what's going on Bessie this is your comprehensive all-in-one everything that you need to know for the ait's version 7 math portion of the exam let's get started so let's start by taking a closer look at breaking down a numerator and denominator so the numerator is the number that is above the fraction line and represents how many parts of the hole that you have so here we have an example of three fours the top number three is our numerator the denominator is the bottom number below the fraction line and it represents how many equal parts of the hole are being divided so in this particular case our denominator would be four to make it easy the denominator helps us understand the total parts that we're considering while the numerator tells us how many of those parts that we're actually focusing on to understand fractions better let's visualize a pizza often served in eight uniform slices to mathematically model this we can represent the whole pizza as the number one and divide it into eight equal parts indicating the division of those parts this means that each piece represents 1/8 of the entire pizza if you consume two slices that equates to 28 now let's imagine a scenario where the pizza is sliced into quarters instead of es in this Arrangement each piece accounts for 1/4 of the pizza interestingly consuming 1/4 of a pizza is equivalent to eating 28 because these fractions represent the same quantity if you divide the numerator and the denominator by two 28 would be equivalent to 1/4 this is a demonstration of equivalent fractions illustrating that despite having different denominators the fractions can express the same value once they are simplified now that we have a better understanding of fractions let's focus on the concept of place values and understand their application when it comes to decimals as a quick refresher consider the number 1,234 it's a familiar territory where many of us recognize the rightmost digit before the decimal as being the ones place the four stands firmly in the ones place truly representing its face value moving one digit to the left we find ourselves in the 10's place where each value is 10 times greater than the one the three in this place value represents 30 as it occupies the 10's Place progressing further to the left another leap by 10 brings us to the hundred's place and in this place that digit is not just a two it also equals 200 and finishing out moving further left brings us to the thousand's place making the one in this place actually 1,000 now as promised we're going to deepen our exploration by introducing a decimal point to the right of the ones place the decimal serves as a critical role for deline ating where the one's Place lies so moving to the right of the decimal allows us to expand our numerical landscape indefinitely a combinating as many decimal places as needed each with its own significance so let's consider this when we're transitioning from the hundreds to the 10's place we're ultimately dividing by 10 moving from the T to the one place we're also dividing by 10 so what name would we give the position immediately to the right of the decimal what result results from dividing 1 by 10 you end up with 1 over 10 or 1/10 leading us to understand this next position as the Ten's place if we move one position further to the right from the T's Place what do we call this new position it results from dividing a tenth by 10 effectively giving us 1/10th of a tenth which leads us to identify this place as our hundreds place pushing even further to the right we advance one more space what designation does this position hold well dividing aund by 10 is considered as 1/10th of 100 bringing us to 1,000 place so in this example instead of us having just the number 1,234 let's expand it to 1, 1234567 what does these additional digits signify well the digit five here doesn't simply stand for five it symbolizes 510 moving on to digit 6 it embodies 6 hunds to make it Crystal Clear 6 hunds can also be represented by placing a six in a hundred's place which is two positions to the right of the decimal lastly what does seven signify it stands for 7,000 to write this in decimal form you would place the seven in the thousand's place which is three positions to the right of the decimal so zooming out the concept of place values allows us to extend indefinitely to the right of the decimal enabling us to represent mathematical numbers with more Precision next let's talk about percentages so percentages are common parts of everyday life we encounter them in numerous situations whether it's a 30% discount during a sale leaving a 20% tip at a restaurant or hearing that there's a 50% chance of rain in the forecast percentages are everywhere but what exactly do they represent the term percent derives from the Latin word Centum meaning 100 so essentially percent translates to per 100 or out of a 100 it serves as a method to express quantities by scaling a total of 100 and then describing a specific value as a portion of that 100 so for example scoring 90% on the ait's exam implies that in terms of ratio you have achieved a 90 out of 100 this could also mean 9 out of 10 questions correct 18 out of 20 or even 450 out of 500 all reflecting the same proportionate relationship to 90% to adjust larger numbers down to 100 we apply the same operation to corresponding smaller numbers so for instance to scale the denominator of 10 up to 100 we simply multiply that by 10 similarly we would multiply nine the numerator also by 10 as well this would give give us 90 over 100 thus percentages function as ratios representing a particular type of ratio that benchmarks numbers against 100 while seemingly arbitrary that Benchmark of 100 provides a convenient standard for comparison making it easier to understand and compare different quantities viewing percentages as basic fractions offers a clear perspective for instance 80% translates to 80 out of 100 as we've learned simplifying fractions involves dividing the numerator and denominator by a common factor here dividing each side of the fraction by 20 we arrive at 4 over5 illustrating that 80% is equivalent to 4 fths exploring another example 50% is the same as 50 out of 100 reducing the fraction by dividing both numbers by 50 yields 1/2 logically aligning with the fact that 50% represents half of 100 for smaller percentages like 3% represented by 3 out of 100 further simplification is impossible indicating that some percentages like 3% are already in their simplest fractional form converting between percentages fractions and decimals should be a pretty straightforward process when using this method Let's test our understanding to ensure that we have fully grasped the concept we've explored how fractions decimals and percentages can each represent the same numerical value highlighting the flexibility of numerical representation now how do we convert between them I've included a memory trick that is actually found in my study guide to help you bridge this concept of converting between fractions decimals and percentages so let's take a closer look at the conversions going either way so if we start out with a fraction and we want to make it into a decimal all we have to do is divide the numerator by the denominator 34s would give give us 0.75 now if we had to convert between a decimal to a percentage we would simply multiply by 100 and move the decimal to the right two places the same thing if we wanted to go the other way if we wanted to go from a percentage to a decimal we would simply divide by 100 and move that decimal place to the left two places and if we had to take that decimal and convert it to a fraction we would divide by the place value we know that 0.3 is an in our Ten's place so this would give us 3/10 let's consider another example of 30% converting this into a fraction is pretty straightforward since we understand that percentages are based on 100 thus 30% translates to 30 out of 100 to simplify this fraction both the numerator and the denominator can be divided by 10 here's a quick trick though when you're taking your test you can simply just remove the trailing zeros to help you get your answer when you have fractions that are like this so 30 out of 10 is going to equal 3/10 in its simplest form now to express this as a decimal all we need to do is identify the Ten's place for 310 this means that we're going to place the three in the Ten's position resulting in 0.3 this demonstrates three distinct methods for expressing the same numerical value showcasing the interchangeability between fractions decimals and percentages so our first practice question States convert the fraction 5/8 to a percentage so we know that we are starting with the fraction 5/8 and we need to convert it to a percentage so this is really simple we're going to divide 5 by 8 and that's going to give us 0.625 now as we said before we're going to multiply by 100 and move the decimal to the right two times so we would go one 2 this would give us 62.5 leading us to believe that our correct answer is C 62.5% what is 0.2 expressed as a fraction simplify the fraction so we know we're going to have to do a little extra math to come up with our answer so we start with 0.2 we know that for certain as our decimal and we need to convert to a fraction in order to do that we're going to divide by the place value so as we learned in decimals that number immediately to the right of the decimal place is our tth place so that means that we are going to convert 0.2 to 210 but again we have to simplify our fraction so in order to do so we need to find the most common number that can be divided in our numerator and our denominator that common number is going to be two so we're going to divide both sides of our fraction by two and that's going to give us 1 fths do we have an answer that's 1/s we do the correct answer for this equation is going to be B5 moving on to our last practice question for this section we have convert the percentage 25% to a decimal so we start off knowing that we have a percentage of 25% and we need to figure out the decimal we do this by dividing by 100 or simply moving the decimal to the left to two times so in this we know that our decimal would be behind the number 25 and we're going to move it to the left two times this is going to give us .25 but you cannot have a decimal without having a whole number in front of it so in this case we want to add a zero so it's going to equal 0.25 do we have that answer we absolutely do the correct answer is going to be a so let's start with order of operations we all understand how to perform addition subtraction multiplication division and so on now it's time to explore scenarios where these operations are going to be combined and applied simultaneously to