choice do not do this it will take way too long just put it in a calculator and then plug in a through D and find the right answer sir here's how to solve SAT Math questions that deal with the number of real solutions question 26 states in the given equation C is a constant the equation is exactly one real solution what is the value of C well if you're unfamiliar with how to calculate the number of real solutions you're going to use this formula called B squared minus 4AC okay and keep in mind you use this when you have a quadratic equation like here set equal to zero okay and in this equation your a value is what's in front of your x squared value your B values what's in front of X and then your C value is your constant which in this case is just being represented as C now if B squared minus 4 AC is greater than 0 then that means that there are two real solutions so that means there are two real solutions in this case we want one real solution so when B squared minus 4AC is equal to zero that's when we get one real solution okay now there is a third case which is where B squared minus 4AC is less than zero and in that case there are no real solutions or in other words there are zero real solutions okay now in our case we know that we have to set this equal to zero since we're looking for where there is one real solution well we know our a value is going to be negative nine when our B value is 30 and we obviously have C so B value is 30. we're going to take that we're going to square it so well 30 squared minus 4 times a we know our a value is negative nine times C which we're just representing as C and that's got to all equal zero 30 squared is going to be 900 12 900 minus 4 times negative 9 which is going to end up being positive 36 so a positive 36 C is going to equal 0. from here we'll subtract 900 from both sides and we'll end up with 36 C is equal to negative 900 and then from there we'll divide both sides by 36 to get that c is equal to 900 or I'm sorry negative 900 divided by 36. okay now negative 900 divided by 36 is going to give us negative 25 so we know our value for C must be negative 25. this SAT Math problem may seem scary but it's actually really easy and today I'm going to explain why question 27 states in the given system of equations p is a constant if the system has no solution what is the value of P so this is a system of linear equations and in order for a system of linear equations to have no solution the slopes must be the same but the y-intercepts must be different now these are going to be in standard linear form now they aren't really in it yet but we're going to put them in it and that'll make it a lot easier so let's start with the top equation now standard linear form is where you have a coefficient in front of your Y and then a coefficient in front of your X and it's not equal to a constant all right so let's go ahead and do that we'll start with equation one which I'm just going to Mark with a 1 right here now we have 3 over 2y minus 1 over 4 x equals two-thirds minus 3 over 2y now we need to move this right here over here and we're going to do that by adding it since it's minus 3 over 2y we're going to add 3 over 2y to both sides now that's going to end up leaving us with 3 over 2 y plus 3 over 2y which will ultimately simplify down to 3y so now we have 3y minus 1 over 4 x is equal to 2 over 3 for our first equation now like I said we have to have the same slope so let's go ahead and find the slope for this first equation well when we have standard linear form which is where we have a coefficient in front of our Y and a coefficient in front of our X that equal to a constant our slope is going to equal negative A over B where okay and M is going to represent our slope where the a value is the value that is in front of our X which in this case is negative 1 over 4. okay so we're going to end up with positive 1 over 4 because we have negative negative 1 over 4 which gives us that positive 1 over 4. we're going to divide that by b b value being 3. so we're going to divide that by 3. now keep in mind when you are dividing by 3 right divide by 3 over 1 okay we multiply by the reciprocal right so that'd be the same as multiple by one third okay so that makes it a lot more simple so we got 1 over 12 now okay so now we know that our slope is 1 over 12. now like I said we need to have the same slope in this next equation in order for there to be no Solutions all right so let's get to that same slope once again we want to put it in the same form so let's go ahead and bring this py over by subtracting it okay so we're going to end up right here this is me our second equation one half X we have this plus three over two which we can actually just go and move to the other side so we'll move it to the other side okay but we're gonna have minus py okay and then we have 9 over 2 minus 3 over 2. okay which is also going to leave us with 6 over 2 which is three so let's put that equal to 3. okay so now we got to solve for what p is well we know we need to get the same slope so we're going to have 1 over 12 1 over 12 must equal now we got negative a our a value here is one half so that'd be negative one-half okay and then we have to divide that by our B value so we have to divide that by this negative P so to divide that by negative p over one now like I said we can just change this division multiplication by doing multiplying by 1 over negative P so multiply by 1 over negative P multiply it by one over negative P okay so now we see that these negative signs are going to cancel so we'll just go ahead and get rid of those negative signs all right so now we have ones in all of our numerators and then we have 12 is equal to 2p so we have 12 equals 2p we divide both sides by two and we're going to end up getting that 6 is equal to P so we see our answer there will be 6. so even though this problem looks really scary it's actually pretty simple it's just a matter of knowing how to set it up understanding standard linear form how to get your slope and the fact that in order for there to be no solution you have to have the same slope and a different y-intercept which as you can see here we do have different y intercepts as well here's how to interpret bar graphs on the SAT Math section question 1 States a group of students voted on five after school activities the bar graph shows the number of students who voted for each of the five activities how many students chose Activity three in this case we'll take a look at our y-axis we see that that's number of students therefore we want to know the number of students who voted for Activity three we find that bar and if we go ahead and go across we see it somewhere around 38 or 39 which we see is answer Choice B here's how to answer this SCT math question about percentages in under 60 seconds question two states what percentage of 300 is 75 all you got to do is do 75 divided by 300 and then that will get you your decimal to convert that to a percentage you multiply it by 100 and that would get you 25 percent so your answer is answer Choice a here's how to solve this sat math question in under 60 seconds the question States what is the solution to the given equation in order to solve this we need to isolate X we'll start by multiplying both sides by 25. next thing we have to do is get X all alone so we have to take the square root of x squared which means we also have to take the square root of 36 times 25 this leaves us with X is equal to the square root of 36 times 25 which will equal out to 30. so we know our answer there is going to be answer Choice B here's how to efficiently solve this ICT math question the question States three more then eight times a number X is equal to 83 which equation represents the situation so one thing you'll note is as I read the question I started writing down the equation I notice we have three more that means 3 plus something eight times a number which is represented as X so plus 8X is equal to 83 we're asked for the equation we've already got it so 3 plus 8X equals 83 we see that we have that in answer Choice D so D will be our correct answer there here's how to interpret this word problem on the SAT question five states hand deposited a fixed amount into her bank account each month the function f of T equals 100 plus 25 T gives the amount in dollars in Hannah's bank account after T monthly deposits what is the best interpretation of 25 in this context well we know F of T represents the amount in her bank account this 100 since it doesn't have any variables next to it is her initial amount and then she's adding on 25 every a month so that that 25 is the amount that she's depositing every month so we have option A with each monthly deposit the amount enhance bank account increases by 25 yes that is a correct interpretation so our answer is a here's how to quickly solve this sat math question and a trick to check your work question Six States a customer spent twenty seven dollars purchase oranges at three dollars