establish protocol for executing multiple mathematical operations we use the proper sequence known as pemos This Acronym represents parentheses exponents multiplication division addition and subtraction guiding us through the sequence which these operations should be carried out moving from left to right as we work our way through initially we are going to address any calculations within parentheses next we're going to handle exponents and then after this we're going to look at multiplication and division now it's really important to understand that we are going to perform this by moving from left to right neither multiplication nor division takes presidence over the other so if our first operation is division we would perform that op operation first finally we conclude with addition and subtraction following those exact same rules neither addition nor subtraction are going to take precedence over the other so if our first operation is subtraction on the left we are going to perform that operation first let's take a look at an example of what this will look like on your te's so our first equation is four plus our parentheses 3 * 2 - 8 / 2 so the very first thing we want to do following PM dos is perform the operations within the parentheses we do this by multiplying 3 * 2 which is going to give us 6 next we're going to move on to exponents we don't have any so we can automatically skip this step then we're going to perform multiplication and division remember we're moving from left to right so the only multiplication and division that I can see is the 8 / 2 we're going to perform that operation and it's going to give us four and then lastly we're going to finish up with our addition and subtraction again moving left to right nothing takes presidence over anything else so in this particular case we have 4 + 6 that's going to give us 10 we're going to drop down our minus 4 and that is going to give us 10 - 4 is equal to 6 so the correct answer for this equation is going to be six let's take a look at another example before we move on to our practice questions so we have 15 + parentheses 3 + 2 raed to^ 2 - 9 * 6 + 2 raed to the^ 3 so it looks a little complicated hang in there with me we're going to get through this so first we're going to perform the operations with the parentheses first so 3 + 2 is the only operations we see with inside our parentheses and that is going to give us five moving down next we're going to calculate our exponents that's our powers and our square roots so we have two exponents here we have 5 ra^ 2 and 2 ra^ 3 5 ra^ 2 is going to give us 25 and 2 raed to the^ 3 is going to give us 8 now we're going to perform our multiplication and division moving from left to right so as we can see here we only have one multiplication and that is N9 * 6 so 99 * 6 is going to give us 54 and that'll end that portion of pen does lastly we are going to add and subtract again moving left to right starting from the left we have 15 + 25 that's going to give us 40 40 minus 54 is going to give us -14 and then we're going to add 8 and that is going to give us our correct answer of -6 let's start with our first practice question calculate the following operations we have 5 * parentheses 2 + 3 - 4 raed to^ 2 + 6 so we're going to start out with our pendos with our parentheses so in parentheses we have 2 + 3 that's going to give us 5 so our new equation is going to be 5 * 5 - 4 raed the^ 2 + 6 next we're going to move down to our exponents so we only have one exponent here right that's 4 ra to the^ of 2 so our new equation is going to be 5 * 5 - 16 that's our equation we just did + 6 now we're going to move on to multiplication and division again moving from left to right so if we take a look here we only have one multiplication that's 5 * 5 so that's going to give us 25 - 16 + 6 and in our last step we are going to do addition and subtraction again moving left to right so we have 5 - 16 is going to give us 9 so we're going to drop down our our plus six that we have left over up here and 9 + 6 is going to give us 15 so the correct answer for this equation is 15 and yes there it is our correct answer is C next let's delve into the concept of rational and irrational numbers so at its core a rational number is simply any number that can be expressed as a ratio of two integers in the denom minator cannot be zero right it's right in the name ratio rational it's going to be really hard for you to get forget that when you're taking your te's let's take for instance the number one it can be depicted as 1 over one or -2 over -2 or even 500 over 500 these examples all demonstrate various ways to represent the number one as a ratio of two integers and theoretically there's an infinite amount of ways that you can do so so long as the numerator and the denominator are the same similarly the number -7 can also be expressed as-7 over 1 7 over -1 or even -28 over4 the list of representations when it comes to -7 is also endless it's clear that -7 is a rational number because it fits the definition of being expressed as a ratio of two integers so what about numbers that aren't whole integers consider for example the number 3.75 how can we express this as a ratio of two integers one way we can do this is to look at 3.75 and convert it as a fraction that would give us 375 over 100 which is the same thing as 750 over 200 alternatively we can understand that 3.75 is 3 and 34s which converted into an improper fraction would give us 15 over 4 so how do we do that what we would do is we would multiply our three by our denominator of four that would give us 12 and then we're going to add our three that is in our num our numerator giving us 15 over 4 what happens when we encounter repeating decimals so consider one of the most well-known repeating decimals 0.33333 33 which continues indefinitely we signify this reputation by adding and placing a bar over the number of three indicating 0.3 is a repeating decimal there are countless other examples of repeating decimals but the key take away is that any decimal with a repeating sequence no matter how long can be represented as a fraction this is going to hold true even with sequences with millions of digits that repeat endlessly so you might be wondering at this point hey nurse Chun with everything that we've covered so far including integers finite non-re repeating decimals and repeating decimals what's left are there any numbers that don't fit into this category of rational numbers some of the most renowned numbers in the entire field of mathematics fall into this category and we call them irrational numbers these numbers cannot be Express as a simple Ratio or two integers I've highlighted a section of significant examples that you're probably going to see on your te's so take Pi for example the ratio of a circle circumference to its diameter is known as an irrational number its digits extend indefinitely without termination and without forming any repeating patterns similarly e which arises from concepts of continuously compounding interests in complex analysis never ends or repeats itself also making it an irrational number the square root of two is also irrational you might wonder if these examples are merely exceptions or perhaps most numbers are rational with only a few rare irrational cases that we're highlighting here however it's crucial to recognize that while these numbers might appear to be exotic due to their unique properties and applications they are not as rare as you might think consider this perspective I mentioned the square root of two as an example but if you calculate the square root of any number that isn't a perfect square you're going to find yourself with an irrational number moreover when you combine any irrational number with a rational number whether it's an addition or multiplication it's going to result in some kind of irrational number this demonstrates just how abundant irrational numbers actually are when we're looking at the mathematical landscape let's take a look at our first example which of the following numbers is irrational so let's break each one of these down so we have 8 / 2 is going to give us 4 we have the square root of 16 also gives us four so we can automatically eliminate A and B as being the correct answer because remember we're looking for an irrational number next we have the square < TK of 15 oh look at that we have a very long number right we have 3.87 298 so on and so forth but let's finish out we have 2.75 2.75 is a rational number so out of all of the options that we have available to us C is going to be the most correct number because there's a lot of repeating deci deal but they don't follow a pattern our next question states which expression results in a irrational number so again we're looking for irrational so starting off with our handy dandy calculator we have 25 / 5 is going to give us 5 so we know that that is rational we can eliminate that 2 raed to the^ of 2 is 4 that's also again a rational number so we can eliminate that we have the square root of 49 the square root of 49 gives gives us seven so again we know that that is also rational we can eliminate that and our last number is the square root of three and again look at that we have a very long number we've got 1.73 250 so on and so on and so forth so out of all the options that we have available to us D is the most correct answer because remember we have a non-terminating decimal with no repeating patterns so when we see those kind of num numbers we know we have an irrational number now that we have a better understanding of rational and irrational numbers let's discuss how we're going to order and compare rational numbers we're going to sequence these six numbers from smallest to greatest with the smallest number positioned all the way to the left of our number line and the largest number all the way to the right I encourage you to pause this video and try to arrange them yourselves before we tackle it together let's proceed with how we're going to get the solution once you've given it a shot we'll utilize a number line to assist us with this task which I prepared for your convenience let's begin by examining each number one at a time so when it comes to your te's the best option is to plot one option at a time and convert any fraction that you have to decimals to make it easier to compare on the number line so let's examine this process so we start with the number-3 over 4 so when we do our conversion -3 over4 is going to give us ne0 0.75 so we're going to mark it on our graph right here next we have 0.5 0.5 is going to be more on the right side of our number line next we have -1.5 again looking at our number line it's going to be placed right in here 2 over three so again we're going to have to do a little math 2 / 3 is going to give us 0.6 and again we add our little bar on top of our six because it is indefinite inite six number pattern it's just 66 66 666 so on and so forth down the line so 2 over 3 is going to be right about here on our number line next we have - one2 so - one2 is going to give us 0.5 and we are going to put that right here on our number line and then our last number is 1.75 so 1.75 is going to be way over here to the right of our number line so we're taking a look at this we've already plotted it we can get our numbers from least to greatest so least is all the way over here on the left and greatest is all the way over here on the right so we're just going to move our way down so I have the number I have the answer down here for you already so we have 1.5 34 12 0.5 2 over 3 and 1.75 that is the correct answer for least to greatest with this numerical data set so we just learned about the number line method when it comes to ordering and comparing rational numbers but there's a lot of people that also use the stacking method so I did want to address it in this video so the question is asking order the following set of numbers from least to greatest as we know when we're looking at a number line when it comes to the negatives the farther away that we get from zero the more negative it becomes right so what we do in this method is we're going to stack all the numbers vertically and line them up based on their decimals so if you look here all of the decimals are lined up so that we can compare our numbers now the reason why this works so well is I want to draw your attention up here to the right of your screen so if we're looking at a set of numbers that are lined up by their decimal points it really helps us understand how each number is greater than or less than so looking at our two examples we have 5347 and 5582 so we have to figure out which one is great which one is least so we start with the first number to the left of our decimal so they're both five so we automatically those are equal to each other we can move on to the right so our next set of numbers we have three and we have five well is three greater than five no absolutely not right so we can say that 5.82 is greater than 5347 so it just helps kind of visualize a little bit better if this method works for you so let's take a look back at our example Le here so again whenever I'm doing calculations with comparing and ordering rational numbers I always want to start with our negative num so I'm only going to focus on our 0.75 and our - 1.5 and our .5 so let's take a look so we start with our first numbers we have zero 1 and zero so as we discussed as we move further away from zero the more negative the number becomes and it also becomes the least So based on this we have have -1.5 so we know that our first number our least number out of this data set is going to be 1.5 so next we're going to move on to our next numbers because both zeros are equal we can automatically eliminate those so now we have seven and we have five of our negative numbers again the further we move away from the zero on the negative side of our number line the more least it becomes so seven is greater than five which means that 0.75 is going to be less than 0.5 so now we've moved through all of our negative numbers we can automatically eliminate those now let's take a look at our positive numbers so with our positive numbers again we're going to be looking at the first number so we have zero zero and one so one is obviously greater remember the further we move to the right of our number line the more greater the number becomes so we know that 1.75 is going to be our greatest number now we have to figure out which of these two remaining numbers are going to be greater than so we have five and we have six is five greater than six absolutely not so we can put 0.5 here and 0.6 here so out of our data set this is going to be our least to greatest all right our practice question States arrange the following rational numbers from least to greatest so as we know based on our number line as we move further to the left it becomes more least as we move further to the right it becomes more great so let's take a look at each individual number so our first number we have is -34 so we know -34 is equal to 0.75 so we can automatically put that right about here and I always just write the number on top so that way I know let's take a look at our next number we have 0.5 you can go ahead and put that there we've got 1.25 so 1.25 next we have 23 so as we know 2 / 3 is equal to 0.6 with a bar above it because again it goes indefinitely to six in the same pattern so 0.6 is just going to be Aidy over here to the right of our 0.5 we'll put 2 over three and then lastly we have the number 1.2 so our 1.2 it's going to go right about here 1.2 So based on the numbers we have have we know the correct answer has to be -1.25 -34 0.5 23 and 1.2 do we have any of these answers that make sense based on what we know yes we do the correct answer is going to be a something else the tease is going to test you on is comparing numbers using greater than less than and equal to when it comes to your rational numbers when we perform Solutions like inequalities we're going to start to see things like like less than or equal to greater than or equal to as well but in this particular cases we're only focusing on three of them less than greater than and equal to here's a quick tip that I want you to remember when you're taking your te's when it comes to less than or greater than and remembering which sign you need to use I want you to think of a very hungry alligator or Hungry Hungry Hippo or maybe even Pac-Man they're always going to want to eat the larger number because I live in Florida I'm going to use the alligator analogy so so the alligator is going to want to eat the greater number so we have three and we have five here as an example so is three greater than five absolutely not right three is less than five so our little Gator here is wanting to eat that number five because it's very very hungry so we're going to use the less than sign right his mouth is going towards the higher number so let's take a look at our numbers so we're comparing 0.75 to 34s so when we look at our number line we have 0.75 right about here and we divide three by four it's also going to give us 0.75 so in this case we can say that 0.75 is equal to 34s our next example we have -1 1110 and 25 so - 1 1110 is going to give us about - 1.1 and 25ths is going to give us about 2.2 so is our -1.1 greater than 2.2 absolutely not right so we can say that 1110 is less than 25s now looking at our last example we have 1.2 and 0.8 so 1.2 we're going to plot her about right here and 0.8 is going to go right about here so is 1.2 greater than 0.8 absolutely so we're going to go ahead and use the greater than sign as we know that 1.2 is greater than 0.8 so our question states which of the following is not true so we're looking for the one that's not true so we're going to break each one of these down so we have 34 is greater than 0.75 well if we convert 34s to a decimal that gives us 0.75 0.75 is 34 greater than 0.75 absolutely not right we know that this one is not true but let's go ahead and move through everything else so next we have -2 is greater than 0.6 so using our number line we know that - one2 when we convert it to a decimal is 0.5 and we have 0.6 so 0.5 would be here 0.6 would be right about here so is - 12 greater than 0.6 yes absolutely so we can automatically eliminate that answer so next we have 0.5 is greater than 1/3 so as we know we have 0.5 and then 1/3 is going to be 0.3 with a little line above it so is 0.5 greater than 1/3 yes absolutely so we can automatically eliminate that and then our last is 2.5 is greater than and 5/3 so when we convert 5/3 we get 1.66 so on and so forth down the line and we have 2.5 so 2.5 would be about right here 1.6 would be right here so is 2.5 greater than 5/3 yes absolutely so the correct answer is a 34s is not greater than 0.75 so on the at it it's going to be crucial that you understand some vocabulary terms when it comes to algebra here you can see we have an algebraic expression 15x + 5 there going to be four key terms that are consistently going to be used when you're taking your test it's going to be coefficient variable term and constant so let's begin with the term variable so variables are elements within an expression whose values are not fixed they represent letters of unknown quantities in our given example that we have here the one variable we have X represents the quantities that are not immediately known without needing additional information next up we have our constant so in our example the constant is the number that is represented by the number five it's labeled as a constant because it stands alone without any variable attached to it unlike variables who values might change depending on specific conditions or inputs a constant Remains the Same within the expression therefore any number that appears by itself not accompanied by a variable is considered a constant in algebraic expressions next up we have coefficient and a coefficient is a number that multiplies by a variable within the expression so for instance in our example our coefficient is actually 15 coefficients and variables are paired together in multiplication relationship ship and when they are written that number our coefficient always precedes the variable and finally let's talk about terms so in an algebraic expression a term refers to a component that is delineated by addition or subtraction symbols so for example in our expression 15x and 5 each represent distinct terms so essentially terms are the building blocks of an algebraic expression excluding the operators that is our addition and subtraction themselves although the concept of terms might not be the primary focus it's crucial to understand because it underpins the structure of algebraic expressions another important concept that you're going to need to know is inverse arithmetic operations so this principle revolves around the question what do we add subtract multiply or divide into a number for it to result in zero to illustrate this concept consider an example example where we're looking to find what can be added to a positive number in order to result in zero given that X is a positive number the additive inverse would Bex it's negative Counterpoint therefore if you add X tox you're going to get zero they essentially cancel each other out in our next example we're dealing with a negative value X to find what needs to be added to achieve zero we use that additive inverse concept meaning that we need to add the opposite Sox + positive X is going to equal zero