per pound how many pounds of oranges did the customer purchase well we know that she spent twenty seven dollars so that's twenty seven dollars now we have to divide that by that three dollars per pound okay so that's three dollars per pound I'll represent pound is lb okay now what we see here is that our dollars are going to cancel and what we're left with is 27 over three which is nine pounds okay so we have nine pounds and obviously you're seeing that this is the correct sort of equation to set up because we end with pounds we don't know how many pounds of Orange Is the customer purchased so that's one way you can check is by making sure you end with the correct units um and as you can see our answer there will be nine here's a tip to help you be more efficient on the SAT Math section and how to apply it so when you glance at question seven you see it's gonna be more than three lines now because it's more than three lines I'm probably gonna be dealing with some sort of equation or system of equations or system inequalities what I'm going to do is as I read through I'm going to start writing down coefficients and variables so that I can kind of get my equation on my initial read through hopefully which should save me time so let's go and read through let's hear about nine storage bins that were each the same price he used a coupon for 63 off the entire purchase the cost for the entire purchase after using the coupon was 27 okay and we know that he got 63 off and it's nine times some variable we'll see if we get what that is what was the original price in dollars phone storage bin we're not giving it so we'll just call that X okay so now we have to solve for x x is the cost per storage bin so all we got to do here is isolate X We'll add 63 both sides okay we're gonna end up with 90 is equal to 9x we divide both sides by 9 and that will give us X is equal to 10. so our answer there will be 10. so as you can see by writing down our coefficients and variables on our initial read through we're able to save time because we don't have to go back and find these coefficients and variables we're able to get that equation set up immediately after our first read-through avoiding a second read through and saving us time here's how to turn a table into a linear function function on the SAT question 8 States for the linear function at the table shows three values of X and their corresponding values of f of x which equation defines f of x in this case all my answer choices are in slope intercept form so what I'll start by doing is finding Intercept in this case since I'm given the x coordinate of 0 I know that my y-intercept will be at 29 which means I can get rid of B and I can get rid of D next thing I'm looking at is slope okay I see that when I go over 1 I go up by 3 which means my slope is going to be 3x So my answer there be answer Choice a here's how to quickly solve this sat math question about similar triangles question nine states right triangle pqr and Stu are similar where P corresponds to S if the measure of angle Q is 18 degrees what is the measure of angle s okay so we know Q is 18 degrees okay we also know then that t is going to be 18 degrees since these are similar triangles okay now we want to solve for the measure of angle s the measure of angle s is the same as the angle measure of P so what we can do here since this is a right triangle okay we know that 90 is going to equal our angle measure s so S Plus our angle measure that is 18 degrees so that would be angle measure t Okay so Plus 18 degrees 90 degrees equals 18 degrees that means that s is going to equal 90 minus 18. 90 minus 18 will leave us with 72 degrees so 72 degrees will be our answer for the value of angle s or answers B here's how to answer this question about Scatter Plots on the SAT question 10 states the scatter plot shows the relationship between two variables X and Y which of the following equations is the most appropriate linear model for the data shown in this case since we're looking for a linear model I'm going to go ahead and find about where the y-intercept is it's right around 10. I see my slope is going to be negative so if I'll go down to my answer choices anything with a positive slope I can get rid of so that means answer choices into choices a I can get rid of and answer C I can get rid of because they both have positive slopes then I'm looking at my y-intercept my Y intercept needs to be right around positive 10 which I see I have in answer Choice D so D we have that positive y-intercept somewhere around 10 then we also have that negative slope of around one so D will be our correct answer there here's how to quickly solve this sat math question question 11 states the given equation describes the relationship between the number of birds B and the number of reptiles are that can be cared for at a pet care business on a given day if the business cares for 16 reptiles on a given day how many birds can it care for on this day well we know we can plug in 16 for R then if we plug in 16 for R we're going to end up with 5 times 16 which will equal 80. so then we would have 80 is equal to 80 okay which means that the value of B has to be zero okay so our answer is a here's a trick to help you move faster on the SAT Math section if we take a look at question 12 it asks what is an equation of the graph shown if I glance at my aunt's choices I see all my y-intercepts are the same so I'm really worried about my slope here I see my slope is negative which means I can get rid of anything with a positive slope the next thing I see is I've got to calculate the negative slope and see what it is well I see that right here I go down by 2 and I go over by 2. now the key thing that they're testing you on is whether or not you're taking a look at what the units are okay because if you look here even though we only have one dash that we are changing we are actually moving over to the right by two units so the key thing here is to make sure that you're paying attention to what your units are on your x-axis and your y-axis so as we can see we are going down by two and we are going to the right by 2 which means that our actual slope here is just Negative X okay so our answer there is C here's how to answer this sat math question super quickly question 13 states if x over 8 is equal to 5 what is the value of 8 over X since we know that X over eight and eight over X are just flipped all we got to do is take this five over one flip it to one over five and that'll be our answer here's a trick to help you be way more efficient on the SAT Math section question 14 States the solution to the given system of equations is X Y what is the value of y the key thing to pay attention to on questions like this is what you're asked to answer with in this case it's the value of y which means there's a chance we don't have to calculate the value of x if we take a look at this system of equations whenever you see stacked equations like this you want to look to add or subtract this bottom equation from the top one and sometimes that might be mean that you have to multiply the bottom of the top equation by some number in order to get to the value you want to answer with which in this case is y so to get y we want to get rid of X so let's take a look at if we can do that well if we multiply this bottom equation by four we're gonna end up with 24x and then we can go ahead and subtract this bottom equation from the top equation and then we'll be left with a Y value and then a constant so let's go ahead and do that so we're going to have 24x minus 4 times 6X so minus 24x our X's are then gone then we have y minus 4y because we have to distribute this 4. so y minus 4y is going to leave us with negative 3y is equal to 48 minus 4 times 72 which when we put in our calculator will give us negative 240. so from there we isolate y by dividing both sides by negative 3 so we can go ahead and divide both sides by negative 3 and that's going to leave us with positive 80 as the value of y so our answer there is 80. here's a trick to solve this sat math question way faster question 15 states line T in the X Y plane is a slope of negative 1 3 and passes through the point nine ten which equation defines line T first thing I'm seeing is I've got this slope of negative 1 3 which means I can get rid of answer choices A and B and I'm left with c and d the only difference with the 2 being their y-intercepts now since I know that this is ultimately going down and I've got these points 9 and 10. if I plot that point just roughly I know my y-intercept is going to have to be somewhere above 10 right because we've got to be going down throughout this time so because we have these points that's why we know that thus our answer cannot be C because C would require us to have a flat line in order to hit that point so we know that that cannot be true and we know our answer there has to be answer Choice D here's how to interpret exponential growth word problems on the SAT Math section question 