demonstrating the inverse property of addition not too bad right now let's dive into inverse property of multiplication this principle is centered around identifying what number when multiplied by another is going to yield one essentially any number times its multiplicative inverse or reciprocal equals 1 so for our first example consider the letter B where B is a whole number and not a fraction it's actually quite helpful to represent B in its fractional form as B over one where any whole number can be expressed as a fraction over one to find the reciprocal we invert the fraction making the denominator the new numerator and the numerator the new denominator therefore the reciprocal of B is actually 1 over B to verify this let's multiply B in its fractional form as B over 1 by its reciprocal 1/ B when multiplying fractions we multiply the numerators together and the denominators together starting with our numerator we have B * 1 is going to give us B and with our denominators we have 1 * B is also going to give us B this is ultimately going to give us B over B which further simplifies into the number one we may also encounter a fraction of 1/ B in order to identify the reciprocal we simply flip the fraction resulting in B over 1 and just like we did in our previous example when we multiply our numerators and our denominators together we are going to simplify it to b or that's also going to equal one let's take a look at some examples for solving equations using one variable so when we're performing these operations we want to perform the opposite or inverse operation of whatever we are doing on one side of the VAR variable to the other side of our equation so our first example we have x - 6 = 13 so we want to isolate X in order to find out what that number is so what we're going to do is just like we did before we're going to do the inverse so we have a -6 we're going to add a positive 6 to both sides adding a positive 6 is going to give us x = 19 we just solved our first equation and our next example we have 3x + 9 = 0 so again we want to isolate X first so we do that by minusing 9 from both sides cuz here we had a positive 9 we want to make sure that we minus 9 in order to achieve zero once we do that we're going to get 3x is equal to -9 and now we need to again isolate that X so how are we going to do that we are going to go ahead and divide by three on both sides in order to get R of that three coefficient before our variable that is going to give us X is equal to3 our next example we have 15 = 5 m so again we want to isolate that variable that's our M so we're going to divide five by both sides and that is going to give us 3 is equal to M and our last example may seem a little bit tricky but actually it's really quite simple we have 11 is equal to W / 3 so again we want to isolate our variable we want to get that W on one side by itself in order to do that we're going to multiply each side by three because that is the inverse operation and that is going to give us 33 is equal to W not too bad right so next up we're going to solve proportions with one variable so you're going to see two fractions with an equal sign between them and you're going to have to solve for whatever the variable is whether X or another letter so in order to do this we're going to cross multiply fractions to obtain an equation with one variable so we do this by cross multiplication of our fractions our first problem is going to give us 10 x is equal to 60 so now we have an algebraic expression and we just need to solve for x so we do so by dividing each side by 10 and that is going to give us X is equal to 6 let's try another example so down here we have x 3 is = to 4 6 so again we are going to cross multiply our fractions and that is going to give us 6 x is = to 12 we have our algebraic expression so we're going to go ahead and we're going to divide each side by six to isolate X and that is going to give us our correct answer of X is equal to 2 for our next example we have 7 7 /x is = to 14 over 28 so again we're going to cross multiply our proportions and that is going to give us 14x is equal to 196 we're dealing with big numbers now right we are again going to isolate our X by dividing each side by 14 and that is going to give us our correct answer of X is equal to 14 hopefully this has helped bridging some gaps and understand how we cross multiply proportions so let's move on to our final example so we have 5x 5x is equal to 10 over 20 so we're going to go ahead and cross multiply this is going to give us 10 x is equal to 100 and of course we're going to divide each side by 10 in order to isolate X and that is going to give us our final example of X is equal to 10 so let's talk about estimation on the atit you're going to have to use your best judgment in regards to length weight and capacity when it comes to various items so starting with the estimation involving length you're going to be using metric system conversions it is a crucial skill that you're going to need to know for the atits as well as Healthcare so choosing the appropriate unit of measurement depends on the object that you are measuring if you haven't already done so I highly recommend that you go check out my video on metric conversions because it's going to help you break down the differences between each unit and how we establish each one but let's take a look at how the atits is going to test you when it comes to this so when it comes to length there's a couple different things that you will need to know you're going to need to know meter centimeter millimeter and kilometer so starting with the meter just to give you real life examples 1 meter tall is about the height of a doorknob so when you look at your doorknob that is about the height of 1 meter when we're talking about centimeter we're talking about the width of a paper clip that's approximately 1 cm when we're talking about millimeter that is approximately 1 millimeter is the diameter of a grain of salt it's really really really really tiny it's not the length of the grain of salt it's that diameter that cut through and then lastly with kilometer when we think of kilometer 1 kilometer is equal to an airport Runway so if you've ever been in an airplane you know that runways very long that is approximately 1 kilm next up we have weight so approximately when it comes to weight a gram is equal to the weight of a paper clip so that paper clip's not very heavy right so it's very very light so that's approximately 1 gram next up we have millgram so 1 millgram is essentially the same weight as a pinch of salt so not very big and then lastly kilogram so 1 kilogram is approximately the weight of one bag of rice and last up we have capacity so one liter if you look at those large water bottles even like this water bottle right this water bottle is approximately 1 liter then we have milliliter so this is huge in healthcare so 1 milliliter is just like a small medication drop or amount of liquid and then lastly we have a Kil so a kiler is approximately the entire capacity of a pool of water so let's take a look at some examples of how this will be tested on the teas so starting with question one what is the best essence for a length of a standard kitchen knife so we're talking about the length so is it 20 mm 200 CM 30 cm or 3 kilm so taking a look back at our examples we know that a standard kitchen knife is not going to be as small as 20 mm right that's that diameter of our grain of rice so we can automatically eliminate a and it's definitely not as long as 200 CM think about it 2 200 paper clips wids that's way too long so we can automatically eliminate that as well and then of course 3 kilometers that would be three airport runways so we definitely know that that is also not our correct answer so all the ones that we have available to us C 30 cm is going to be the most correct answer our next question is what is the most appropriate unit an estimate for the weight of a car is it A500 G B 1.5 kilg c500 kilg or D 15,000 MGR so in order to answer this question we have to know that cars are significantly heavier than small objects measured in grams and milligrams so we can automatically eliminate anything that has grams or milligrams from our question so now we're just left with kilograms so if you didn't already know one kilogram is equal to 2.2 lb that's actually a conversion that's going to be really important for you to know so if you think about it 1.5 kog is just a little over 3 lb so we can automatically eliminate that based on all of the answers that we have available to us c500 kg is going to be the most correct answer and our last question states which is the closest estimate for the total volume of a bathtub filled with water is it a 150 milliliters B 150 L C 15 Cen or D 1.5 kilers so again using our deduction of real world examples of how to measure objects we know that a bathtub with water has to contain way more than 150 milliliters so we can automatically eliminate that and of course 15 C isn't going to cut it so we can eliminate that so we're left with 150 L and 1.5 Kil well when you think about it 1.5 Kil is going to be excessively more than what we're going to be able to fill up a bathtub with remember 1 Kil is equal to the size of a swimming pool the amount of water that's in that swimming pool so we can automatically eliminate that so based out of the all of the answers that we have available to us B 150 L is going to be the correct answer so as you can see on your screen over here this is a little cheat sheet of everything that you're going to need to know in regards to how to translate words into algebraic expressions so let's go over those crucial Expressions that are going to be pivotable in solving algebraic word problems so when you come across the phrase four times a number you're going to automatically translate this into 4 * X where X is going to represent that unknown quantity that we have to figure out for expressions like four more than a number that is translated into x + 4 something to make sure you have careful attention to is when you see things like four less than a number it can be easy to interpret this as 4 - x but that's actually incorrect the correct translation is x - 4 despite that initial position of four in the sentence so be very careful when you come across word problem problems like this for a phrase such as four * the difference of two numbers it's understood that four is going to be multiplied by the difference between X and Y or it's going to be four parentheses x - y highlighting the importance of placing that xus y within parentheses and the four on the outside when you see terms like sum and N mentioned within our word problem we automatically associate that with the sum of two numbers so this