16 States the function f of x is equal to 206 times 1.034 is the power of X models the value in dollars of a certain bank account by the end of each year from 1957 through 1972 where X is the number of years after 1957. which of the following is the best interpretation of f of 5 is approximate to equal to 243 in this context well we know that 243 is ultimately what the value of f of 5 is therefore that 243 is the amount that is in the account after five years after 1957 so that would put us at 1962. okay so let's go and finance choice that matches up with that interpretation we have option A the value of the bank account is estimated to be approximately five dollars greater in 1962 than in 1957. no B the value of the bank account is estimated to be approximately 240 three dollars in 1962 yes 1962 is five years after thus the F of five and it's equal to that 243 dollars in value here's how to quickly answer this sat math question using proportions question 17 States for a certain rectangular region the ratio of its length to its width is 35 to 10. if the width of the rectangular region increases by 10 by 7 units how must the length change to maintain this ratio okay so what we know here is that our length is going to equal three and a half times our width and the way we get that is by having 35 over 10 right if you were to rewrite 35 over 10 you get three and a half all right so now from here we want to know if we increase our width by 7 units how much will our length change well we can just use this proportion right here so all we would have then is three and a half times seven which will equal 24 and a half so we know that we must increase by 24 and a half since we are increasing our width we must also increase our length in order to maintain that Ratio or or proportion so our answer there's B here's how to break down hard sat math problems to make them easy and simplistic question 18 states Square P has a side length of X inches Square Q has a perimeter that is 176 inches greater than the perimeter of square P the function f gives the area of square q and square inches which of the following defines F okay well we know that ultimately the perimeter of square Q is going to be 4X plus 176. all right that is equal to perimeter of square Q okay now the key thing here is we need to understand what we have to solve for and that's the area of square Q so in order to find the area of a square we need to know the side length so in order to find the side length here what we can do is we can take this perimeter and we can divide it all by four so if we go ahead and take 4X plus so 4X plus 176 and we divide all of that by four that's going to leave us with X Plus 44 as the value of the side length now the area of the square then is going to be the value of the side length squared so we know our answer there will be answer Choice a here's how to quickly rearrange variables on the SAT question 19 States a given equation relates to the sync positive real numbers W X and Y which equation correctly expresses W in terms of X and Y well in this case all we got to do then is isolate W as you can see in all our answer choices so what we'll do here is we'll go ahead and divide both sides by 2 when we divide this side by two we see we have 2 times 7 which is going to equal 14 so we can go and cancel both these 14s now we're left with X over Y is equal to the square root of W plus 19. so we're going to go ahead and square this side so we can get that W out from underneath that square root and since we're done since we're squaring that side we also have to square this side so we're going to end up with then is going to be X over y squared is equal to W plus 19 to isolate w we got to subtract 19 from both sides so we're going to end up with W is equal to X over Y and that's all going to be squared minus 19. if we look at our answer choices we see that that's going to be answer Choice C so c will be our answer for question 9 19. here's something really simple about circles that you need to know for the SAT question 20 States Point O is the center of a circle the measure of Arc RS on the circle is 100 degrees what is the measure in degrees of its Associated angle RS now what you need to know here is that if you have an arc for example RS and you have its Associated angle that Associated angle is going to be the same degrees as the arc so the answer here is just 100 and if you want to see this sort of drawn out I'll do that really quick okay we have Center o and we're drawing r o s so R to O to S okay we're told that the arc RS is 100 degrees well that means that the angle also has to be 100 degrees here's how to quickly solve this difficult sat math question question 21 states the expression and then I'm not going to say that but you can read it it's equivalent to a x to the power B where A and B are positive constants X is greater than one what is the value of a plus b pay really close attention to here this is free response it'd be very easy for someone to answer the value of a or b or a times B we need to answer with a plus b okay that means let's go ahead and solve so what I see here is that I'm taking the fifth root and then the eighth root so I've got a six and then we've got fifth root of three raised to the power five which would leave us with just three since that's five over five which is to the power of one okay so I'll just write that as power to one so you understand that next thing we have is X to the 45 over five okay so x to the 45 over 5 is the same as x to the power of nine okay so we have x to the power of nine now we have to multiply that by this root eight times or root eight and then two to the power of eight which is the same as 2 to the power of eight over eight or two to the power of one and then we have to multiply that by x to the power of one over eight all right so from here it's pretty simple we're going to take the six times three times two and six times three gives us eighteen eighteen times two gives us thirty-six now we have 36 okay so now we have 36. now the next thing we've got is this x to the power of 9 times x to the power of one eight so what I'm going to do is I'm gonna go ahead and convert this 9 into something where the denominator is eight as well so I'll do 9 times 8 and that's going to give us 72 so we'll have x to the power of 72 over 8 and now I can go ahead and add 72 and 1 to get me x to the power of 73 over 8. okay now we know that our value for a is going to be 36 okay so a will equal 36 and we know that our value for B is going to equal 73 over 8. so from here what we have to do is we have to sum these two values together now in order to sum these together I need to convert this 36 into also being over 8. so I'll go ahead and convert that so that's going to be 36 times 8 which will give us 288 288 over 8. so now we have 288 plus 73 288 plus 73 is going to leave us with 361 all over 8. so our answer there is 361 divided by 8. here's how to answer this difficult question about calculating the area of a triangle on the SAT question 22 States a right triangle has side lengths of 2 root 2 6 root 2 and root 80. what is the area of the triangle in square units key thing we have to figure out here is what is the hypotenuse out of these three lengths because we know the formula for the area of a triangle is equal to one half the base times the height so what I notice here is that I've got two root 2 6 root two so this is obviously the greater so we know one of our legs would be that two root two now between six root two and root eighty what I know is if I take six and I Square it I get 36 okay so what I could do is I could rewrite 6 root 2 is equal to root 72 and I get root 72 because I take 6 squared that gives me 36 and then I multiply that by that 2. okay so I know that ultimately this root 80 is what would be either hypotenuse so therefore I know that my base and my height values can be represented then as that 2 root 2 and then that 6 root 2. okay from here it's really easy because we see this 2 is going to cancel with this 2 and we'll be left with a is equal to 6 times root 2 times root 2 which we know will end up equaling 6 times 2 which will equal 12. okay so our answer Advanced Choice B here's how to solve one of the most difficult questions on the SAT Math section and even though I think this question is kind of pointless and stupid I'll still show you how to solve it question 23 States the expression 4x squared plus BX minus 45 where B is a constant can be Rewritten as HX plus K times X Plus J where H K and J are integer constants now if you ever come across on the SAT then saying that something is integer constants that probably means it's going to be required in order for you to solve the problem so pay close attention to it which of the following must be an integer so once again if they're telling you that something needs to be an integer pay attention to that always so in this case what I'm going to do is I'm going to start by expanding this right here this HX