can be written as x + y in contrast the difference between two numbers would be xus y if you encounter words like product of two numbers it's going to refer to the multiplication of X and Y and then lastly if you see words like is or equals that's just a fancy way of saying our equal sign these expressions are going to be fundamental when it comes to algebra and mastering that translation into mathematical terms to effectively solve your word problems let's take a look at some examples so our first practice question is if six more than a number is 14 what is the number so let's break this down starting from left working our way to the right so six more than a number is x + 6 so then we have the word is is is just of course just a fancy way of saying our equal sign and then then lastly we have 14 and of course 14 is going to be 14 so now we just need to put our problem together so we have x + 6 is = to 14 now we just need to solve for x so we do so by minusing six from both sides and that is going to give us X is equal to 8 our next example States 3 * a number equals 45 what is our number so again and we're going to break it down so 3 * a number is going to be 3x then we've got equals I mean it pretty much just tells us what it is right equals is equal to equal sign and then lastly 45 again 45 is just 45 so now we just need to put our equation together so we have 3x is equal to 45 we need to isolate our X in order to find the number so we're going to divide both sides by 3 and that is going to give us X is equal 25 all right now that we get the hang of this let's see if we can figure out these more complicated problems so we're going to start off with again solving our equation from left to right breaking it down so five times the difference of two numbers is 35 if one of the numbers is 10 what is the other number so again we start with with five times is the difference of two numbers so that is going to be 5 with parentheses x - y so we've got our first part we have the word is so that is our equal sign and then we have the word our the number 35 well it also tells us one additional thing right it says one number is equal to 10 so that gives us our new equation so we can say 5 parentheses x - 10 is equal to 35 now we just need to solve so in order to solve this we have to multiply our five by everything with inside our parentheses so we're going to multiply five by X and we're going to multiply five by 10 so we've got 5x - 50 is equal to 35 now we need to start isolating our X so we are going to add 50 to both sides that is going to give us 5x is = to 85 and now we are going to divide five by both sides in order to isolate our X and that is going to give us X is equal to 17 not too bad so our next question States the product of eight and the sum of a number in two equals 64 what is our number so let's break this down working left to right so the first thing thing we have is we have the word product so as we know product is going to be our X and it's going to be our y multiplied together so we know that we have the product of eight so that's the start of our equation and it's going to be the sum of a number in two so that's going to be X as our number plus the number two and we're going to close our parentheses sign next we have the word equals so of course equals is equal to equal and then last we have the number 64 so 64 is 64 so now we're going to do the same thing like we did before and we're going to multiply our eight by everything within our parentheses so we're going to multiply 8 by X and we're going to multiply 8 by 2 so that's going to give us 8x + 16 is equal to 64 now we need to start isolating our X right so we're going to minus 16 from both sides that's going to give us 8X is equal to 48 and then we are going to divide 8 by both sides to isolate our X so X is going to be equal to six which is the answer to our word problem so let's work on a little bit of a longer word problem so this word problem States a hospital uses 2.5 lers of sanitizing liquid across five examination rooms every day if a bottle of sanitizing liquid contains five 00 Mill how many bottles are used per day so first right off the bat I'm just going to eliminate the fact that we have across five examination rooms every day because has nothing to do with our problem so we're solely focusing on the fact that we have a use of 2.5 lers of sanitizing liquid and the fact that each bottle contains 500 milliliters so we have to figure out how many bottles are used per per day so the very first thing when we're looking at this problem is we have liters and we have milliliters so we're going to have to convert our liters to milliliters in order to solve this equation so again if you haven't watched the metric conversions highly recommend that you watch it so this makes sense so 2.5 L is equal to 2,500 milliliters so now we're just going to Plug and Play so we know that we have five 500 millit and that is going to be multiplied by B because we need to know how many bottles we need right and that is going to be equal to 2,500 milliliters this is our equation we have 500 milliliters per bottle that's represented by the B that we added into our equation and it's going to be equal to 2,500 M now all we have to do is isolate our B so we we do this by dividing both sides by 500 and that is going to give us B is equal to five bottles so they need a total of five bottles in order to make it through every day our next question States Sarah trains for a marathon by running 15 km every day if a marathon is 42.195 KM long how many days of training does it take Sarah to run at least the distance of a marathon so we have two numbers is here we know that she runs 15 kilm every day and then our second number is is we know that a marathon is 42.195 kilomet and we need to figure out how many days it's going to take Sarah to run this so we're just going to plug this into an equation so we're going to have 15 km over 42.195 km is equal to 1 day over X day so what I like to do when I'm doing word problems especially proportions and ratios is I like to keep the same terms on each side so on this side we know that these are the kilometers that she's going to run and on this side this is the time right how many days so this is like I had a kind of keep it simple and to keep it organized so as always we're going to cross multiply so we're going to do this by cross multiplying our numbers that's going to give us 15 x is equal to 42 2.95 and now we're just going to divide each side by 15 in order to isolate our X and that is going to give us 15 is equal to 2.83 days or if you were required to round up it's approximately 3 days so the correct answer is 2.81 3 or 3 days so next up let's talk about percent proportions where relationships between a part and its whole matches the Rel relationship between percent and 100 this allows us to create an equation or proportion to represent that relationship so the formula that we are going to use as you can see here is is over of is equal to percent over 100 with this formula in hand we can tackle a lot of problems focusing on percents let's take a look at our examples so looking at our first practice question we have 18 is what percent of 42 so we break this down we have our is that's 18 we're trying to figure out what a percent is because we have the words what percent and we have our of which is of 42 so now we're just going to plug in our numbers we have 18 over 42 is equal to what percent right so that's going to be our X over 100 now we're just going to cross multiply so we're going to multiply 42x is equal to 1,800 we need to isolate our X by dividing both sides by 42 and that is going to give us X is equal to 42.86% so moving on to our next question we have 20 is what percent of 45 we're going to use the same formula so we have our is that's 20 we're trying to figure out what percent that's our X and then lastly we have 45 so now we're just going to plug in our numbers so we have 20 over 45 is equal to what percent that's our X over 100 so now we're just going to cross multiply so we're going to multiply here we're going to multiply here that's going to give us 45x is equal to 2,000 then we're going to isolate our X by dividing both sides by 45 and that is going to give us X is equal to 44.44% and now with our last initial example we have what percent of 12 is 27 so this one's a little bit different right let's just break it down working left to right so we have what percent that's our X that's what we're looking for 12 so that's our of is 27 so now we just need to plug in our numbers we have 27 over 12 is = to X over 100 now we're going to cross multiply that's going to give us 12x is equal to 2700 we're going to divide both sides by 12 and that is going to give us our answer of 22 sorry 225% let's take a look at some longer word problems when it comes to percentages so our first problem is in a town survey 250 people said they prefer tea over coffee out of the total 500 respondents what percentage of the respondents prefer tea so again we're just going to work this out working left to right so we know that our is is our 250 people those are the people that prefer tea over coffee our of is the total amount of people that were surveyed right so that's 500 respondents and then what percentage of the respondents prefer T so again what percent that is what we are looking for that is our X so now we're just going to plug in our numbers 250 over 500 is equal to X over 100 now we're going to cross multiply so we're going to cross multiply here and cross multiply there that's going to give us 500x is equal to two uh 25,000 we're going to divide both sides by 500 in order to isolate our x and x is going to give us 50% our next example States a store sold 120 out of 150 chairs they had in stock what percentage of the chairs were sold so again we're just breaking this down so working left to right a store sold 120 so that is our is out of there's our word of50 chairs what percent is what we're looking for that's our X so let's go ahead and plug this into our equation we have 120 over 150 is equal to X because we're trying to figure out what percent over 100 so now all we have to do is cross multiply that gives us 150x is equal to 12,000 we're going to divide each Side by 150 and that is going to give us X is equal to 80% so 80% of the chairs were sold now let's take a look at word problems using ratios and proportions so ratio and proportions are problems that are one and the same and they can be solved with the same cross multiplication that we've been using this entire video so starting with our example we have Carmen can read four fictional books in 10 hours assuming all the books are the same length how many hours would it take her to read 5050 fictional