plus K times X Plus J I'm going to expand it out because as you can see we're going to need to end up getting the values for you know h k possibly J as well in order to get our answer so let's go ahead and do that so I'm going to end up with 4x squared plus BX minus 45 is going to equal and now we'll go ahead and expand on this so we'll have h x squared h x squared and then we'll have plus KX and then we'll also end up with a plus JH X as well so we're going to have plus j h x plus k x and then we'll have a plus k j as well from this K times J okay so from here let's go ahead and try to simplify this right side so we'll still have h x squared but I see I have this common factor of X so I'm gonna go ahead and pull that out so we'll end up with plus x times j h plus K and then plus KJ all right from here let's start getting what things equal we see that we're going to have a negative 45 is equal to the value of K times J the other thing that we'd see is that we have this this BX not necessarily anything we can really do with that at this point we know that 4 is going to equal the value of H since we have 4x squared and HX squared so we also know that 4 would equal the value of H from here this is the point where it really becomes important to pay attention to the fact that h k and a j have to be integers because they have to be integers I can then state that uh K must equal negative 45 over J if we see this that negative 45 over J then we know that that's an integer if we also see that J is equal to negative 45 over K we know that would have to be an integer as well and then obviously if we see you know the value of just H so let's go and run through these we have option A B over H obviously we don't have the value of B so that's not going to be our answer uh at least you know not at this point B uh b b over K once again not really looking for the value of B uh down here 45 um over H we know H is 4 45 over 4 isn't an integer um it is a decimal but it's not an integer and then we have C 45 over K which we know would actually have to be our correct answer here and I'll go and explain why right like I said we have J is equal to negative 5 45 over K we know that J is an integer just because there's a negative doesn't mean that it's not an integer right so our answer there would have to be answer Choice D because this ultimately has to be negative and then 45 over K has to be an integer keep in mind that if we have negative 3 for example that's an integer as well so our answer there would be D and that's how you solve this really difficult SAT Math problem here's how to efficiently solve this difficult SAT Math problem question 24 say in the given system of equations a is a constant the graph of the equations again in the given system intersect at exactly one point anytime you're told that a system intersects at exactly one point or exactly two points or that it never intersects pay close attention to that and the X Y plane what is the value of x always pay close attention to what you're asked to answer with as well in this case it is the value of x okay so that means we might not actually have to solve for a okay we just need to solve for the value of x all right so immediately what jumps out to me is I have y equals y equals so I'm going to go ahead and substitute this 3x plus a over here for y and then I'm going to work to set it all equal to zero and the reason why I want to set it all equal to zero is because I want to end up using B squared minus 4AC is equal to zero because if this value is equal to zero then that means that we have one real solution or in other words we intersect at exactly one point so that's ultimately what I'm trying to get in terms of the form so I want to set it equal to zero to do that I'm going to have 2x squared and then I have this minus 21x but I also have to subtract 3x from both sides so once I do that that's going to end up leaving me with negative 24x and then I have plus 64 but I also have to subtract a from both sides so that's going to give me plus 64 minus a now what I'm going to do here is I just want to deal with a c value I don't actually want to deal with 64 minus a so I'm just going to go plus C and then I'll stay up here that c is equal to 64 minus a all right from here we know that our B value is negative 24 so we'll have negative 24 squared minus 4 times our a value which is 2 times C and we know that's all has to equal 0 for there would only be one real solution now negative 24 squared is going to give us 576 when we put it in our calculator so we have 576 minus 8 C is equal to 0. we can rewrite this as 576 is equal to 8C by adding 8C to both sides which therefore means that c is going to have to equal that 576 divided by 8 which is going to equal a value of 72. okay so now I'm going to go ahead and erase the value of C here and I'm going to put in 72 we'll put in that equals 0 as well and then I'm going to erase everything down here okay so from here we're going to look to factor out what we can and solve for the value of x so I see that 2 is a common factor in all these so I'll factor out a 2. I'll be left with x squared minus 12x plus 36 is equal to 0. from there I'm looking for numbers that multiply to get 36 that will have a sum of negative 12. so from there I'm looking at 6 as the value for that so that would be x minus 6 times x minus 6. okay now we factored this out we see that X will have to equal 6. so our value for the value of x then would be 6 so our answer there would be C here's something you need to know about isosceles right triangles for the SAT question 25 States an isosceles right triangle has a hypotenuse of length 58 inches what is the perimeter in inches of this triangle all right so let's go ahead and write this out anytime you're given a triangle or a circle question on the SAT it's usually helpful to write it out so we're given our hypotenuse now keep in mind that since this is an isosceles right triangle we know that it's a 45-45 90 triangle which has this special property where the side length x will be the same side length and size and then the value of the hypotenuse would be x times the square root of 2. now this is important because when we have our given hypotenuse of 58 inches I would elect to rewrite this actually as and I'll just write that this is equal to 29 times root 2 times root 2. and you'll see why I'm writing that in a second here so times root 2 times root 2. now the reason I'm choosing to write it like that is because then we know that our values down here would be 29 times root 2. since they just don't have that extra root to add on to them and since they're isosceles they're going to be the same now from here we want to find the perimeter so the perimeter is pretty easy to do we're just going to add in all these side lengths we'd have 29 times root 2 and we'd actually multiply that by two since we'd be adding it to itself so it's the same as multiplying by 2 which is ultimately I'll go over what that will equal out to in a second but we also have to add on to that then our variable hypotenuse which we know is 58. okay so from here we have 29 root 2 times 2 that's going to end up leaving us with 58 root 2 plus 58 as a value of our perimeter which we see is going to be equal to answer Choice C so c will be correct answer there here's how to solve one of the most difficult questions on the SAT question 26 states in the XY plan a parabola with vertex 9 and negative 14 intersects the x-axis at two points if the equation of the parabola is written in the form y equals ax squared plus BX plus C where a B and C are constants which of the following could be the value of a plus b plus C now keep in mind we always want to pay attention to what we're asked to answer with in this case that is the value of a plus b plus C so we need to know the value of all three values so first thing we know is our vertex so if we're given our vertex in a parabola question like this we're probably looking to use vertex form now vertex form would be Y is equal to a times x minus H squared plus K where our vertex is where vertex is h k all right well we know our vertex let's go ahead and substitute in so we're going to have y is equal to a x minus 9 squared minus 14. okay from here we want to get these values a b and c which means we need to get into the form of ax squared plus BX plus C so to do that let's go ahead and expand this X minus 9 squared so we're going to get Y is equal to a times x squared minus 18 minus 18x and then we would have plus 81 minus 14 or that'd be plus 81 um yeah plus 81 minus 14. all right so now from here we need to distribute this a so we got to distribute the a to the x squared to that negative 18x and to that plus 81. so I'm going to go ahead and raise what we have here so we're going to have y is equal to ax squared minus 18 a X Plus 81 a minus 14. all right so now from here what I'm going to do is get the values of a b and c so let's start with a so we know that a is going to equal just a it is what it is