books so Step One is we want to identify our knowns and our unknowns so what we do know is that Carmen can read four fictional books in 10 hours we know that right what we don't know is how many hours it's going to take her to Read 50 fictional books so our gnomes are she can read four fictional books in 10 hours and we're trying to figure out how long it's going to take her to Read 50 fictional books are unknown is that amount of time that it's going to take to read those 50 so for step two we need to plug in our proportions into our equation so we have two proportions and I've written them out here we have four books in 10 hours and we have 50 books in X hours so like I said before what I like to do is keep everything organized so on the left side of our equation I have all of our books and on the right side of our equation we have our time so what I did is I plugged in four books into our numer Ator on the left and 10 hours into our numerator on the right on the bottom we're trying to figure out how long it's going to take her to read those 50 books so I put 50 books underneath our four books in the denominator on the left and I put X hours which is what we are trying to figure out in the denominator under our 10 hours numerator on the right so now all we have to do is we have to solve for the proportion so we use that cross multiplication that we've been doing so we would have 4 * X would give us 4X and we have 50 * 10 would give us 500 and then of course in order to isolate X we would just divide each side by four and that would ultimately give us X is equal to 125 hours let's take a look at some examples of this in real time so our first problem States the ratio of boys to girls in a classes 8 to 7 if there are 48 boys how many girls are in class class so what we know is we have an 8 to7 boys to girl ratio and we have 48 boys what we don't know is how many girls are in the class so that's what we're trying to figure out so like I said before what I want to do is organize everything it is that I need to know so I'm going to put the boys on one side and I'm going to put the girls on the other side so I know that there is a 8 to 7 ratio of boys to girl those are going to be my numerator on both sides of my equation I know that I have 48 boys what I don't know is how many girls I need so now that I have my equation it's again just a sense of cross multiplication so that's going to give us 8X is equal to 336 we need to isolate for X on both sides by dividing by8 and that is going to give us the number 42 so we know we have 48 boys in class and the number we didn't know is now we have 42 girls in that class our next example is Calvin can make 16 cakes in 8 hours how many cakes can he make in 16 hours so again what do we know we know that Calvin can make 16 cakes in eight hours but we don't know is how many cakes he can make in 16 hours so we're just going to plug this in again organizing it I like to keep everything organize we're going to put cakes on one side and we're going to put our time on the other side so starting with our ratio we have 16 to 8 right so we're going to put 16 in our numerator on the left and we are going to put eight in our numerator on the right and now we know at the bottom he needs to make a certain amount of cakes within 16 hours so we know what the hours is on the bottom we just need to figure out how many cakes he going to be able to make so now that we have our equation we're going to cross multiply and we do this by saying 8X is equal 2 256 we need to isolate our X on both sides by dividing by 8 and that is going to give us X is equal to 32 cakes So within 16 hours Calvin should be able to make 32 cakes so in a hospital board the ratio of nurses to patients on a particular day is 1 to four if there are eight nurses on the ward how many patients are they going to be responsible for so we know we have eight nurses and we're trying to figure out how many patients they're going to have so again we're going to organize it so we're going to put our nurses on the left and we're going to put our patients on the right and we're just going to plug in our ratios so we have one nurse to four patients we know that we have eight nurses we're trying to figure out how many patients they're going to have on the floor so again we're going to cross multiply that is going to give us X is equal to 32 there's no additional math that needs to take place so we know that if we have eight nurses on the floor we should have 32 patients as well so here's a really good one for those of you who are going to nursing school so a prescription requires medication to be given at a ratio of 2 mg per 10 kg that's our first ratio if a patient weighs 70 kilg how many milligrams is that patient going to receive so again we're going to organize our information we're going to put our Med here on the left hand side and we're going to put our kilograms which is our patient's body weight over here on the right hand side and we're just going to plug in our ratios so we have 2 Mig of Med for 10 kg of patient we know that we have a 70 kilogram patient so we need to figure out how much Med they're going to receive and now we're just going to cross multiply so that's going to give us 10 x is equal to 140 we're going to divide each side by 10 in order to isolate X and that is going to give us X is equal to 14 mg so for this patient we need to make sure they get at least 14 milligrams based on their body weight next let's talk about rate unit rate and rate change I group these Concepts together with ratio and proportions because they follow the same principles so rate is essentially a ratio that illustrates the connection between two quantities with different units effectively demonstrating how one quantity relates to another one when the units are going to differ whereas with a unit rate this particular particular form of rate is going to be where the denominator is one and is essentially expressing the same amount of a single unit so let's explore how each example clarifies this concept so with our first example rate we have 36 miles in 4 days how many miles per day so again we organize by putting our miles on one side and we put our days on the other side so we know that this individual is going to be able to drive 36 miles in 4 days that is our first ratio our next ratio is how many miles per day so we know that we have one day and we're trying to figure out how many miles so we plug this into our equation and of course we're going to cross multiply so that's going to give us 4X is equal to 36 we need to isolate X in order to figure out our equation so we're going to divide 4 by both sides and that is going to give us X is equal to 9 miles per day next up we have unit rates and we're usually talking about unit rate we're talking about money so in this case we have $968 hours how many dollars per hour so we know we get $96 in8 hours now we've got to figure out in each hour how much money am am I getting so we're going to go ahead and plug that into our equation remember keep everything the same we're going to have dollars on one side and we're going to have hours on the other side and we're going to get 96 overx is = to 8 over 1 we're going to cross multiply that's going to give us 8X is equal to 96 we have to isolate X so we're going to divide each side by eight and that is going to give us X is equal to $12 per hour so how do we solve word problems with inequalities so inequalities is a mathematical expression that describes relationships between two values when they are not even so instead of saying two values are exactly the same like we do in equations inequalities is going to tell us plus if one value is greater than less than greater than or equal to or less than or equal to another value this helps us understand the range of limits within a certain quantity that can exist there are four main symbols that you're going to see when it comes to inequalities first up we have the greater than symbol and that's going to indicate that the value on the left side is larger than the value on the right side next we have the less than symbol and this is going to indicate that the value on the left side is smaller than the value on the right side next we have greater than or equal to and that's going to indicate that the value on the left side is either greater than or equal to that value that's found on the right side and you've guessed it you're right that less than or equal to is the exact same thing it's going to indicate that that value on the left side is either less than or equal to the value on our right side so here's a quick tip for remembering less than and greater than signs think of each sign as either a Hungry Hungry Hippo or a Pac-Man or for me because I live in Florida I always think of an alligator they're all hungry right they all want to eat as much as they possibly can so their mouth if you think about that little sign as a mouth is always going to eat the greater number right so it's either going to be greater than or it's going to be less than depending on which way they're eating it so hopefully that kind of helps trigger your memory when you're trying to figure out which one is which so let's take a look at our first example we have a hospital elevator has a maximum weight capacity of 2,000 lb if there are already three people on the elevator with a combined weight of 460 lbs write an inequality expressing the possible weight that may be added to the elevator use W to represent the weight so we're given that the elevator has a max capacity of 2,000 lbs that's the total amount of weight that can be in that elevator and you cannot go above that threshold so in scenarios where we're dealing with maximum limits the appropriate mathematical symbol to use is going to be less than or equal to because the total weight can reach up to 2,000 pounds but it's not going to be able to surpass it so we either have to have less than or equal to what that weight is next we want to make sure that we add our weight so we know that we have 2,000 lb is going to be acceptable given that there are three people already on the elevator and they have a combined weight of 460 lb we actually have to incorporate that within our equation so I'm going to add 460 over here to the left side of the equation and then of course we have to incorporate W because W is going to denote what that weight is going to represent for that additional weight to not exceed that 2,000 lb so that's going to be w+ 460 is less than or equal to 2,000 lb now we just have to solve our equation so we're going to minus 460 from both sides in order to isolate W and that is going to give you w is less than or equal to 1,540 LBS so at most the most that they can add to that hospital elevator is