now B is going to equal this value of negative 18 ax or negative 18a since that that X is included right here so negative 18a our C value is going to equal 81 a minus 14. so minus 14. all right so from here I want to sum these all together so a plus b plus C is going to equal a minus a plus 80 1A minus 14. now from here what we're going to look to do is go ahead and combine these a terms so what we'll have then is a minus 18a which will give us negative 17A negative 17A plus 81 a will give us positive 64a so we'll have 64a minus 14. now what I'm immediately noticing and that I want to look at is we have to keep in mind that this intersects the x-axis at two points now because we're intersecting the x-axis at two points but we know our vertex has a y coordinate that's below that x-axis we know that we must open up so our Parabola must look like this like a smiley face you must open up across that x-axis well that means that the a value must be positive so if our a value must be positive then we're going to have 64 times some positive number let's just pick 0.001 just as an example here okay and then we have to subtract 14 from it so we know that we can't ever get below negative four we can't even reach negative 14 right because this is going to be some positive number we're subtracting that negative 14. so we can't get to negative 14 we can't get to negative 19 and we can't get to negative 23 okay we would have to have that value be negative 12. that's the only possible value so our answer has to be D here's something you need to watch out for on the SAT question 27 States function f is defined by f of x equals negative a to the power of X plus b where A and B are constants and the X Y plane the graph of y equals f of x minus 15 is a y-intercept of 0 negative 99 over 7 or the product of or the product of A and B is 65 over 7. what is the value of a so a couple things you need to look out for here number one what are you asked to answer with in this case it's the value of a so that's the only thing we should really focus on solving unless there's something we need to solve before in order to get to the value of a next thing we are given that the product of A and B is 65 over 7. so let's go and write that a times B is equal to 65 over 7. now the main thing you need to watch out for in order to avoid getting stuck on this question or getting it wrong is the fact that we have y is equal to f of x minus 15. if you ever have a transformation like this on the SAT you really need to pay attention to it because if you don't you're probably gonna end up getting it wrong so in this case we have f of x is equal to negative a to the power of X plus b well we know that y then must equal a negative a to the power of X Plus B minus 15 okay and that's a really important distinction now that we have the value of y let's go ahead and substitute in our points so we have this point of X being 0 so that would equal then negative a to the power of 0 plus B minus 15 is going to equal that value of negative 99 over 7. so we have negative 99 over 7. now from here what we're going to look to do is solve for the value of B and then we can use B in order to get the value of a using this equation up here so to solve for b we can go ahead and write this and I'll try to write out a little bit bigger up top so as you can see we had a to the power of 0 and any number to the power of zero is one so that leaves us with this negative 1 plus B minus 15. now from here what I'll do is I'll go ahead and add one or add 7 over 7 to both sides now once I add 7 over 7 to both sides I'm left with negative 92 over 7 is equal to B minus 15. I'll do the same thing with that 15. so I'll add 15 times 7 over 7 to both sides if I put my calculator here 15 times 7 that's going to give me 105 okay so if I add 105 over 7 to both sides that's going to end up leaving me with B is equal to 13 over 7. all right so now that I have my value of B I can go back and I can solve for the value of a now how can we do that well if we go ahead and divide both sides by B here we would get a is equal to 65 over 7 times B okay well we know that the value of B is 13 over 7. so if we go ahead and plug that in for the value of B 13 over 7. we see that these 7s are going to cancel and we're going to be left with 65 divided by 13 which is going to equal 5. so our answer for the value of a which we're asked to answer with always check that is 5. here's how to quickly interpret this line graph question one States the line graph shows the estimated number of chipmunks in a state park on April 1 of each year from 1989 to 1999. now y-axis we see we have the estimated number of chipmunks XX we have year based on the line graph and which year was the estimated number of chipmunks in the state park the greatest what we see that the greatest or maximum point is right here which we see is going to be in the year 1994 so our answer is B here's a tip for unit conversion questions on the SAT question two states a fish swimming distance of 5104 yards how far do the fish swim in miles what you're going to do is you're going to take your number of yards 5104 and we'll use y to represent yards here we want to multiply in a way that gets rid of the Yards we want our yards on the bottoms we want that 1760 yards per one mile with yards in the bottom now as you can see our yards will cancel and we left with our distance in miles which when we put this into our calculator will give us 2.9 miles and as you can see our only unit that is left is Miles here's how to solve this sat math question in under 60 seconds question three states which expression is equivalent to 12x cubed minus 5 x cubed key thing here since you have X cubed and X cubed you just do that 12 minus 5 that's going to leave you with seven X cubed and your answer there's going to be C here's a tip for the SAT Math section question four states what is the solution XY to the given system of equations if you see that you have 5y equals x you can go ahead and take this X and you can substitute in that 5 5y this will help you solve y and afterwards you can solve x so it's going to do that 5y plus y will give us 6y is equal to 18 from there we divide both sides by 6 and we'll see that Y is equal to 3. now at this point we have to go ahead and take we can actually just take a look at our answer choices and see that our answer has to be a and the reason our answer has to be a is that's the only answer Choice with 3 as our y value so as you can see you don't always have to solve for both X and Y sometimes you can solve for just one of the variables here's how to quickly solve this sat math question question five states the point eight two in the X Y plane is the solution to which of the following systems inequalities well we have our exponent of eight and our y point of two so let's go ahead and check well if we plug in 8 for X we have 8 is greater than zero that's true now is 2 greater than zero that's also true so our answer has to be a here's something you need to know about absolute value for the SAT Math section if you take a look at question six it says what is one possible solution to the given equation well here we have the absolute value of x minus five is equal to 10. so the way that we're going to set this up is first by doing x minus 5 is equal to 10. when we do that we see if we add 5 to both sides we'll get X is equal to 15. so one of the possible answers here is 15 but there is another possible answer when you have absolute value like you do here another way or your other answer here would be negative x minus 5 is equal to 10. from there you distribute that negative sign to your X and also to your minus five that's going to end up giving you negative X plus 5 is equal to 10 you would subtract 5 from both sides and you'd get Negative X is equal to 5 and then from there you would get that X is equal to negative 5. so negative 5 could be your the other answer to this question here's how to efficiently solve this ICT math question question 7 States the function gives the total number of people on a company retreat with X managers what is the total number of people on a company retreat with seven managers well in this case we just plug in 7 for X so we would have F of seven is going to equal seven times seven plus one seven times seven is 49 49 plus 1 is 50. so our answer there is 50. here's how to quickly solve this sat math question by plugging and chugging question eight states which table gives three values X and their corresponding values h x for the given function all right so the first thing I'm going to notice is that all of these have these X's of one two and three so it's gonna be pretty easy to solve this I'll start by plugging in one for X if I put in 1 for X I have 1 squared minus 3 I know 1 squared minus 3 is going to give me a value of negative 2. therefore I can get rid of a b is still a possible answer because that is true C I can get rid of and then D is also possible answer because that's true next I'm going to do is eventually you're going to jump 2 3 just because I know it's really easy 3 squared minus 3 well 3 squared is 9 9 minus 3 would be six so I see that I have 3 and 6 which is true if I look here 3 and 3 isn't true and 3 and 3 isn't true my answer there's got to be answer Choice B and the reason just to point this out really quickly the reason I chose 3 is because I saw that 2 and 1 here are both the same okay because two and one were the same and B and D and those were the two that I had it narrowed down to there was no point in checking them I want them ahead and moved on to using the x value of three here's something you need to know about when the exponent is zero for the SAT question 9 States the function f is defined by f of x is equal to 270 times 0.1 to the power of X what is the value of f of zero well if we plug in 0 for X we have 0.1 raised to the power of zero now 0.1 raised to the power of 0 is like any number raised to the power of zero it's going to equal one therefore you're going to end up with 270 times 1 which is just going to equal 270 and your answer will be answer Choice D here's something you need to know about margin of error for the SAT question 10 states to estimate the proportion of a population that has a certain characteristic a random sample was selected from the population based on the sample it is estimated the proportion of the population that has the characteristic is 0.49 with an estimated margin of the error of 0.04 based on the estimate and the margin of error which of the following is the most appropriate conclusion about the proportion of the population as the characteristics a couple things I want to point out here number one we are taking a random sample it's very important we take a random sample because if we don't we can't apply it to the general population it has to be a random sample in this case it is so that's perfect now the next thing we need to know is that the estimated proportion of the population that has a characteristic is 0.9 with an Associated margin of error of 0.04 so what that means is answer Choice a it's plausible that the proportion is between 0.49 and 0.53 and you get that by adding 0.04 to 0.49 and subtracting 0.04 to 0 0.49 so our answer there has to be answer Choice a here's a tip to help you move faster on the SAT if you encounter a word problem you see it's more than three lines try to write out the equation or system of equations or system inequalities that comes up in the question as you read it on your first read through that way you don't have to do a second read through when you're done reading the question so let's go ahead and Show an example a moving truck can tow a trailer if the combined weight of the trailer and boxes is no more than 4 600 pounds well that means that 4 600 pounds must be greater than or equal to the combined weight of the trailer in the boxes now what is the maximum number of boxes this truck can tow in a trailer with a weight of 500 pounds if each box weighs 120 pounds so we'll use B for the number of boxes okay now we have to keep in mind here that we just got to solve for b so we'll just subtract 500 from both sides that's going to end up leaving us with 4100 must be greater than or equal to 120 B we'll divide both sides by b or I'm sorry divide both sides by 120 okay 4100 divided by 120 we can put that in our calculator when you put that in your calculator you're going to get 34 repeating okay now keep in mind that B has to be less than that so the maximum number of boxes would be 34. so our answer is a this sat math question is a great example of when it can be a great idea to use Desmos on the digital sat question 12 States what is the positive solution of the given equation well what I would do here is I would take this instead of equal to zero so I'd have negative 4x squared minus 7x plus 36 is equal to zero now from here what I'm going to do is I'm just going to go ahead and put it in Desmos and then I'm going to solve for what the positive solution is by looking at where it equals zeros or in other words where it crosses the x axis now keep in mind we need the positive solution so the positive solution here we see is 2.25 or in other words 9 over 4. so our answer there is going to be answer Choice B here's how to quickly answer this probability question on the SAT question 13 states the table summarizes the distribution of color and shape for 100 tiles of equal area if one of these tiles is selected at random that's a key component here what is the probability of selecting a red tile express your answer as a decimal or fraction not as a percent all right so ultimate here we are selecting a red tile we see the total number of red tiles is 30 and this is out of a total of 100 so it'll be 30 over 100 which will calculate out as a decimal of 0.3 so our answer is 0.3 here's something you need to know about parallel lines for the SAT Math section question 14 states with a given function at the graph of y equals f of x and the X Y plane is parallel to line J what is the slope of line J well since they're parallel they have to have the same slope which means that line J must have a slope of two here's how to quickly solve this sat math question that deals with proportions and differences question 15 states a proposal for a new library was included on Election ballot a radio show stated that three times as many people voted in favor of the proposal as people who voted against it so we have three times the number that voted against it which I'll represented as an a is equal to the number who voted in favor of it social media posts reported that 15 000 more people voted in favor of the bill or of The Proposal than voted against it okay so we also know that the number of people who voted in favor is equal to the number who voted against it plus fifteen thousand based on these data how many people voted against the proposal so in this case you want to solve for a what we already have what f is equal to so we can go and substitute in so we would have f is equal to a plus 15 000 so a plus 15 000. from here I'll go ahead and isolate a by dividing both sides by three so I'll have a is equal to a over three plus fifteen thousand over three which would simplify down to five thousand uh from there I'll go ahead and subtract a over three from both sides a minus I'll just write this out a minus a over 3 is going to equal out to two a so two thirds times a which is equal or which is equal to five thousand still next thing we'll do is we'll multiply both sides by three over two so we'll multiply both sides by three over two okay keep in mind that five thousand times three over two will give us 7 500 so we'll have 7 500 then is equal to a so our answer there is answer Choice a here's something you need to know about intersecting parallel lines on the SAT question 16 States and the figure lines M and N are parallel if x equals six K plus 13 and Y equals 8K minus 29 what is the value of Z so always pay attention to what you're asked to answer with in this case that is the value of Z well calculate the value of Z we need to know what this angle right here is now we know that angle has to be the same as the value of angle Y and the value of angle X so we can go ahead and just put y as the angle measure there as well so what we ultimately need to do is we need to get 180 which is the value of this semi-arc right and we'll take that or that you know that angle measure right 180 we have to then subtract angle Y and that'll give us angle Z so we need to solve for angle y first well in order to solve for angle y we know that X and Y are equal to each other so we'll set these two together and we'll solve for K and that'll give us the value of angle y so we have 6K plus 13 is equal to 8K minus 29. now from here I'll go ahead and subtract 6K from both sides once we do that we'll be left with 2K on our right side we'll go ahead and add 29 to both sides so we can start isolating K even further that's going to leave us with 42 is equal to 2K we'll divide both sides by 2 and we'll get that K is equal to 21. so okay we equal 21. from there we can go ahead and solve for the value of y all right so we have K is equal to 21 so we have then the value of we'll solve for x just because it looks like it's a bit easier so we'd have a value of x is going to equal 6 times 21 plus 13. okay 6 times 21 is going to give us 126 126 plus 13 is going to give us 139 then we have to do 180 minus 139 to get the value of a z and that's going to leave us with 41 so our answer there's C here's a trick to help you on the SAT Math section question 17 states in the given equation p is a constant the equation has no solution what is the value of p in this case s to answer the value P always pay attention to what you're asked to answer with and pay attention to key information for instance the equation has no solution well because it has no solution we know it's set equal to a constant we also know that we have X and a value that will ultimately be with X