going to be 1,540 lbs with those additional people that are already on the elevator our next example states to qualify for a scholarship students must score at least 85% on their final exam if x represents the student score in the final exam which inequality represents the score of students who qualify for the scholarship so as we know these students have to have at least 85% on their exam so we're going to put 85% over here on the right side of our equation and now we need to figure out what we need to solve for x so they have to have at least a 85% so that means that they have to have either greater than or equal to 85% so I'm going to go ahead and plug that into my inequality and that's going to give us our equation for this problem X is greater than or equal to 85% in order for the student to get the scholarship let's take a look at a little bit more of a complicated problem so a factory has a fixed operating cost of $3,600 per day plus cost of a doar 40 per product produced if a product sells for $420 what is the least number of product that must be sold per day to make a profit so to make a profit we have Revenue received needs to be greater than the cost needed to make it so that's going to give us our revenue is greater than cost that's very important now we need to create an inequality for this so we already know that X is going to equal our products produced so for cost there's a fixed cost of $3,600 per day so we're going to put that on our cost side $3,600 per day and we know that we have to have a cost of $140 time x for each product that we're making so now we've got our revenue is greater than 3,6 plus 1.4x that's how much it's going to cost us to make these products now we need to figure out our Revenue so our Revenue we have it right here it states that we sell the products for $420 so we're going to plug in that $420 cents on the one side and we also need to plug in X because X is our products that are being produced so now we have our equation 4.2x is greater than $3,600 Plus plus 1.4x so now we need to start isolating for X so how we going to do this well we need to get X on one side so we're going to minus 1.4x from each side and that is going to give us 2.8x is greater than $3,600 we need to isolate for X we do so by dividing each side by 2.8 and that is going to give us X is greater than 1, so starting with direct proportion this is going to describe a relationship where one quantity increases and the other quantity is also going to increase at a consistent rate conversely as one quantity decreases that other quantity is also going to decrease maintaining that consistent ratio so this constant rate is known as constant proportionality and it symbolized with the letter k it quantifies that direct relationship between between two quantities so in mathematical terms when we're looking at this the two quantities of X and Y are going to be directly proportional so we're going to use the equation Y is equal to KX where K is that constant proportionality this means that every unit that increases in our X is also going to increase in our y when it comes to those K units so I have a little graph down here to kind of help that make sense so whatever X is in our first example we have Y = 5x whatever X is going to be it's going to be multiplied by five every single time so you're going to see this kind of linear equation when it comes to these direct proportional equations so imagine that you have a bag of marbles and for every marble that you take out you get two candies if you take out one marble you get two candies if you take out two marbles you get four candies so on and so forth this is direct proportion because the more marblers that you take out the more candies you're going to get and it's always going to increase at the same amount in a non-directly proportional relationship the correlation between two variables isn't going to be linear like we saw with our direct proportional changes this means that the ratios are not going to change at the same time and consistently imagine that you're pouring water on Plants the more water that you pour the taller the plant is going to grow but it's not always going to be the same like we have with our predictability when it comes to the candies and the marbles example sometimes you might pour just a little bit of water and the plant's going to grow really really big right and other times you're going to pour just a little amount of water and the plants aren't going to grow that much this is what we like to use in terms of non-directly proportional relationships because there isn't a steady pattern that tells us exactly how much that plant is going to grow based on the water that it gets in these relationships variables interact in a more complex way we typically see Y is equal to mx + b because these equations tend to be more slope based than we solve with our linear equations in our direct proportional examples in this case only if B is equal to zero would we actually have a direct proportional relationship so I've listed a couple of equations here on your slide to help you kind of identify the differences between these two relationships so it's easy to answer these questions when you're taking the t's so let's identify the differences between direct and non-direct proportional relationships so our first example we have y is equal to 2x so as we know with our direct relationships we're going to have y is equal to KX and with our non-direct proportional relationships we're going to have things like Y is = mx + b so in this case we have y is = 2x so as we increase our X we're always going to have the same number of increase because of that two coefficient before our X so this is an example of a direct relationship next up we have y is equal to 8 well we have no variable here right it's just Y is equal to 8 it's that's constant it's a fixed number it's not going to change so this is actually going to be a non-direct relationship because we need that variable in order to make that change next up we have y isal to 3x + 10 this is a great example of Y is equal to MX plus b because even if we increase our X it's not going to be the same consistently every single time so because we have that additional term that 10 on the end so this is an example of a non-direct relationship and then lastly we have y is equal to 5x so 5 * X as many times as we're going to multiply something by that five with that X it's going to be the same every single time so we're going to have a linear equation so this is a great example of a direct relationship let's start off by talking about mean median and mode so our mean is the average number of a data set our median is the middle number of a data set and our mode is going to be the most frequent number of a data set so when you are taking your te's they are going to give you a list of numbers that is going to be your data set that you have to compare in order to find these Val values what I highly recommend that you do when you're taking your te's is to place those numbers in numerical order from least to greatest trust me it's going to save you a whole lot of time as you're doing your calculations so I've already done that here for you in the data set that was provided so starting with our mean remember that's the average we have to figure out the average of all these numbers our very first step that we want to do is we want to add all of our numbers together So based on our data set if we were to add all of these numbers together we're going to get 127 but we're not done yet we have another step that's very important once we figure out what the overall sum is of all these numbers we need to divide it by the total numbers that are within our data set so if we were to add these numbers together 1 2 3 4 5 6 7 8 9 10 we have 10 numbers so we're going to take 127 which was our sum and divide it by 10 which was the total number of numerical digits that were provided to us that's going to give us our number of 12.7 or 13 if the t's asked you to round it up so the mean of this data set is 12.7 or 13 next let's move on to our median so the median is our middle number of a data set so when you think of median think about when you're driving down a highway right there's always something in the middle that is always protecting you from being able to go into the oncoming traffic lane right we don't want that we don't want accident and when we think of median we think of middle think of that middle median that's down our Highway roads so again organizing those numbers are going to help save you so much time on the t's we're going to have to start kind of Crossing off numbers from each end until we figure out what our median numbers are so we would cross up 8 and 18 10 and 16 10 and 15 10 and 14 and look at that we have an even amount of numbers so we actually have two middle numbers so when we have an even amount of numbers we need to find the average between the two in order to determine what our median is so if we were to add two and 14 that's going to give us 26 you're going to divide that by two and that is going to give us 13 so the median for this data set is 13 now it's way easier if you have an odd number because of course whatever that number number is in the middle would be your median but when you have even numbers make sure you're adding them together and dividing them by two and then lastly we have mode so when I think of mode I think of most m o mode m o most mode is most it's the most frequent number in our data set so again lining them up save you a whole bunch of time so if we look here we have one8 we've got 3 10 11 12 2 14 means 115 116 and 118 what number do we see the most well the number that we see the most is 10 So based on the numbers that we have here our mode most number is going to be 10 as it appears three times in our data set our first practice question is you are a nurse in a postoperative unit test with assessing the pain levels of 10 patients who have recently under gone surgery you are using a standard 0 to 10 Pain Scale where zero indicates no pain and 10 signifies the worst possible pain throughout the day you record the following pain scores for patients after knee surgery so as we can see here we have our patients rooms which really isn't something that we have to worry so much about when we're trying to find mean median mode but we have pain scores these are the ones that going to be important so let's start off with finding our mean again we need to put these in numerical order it's going to save us a whole bunch of time later trust me so let's see the lowest number we have here is three so we've got three we've got another three here and we have another three here next we have four we've got another four here and another four here