so because of this if there's any sort of number next to X then we know that we're going to have at least one solution so in order to be no Solutions we need to get this set equal to zero so ultimately what we need then is negative 3x plus 21 PX to equal zero well in order for that to be the case we can go ahead and add 3x to both sides we'll get that 3x has to equal 21 times P times x we see our X's will cancel and we'll be left then with 3 is equal to 221p in order to solve for p we'll divide both sides by 21 we'll see that P is equal to 3 over 21 which we know is equal to 1 over 7 so our answer has to be B here's a trick you should know about quadratics for the SAT question 18 states the function f is defined by the given equation for what value of x does f of x reach its minimum well if you have a quadratic like this your value of x at which f x will reach its minimum or maximum will be the midpoint between the two x-intercepts we see here our x-intercepts are going to be 10 and negative 13 and the way that we get those is by taking a look at x minus 10 setting that equal to zero adding 10 to both sides and then you see x equals 10. same thing that you'll do for X plus 13 you subtract 13 from both sides when it's set equal to 0 and get that x equals negative 13. now from here we're going to take our two x-intercepts we are going to sum them together and then we'll divide by 2 to get the midpoint of them and that'll be the value of x where f of x reaches its minimum in this case that's going to give us negative 3 over 2 so our answer will be D here's a tip that's extremely helpful for the SAT and that is the vertex form of a quadratic and I'm going to illustrate this with question 19. the question States the function f of x is equal to 1 9 x minus 7 squared plus 3 gives a metal balls height above the ground f of x in inches x seconds after it started a moving track where X must be between 10 and 0 which are both inclusive which of the following is the best interpretation of the vertex of the graph y equals f of x in the X Y plane this question is way easier if you understand what this vertex form is well vertex form at its simplest would be f of x is equal to a which can be some coefficient x minus h squared plus K where your vertex vertex is going to be H comma K so as we can see in this case our vertex would be at 7 for our x value okay so I'll write seven I'll just I'll just rewrite this okay our vertex would be at seven and three okay so that's going to be where our vertex is so understand vertex form is going to help you a lot now we have to keep in mind that 7 is not any sort of height 7 is just the time since it's that x value now the actual height here is three now the other thing we need to understand is whether this is a function that is opening up or down and you get that from the a value okay and our a value is positive which means that our graph is going to open up like a smiley face now because that's the case we know that that vertex will be a minimum it'll be a minimum point or a minimum height so if we look at our answers we have option A the metal Ball's minimum height was three inches above the ground we know that'll be the correct answer because we ultimately have that minimum height at seven three three being the number of inches it is above the ground here's how to quickly solve this City math question about triangles question 20 States in Triangle jkl cosine of K is equal to 24 over 51 and angle J is a right angle what is the value of the cosine of L well with triangles and circles questions I always recommend to go ahead and just draw them out because it's very helpful to visualize okay in this case we know that we have a right triangle we also know that angle J is going to be the right angle so we can go and put J here and then we'll put k l now we're asked for the value of cosine of L okay we know that the cosine is our adjacent over our hypotenuse all right so in order to calculate this we need to know the values of the hypotenuse and the adjacent one now we also know that the value of the cosine of K is 24 over 51. okay well if the cosine of K is 24 over 51 we know our hypotenuse is 51 we know that the adjacent side length is 24. so from here what we'll do is we'll do 51 squared minus 24 squared is equal to l j this line squared all right now what we can do then is we'll just take the square root of all this and the square root of LJ squared and that'll just give us the value of the length LJ okay so once we do that if we put that in our calculator that is going to give us a value of 45 for LJ so we have 45 here now now we're asked for the value of cosine of L so we can go ahead and have the adjacent which is 45 over our hypotenuse of 51. so 45 over 51 which can be simplified down if we divide both sides by 3 to 15 over 17 or you could also put it in your calculator and if you do that you're going to get a value of 0.2 okay so answers are here and here and you could also take the value of 0.8823 and that would be accepted as well by the SAT here's how to solve this sat math question that deals with an equation as no real solution in question 21 it says in the given equation B is a positive integer the equation has no real solution what is the greatest possible value of B so a couple key things here if you're told that an integer is positive or negative always pay attention to it next thing we know that we have no real solution so we're asked for what the greatest possible value of B is well you need to stand here is that for a quadratic to have no real solution the value of B squared minus 4AC to be less than zero so from here let's go ahead and get the values of B A and C and we'll plug them in so we know that our value of B here is just B okay it's just what's in front of the X so we have B squared then minus 4 times a we know our a value is going to be negative one so I'll just change this to a plus four and then I have multiplied by C our C value is negative six seven six all right from here we have 4 times negative six seven six and that'll give us a value of 2704 so we're going to have B squared then minus two thousand or I'm sorry yeah two thousand seven hundred and four okay we know that this has to be less than zero so from here I'll go ahead and add 2704 to both sides and I'll end up with B squared must be less than 2704 at this point we take the square root of both sides to isolate B so once we do that we're going to be left with B must be less than 52 too now keep in mind your answer for the greatest possible value of B is not 52 because we have to have it B value that is less than 52 and it has to be a positive integer so the next highest positive integer that is less than 52 is 51 okay so the maximum possible value of B is 51. here's how to solve this tricky sat math question question 22 States if a new graph of three linear equations is created using the system of equations shown and the equation x plus 4y equals 16. how many solutions X Y will the resulting system of three equations have well let's start by going ahead and plotting this third equation so we see we have X plus 4y equals negative 16. now I'm going to convert that into slope intercept form so it's easier to graph so what I'll do is I'll start by subtracting X from both sides that'll give me 4y equals negative x minus 16 from there I'll divide both sides by 4 and that's going to give me Y is equal to Negative X over 4 or in other words negative one quarter X and then I have that much minus 16 over 4 which is equal to minus 4. so I'll go ahead and plot that on this graph okay so I have this minus 4 and I'm going up by about a quarter per so I'll just go ahead and go over four and then I go up by one all right and if I go ahead and do that I'll just kind of draw a rough line because we don't need it to be perfect and as we can see if we extend these lines okay here's where our Crossings will be now as you can see there's not any single point where all three of these lines intersect now there are points where two of the lines intersect but there is no point where all three intersect at one time so the answer here is zero many people would be tempted to pick something like one or maybe two but the correct answer is zero because it's about where all three of these lines intersect at a single point and that never happens here's something you need to know about the formula for exponential growth on the SAT question 23 States the function f gives the value in dollars of a certain piece of equipment after X months of use if the value of the equipment decreases each year by P percent of its value the previous year what is the value of P now keep in mind we're talking about how much it decreases each year now what you want to take a look at is your growth factor here now your growth factor is 0.6 4 which means we are decreasing in value by 1 minus 0.64 which is 0.36 or in other words 36 percent so our answer here is C and one thing I do want to point out here is that many people might be tempted to take a look at this X over 12 and think that that changes something here but it doesn't okay you're ultimately decreasing each year still by that 0 or by 0.36 right since we have this growth rate of 0.64 which means we are decreasing by 36 percent every year here's something you need to know about medians and ranges for the SAT question 24 States The Dot Plot represents 15 values in data a data set B is created by adding 56 to each of the values in data set a which of the following correctly compares the medians and ranges of data sets A and B well we know that if we add 56 to each of the values in data set a we ultimately are just going to change these values but we're not actually going to change the range or in other words the distance from the minimum to the maximum we are just going to change what the minimum is and what the maxim is but not what the range is or the difference between the two so our range will stay the same so we can go ahead and write that range is going to stay the same I'll just represent that as range is equal now the next thing we need to look at is the medians well obviously if we're adding 56 to each of these values they will ultimately change for example 22 plus 56 would then become 78 and then we'd have 79 and then right here we'd have 80 81 and 82. so ultimately obviously our median then is going to increase significantly okay so our median will increase but our range will stay the same okay so option A we have the median of data set B is equal to the median we know that's not true option b we have the median of data set B is equal we know that's not true option C the median of data set B is greater than the medium data set a we know that's true and the range of data set B is equal to the range data data set a we know that is true so our answer is C here's why you need to know the equation for circle on the SAT question 25 States the equation x squared plus y minus 1 squared equals 49 represents Circle A Circle B is obtained by shifting Circle A down two units in the X Y plane which of the following equations represents Circle B so first I want to show you what the circle equation is the circle equation would be something like is the circle equation is this x minus H squared plus y minus K squared is equal to r squared where R is your radius K is the center y value and H is the center x value so in this case your Center would be h k all right now that I've shown you the circle equation let's go and talk about how you can quickly answer this as long as you know the circle equation well all you're doing here is you're just shifting Circle A Down by two units so you're not affecting you're not affecting the x value of the center and you're not affecting the radius so those need to stay the same and just on that you can get rid of a and you can get rid of C now as you can see the next difference is going to be obviously the K value now you have to understand that if you're Shifting the circle down by two units you need to know that that minus K means your Center is at K so if you need your Center to move down then you have to ultimately add two okay because if you want to move that Center down two units you have to add two because there's that minus sign in front of K all right so then what you do is you take this minus this one you add 2 and you get y plus 1 so your answer has to be answer Choice D here's how to solve this very difficult sat math question about surface area question 26 states that two identical rectangular prisms each have a height of 90 centimeters the base of each prism is a square and the surface area of each prism is K centimeters squared and the prisms are glued together along a square base the resulting prism has a surface area of 92 over 47 K centimeters squared what is the side length in centimeters of each square base all right well we know that the surface area of each prism and keep in mind it says there are identical rectangular prisms we know the surface area of each is going to be K centimeters squared so we know the total surface area if they were kept separate would be 2K centimeters squared okay we also know that when they are glued together then the total surface area of everything then becomes 92 over 47k which is obviously less than 2K right so the difference there is what we want to find so the 2K minus 92 over 47 okay now we know that that's ultimately going to equal 2 times the area of each of those squares so that would be 2 times s squared where s is going to represent our side length of the square which is what we're going to solve for here all right so let's go ahead and get a value for K here because we're going to need a value of K to then plug into another equation so as far as what that other equation would be let's go and find what that is so we know that we have a height of 90 centimeters on the rectangular prisms and we know that the side length is going to be S we'll represent that as s so we then know that the surface area for the individual rectangular prism will then be represented as 90 which is the height times our side length which we're going to represent as s so in other words we can write that as 90s times 4 since we have four of those side lengths for each part of that rectangle okay so 4 times 90s and then we would have that much plus 2s squared okay because we also have to account for the square on the top and the bottom of the rectangular prism and that's all going to equal K all right so if we can find the value of K then we can substitute in so let's go ahead and do that 2K minus 92 over 47k will leave us with 2 over 47k and then from there we need to solve for K so we'll multiply both sides by 47 over 2. now once we do that we'll obviously end up with k then is going to equal we'll have these two cancel out so we'll have 47s squared okay so let's write that neatly K is equal to 47 s squared all right now let's go ahead and substitute in for K so we have 47 s squared now I'm going to scroll down so we can continue writing out this question so from here 4 times 90s will give us 360 s we have this plus 2s squared we have this 47s squared what I'm actually going to do here is I'm going to start moving everything over to this right side so we can get a set equal to zero now the reason that I want to move everything over to the right side is that I have a positive coefficient in front of s squared just because that generally makes questions easier so let's go ahead and do that by subtracting 2s squared from both sides and subtracting 360 s from both sides once we've done that we see that 0 is going to equal 45 s squared minus 360 s from here we can factor out 5s from both that 45 s squared and that negative 360s so once we do that we'll have 0 is equal to 5S times 9 s minus it looks like that's going to be 360 divided by 5 will give us 72. so from here we'll go ahead and set this equal to zero again so now we're going to have 0 is equal to 9s minus 72 from here we'd add 72 to both sides once we do that we're going to have 72 is equal to 9s we divide both sides by 9. 72 over 9 is going to leave us with 8 and we see that 8 will be our value for S which we see is going to be answer Choice B so B would be our correct answer here for question number 26. here's how to answer this difficult sat math question about percentages the question States 210 is p percent greater than 30. what is the value of P well we know that we start with the value of 30 then we want to multiply by 1 because if we don't increase at all we still need to multiply by 1 plus the percent that we are going to increase by so that would be p over 100. now we know that we ultimately need to equal that to 110. from here we distribute our 30 which is our initial value to 1 plus however much greater we are increasing whatever that percentage is so that will give us 30 plus 30 over 100 which can be represented as 0.3 times P that's all going to equal 210. from here we subtract 30 from both sides and we're gonna get that 0.3 3p is equal to 180 from there we'll divide both sides by 0.3 180 divided by 0.3 is going to leave us with 600 for the value of P so the value of p is going to be 600 now one thing that you really need to pay attention here is the wording okay the wording is that it is p percent greater than 30. that's really important because you don't know if sometimes they'll say hey P is uh it's p percent greater than 30 sometimes they'll say it's p percent of 30 sometimes they will say uh it's p percent less than 30 so always pay close attention to the wording with percentages questions because if you don't pay attention to the wording or percentages questions you're going to get it wrong it's really really important if you enjoyed this video please like subscribe and consider sending a super thanks to help support my channel additionally please drop a comment below letting me know what videos you want me to make in the future and if you are looking for additional Educational Services that it offer please check out my website Hayden roadie.com