after that we have five and we have another five here that's all of our fives then we have six and our last number is seven perfect now we got them in order we're going to save ourselves a whole bunch of time so with me and remember that's our average so we're going to add all of these numbers together so if we were to add 3 + 3 + 4 so on and so forth it's going to give us a total of 44 but next we're not done we have to divide it by the total number that's within our data set so we have 1 2 3 4 5 6 7 8 9 10 we have a total of 10 numbers so we're going to divide 44 by 10 and that is going to give us our mean of 4.4 next let's consider our median so remember we're driving down the highway we get a median in the middle of us add is our middle number so again I've already written the numbers out here for you in numerical order so now we got to find our middle so when I'm doing this what I'm doing is I'm just marking off the ends so 37 36 35 4 five look at that remember we have an even number of data points so we have have 4 + 4 well 4 + 4 is equal to 8 / 2 is equal to 4 so our median number here is four our last data point that we have to find is mode remember m o and mode m o and most we're looking for the most frequent number of our data set so I have it written out here but I'll just visualize it for you we have three 3es we have three fours we have two fives one six and 1 7 well look at that we have we actually have two numbers that appear most often within our data set when this happens it means we have two modes there's no additional addition subtraction multiplication nothing else we need to do we just know that our mode is going to be both three and four all right let's do some practice a small bakery tracked its daily cookie sales over one week and recorded the following numbers 12 15 11 12 14 16 and 12 calculate the mean median and mode of the bakery's daily cookie sales now what I did over here in the corner is I already put them in order for you but it's very important remember when you're doing this put them in order save yourself some time so mean we're finding the average what's the average of all these numbers so we have 11 + 12 + 12 + 12 + 14 + 15 + 16 so when we add all these numbers together we're going to get 91 but again we're not done we have to divide these numbers by the total numbers within our data set so we have 1 2 3 4 5 6 7 so we have a total of seven numbers in our data set so 91 / by 7 is equal to 13 cookies next let's figure out our median so again our median is our middle number driving down the highway we're looking at that median so again I'm going to go ahead and write out all of these numbers here and now we have to find that middle number so I'm going to start Crossing some numbers out so 11 and 16 12 and 15 12 and 14 and then look at that remember we have an odd number so automatically our middle number is our median so our median is 12 cookies and then lastly let's talk about our mode so again we're going to write this out all of our numbers here to figure out what our mode is so we have remember mode is most most frequent number we have 111 we have 3 12s 114 115 and 116 so our mode of this data set is going to be 12 cookies because this is the number that appears most within our problem next let's explore how we determine the range of a data set where is the range defined it's the difference between the highest and lowest numbers of our set in our initial example we have a jumble data set so we're going to need to organize that to figure out what our highest number is and our lowest number is so if we organize this we can see that our lowest number is two and our highest number is 10 so as we said as we're calculating range we want to subtract our highest number from our lowest number so 10 - 2 is equal to 8 so our correct answer for range with this data set is eight pretty simple right it's going to be really easy when you're taking it on the t's let's take a look at our other data set so again it's just a bunch of jumble numbers so we want to make sure that we put that in numerical order from least to greatest save ourselves some times later right when we organize this we have 73 as our least number and 96 as our greatest number so we're going to just subtract the two 96- 73 is equal to 23 so our range for this data set is 23 so let's take a look at a practice question to tie all this together so the question States a student scores on five math quizzes are as follows 82 76 91 85 and 88 what is the range of the students quiz scores so very first thing we want to do is we want to sort our data set so we have our lowest number is 76 then we have 82 and what I like to do is I like to just cross out as I'm moving through so that way you're not double counting numbers 76 82 the next number we have is 85 then we have 88 and our last number is 91 now that we've sorted our data set we know what our greatest number is and our least number is so our greatest number is 91 our least number is 76 and we are going to calculate our range we do this by subtracting our greatest number 91 by 76 and that is going to give us 15 so the range of this equation is 15 and look at that b is our correct answer so something else that's important when you're taking your te's especially when it comes to range is going to be distribution of that range when it comes to shape so first up we have symmetry so a distribution is considered symmetric if you can draw a vertical line through the center of its graph representing two halves that approximately mirror each other for our first example let's consider a continuous distribution graph here it appears that the mean is equal to the median that makes sense right our median would be right down the middle our mean is going to be equal to our median and as you can see we have an even Distribution on our left side as we do our right side something else that's important to note here is our graphs Peaks they indicate the frequencies and provide us with our modes as you can see here we can observe two prominent Peaks suggesting that we actually have two modes this leads us to identify this kind of distribution as bi modal meaning that we have two modes or two frequencies it's also symmetric and we have coning mean and median values so in this particular example this is going to be bi modal bi meaning two two modes in our next scenario we have a mound shape or it's also commonly referred to as the bell shape because of the way that the distribution looks drawing a central line down the middle through our graph it reveals again that symmetry with equal distribution on both sides of that line thus again it means that our mean is going to align with our median once more the notable feature here that we didn't see in our previous example was that we only have one highest peak so this is going to illustrate in this scenario that our mean is going to be equal to our median which is also going to be equal to our mode so we would call this kind of distribution uni modal uni meaning one one mode it's also symmetric so we can call this again uni modal I hope you now see the difference between bodal and unimodal examples next let's talk about uniform or rectangular symmetry so this distribution occurs when all categories or classes within a distribution have frequencies that are equal or very close to equal observing such a distribution in our first example we notice that it appears symmetric with an even split of data on each side of our median so the Symmetry suggests that our mean is likely equal to the median in this distribution as well however the concept of mode is going to be somewhat unique when we're looking at this kind of graph in this case we have no mode as we do not have any Peaks as we would have seen in our previous examples in addition the frequencies don't have to be exactly the same in all classes there can be slight variations as we see here in our second example the distribution maintains symmetry implying that our mean is still equal to our median although the mode is less defined in a uniform distribution due to the equal or nearly equal frequencies of all classes one could point to multiple modes or consider the entire distribution mode like for its uniformity so we actually call this roughly uniform while it's not perfectly symmetric a roughly uniform distribution comes close showcasing an even spread across all of the classes lastly we're going to talk about skewness everybody's favorite the most confusing right well what I want you to do when you're taking your test is I want you to look down at your test toes and I want to notice the way that your toes trend on each side of your foot in a left skewed distribution the bulk of the DAT is off to our left side so that is where we are going to find our mean at the point of the slope we're going to find our mode right that's our most frequent number if you look at your toes on the left side of your foot your great toe is at its highest and each toe that follows is going to get slightly shorter giving you kind of like an arch this is the same thing that's happening with a left skewed graph so in this skewed landscape the mean or the average is going to be positioned a little bit more to the left the median which divides the data set into two is going to be found slightly closer to our mode but still between the mode and the mean this Arrangement means that half of the data points are going to fall on the left of the median with the remainder distributed to the right conversely our right skew distribution is going to be in the opposite direction here the data is going to accumulate mostly on the left side before it starts trending off onto the right just like we pictured before looking at our toes the same thing is happening with our right foot our greatest toe is going to be higher on the left side and as each toe follows you're going to see that same skewness take place that same Trend so when you're trying to figure out the difference between your left and right skewed when you're looking at graphs I want you to look at your feet is it your right foot that it's mirroring or is it your left foot that is mirroring again the mode when it comes to right skewed is going to remain the highest point right here our median is going to be slightly positioned to the right very close to our mode and then our mean which is the average of our numbers are going to pull a little bit further to the right indicating the skewness very important for you know to your teeth hopefully now you got it let's take a look at our practice question a researcher is analyzing the distribution of a data set representing a number of daily online orders received by two different products over a month the data shows two distinct Peaks so that is going to be our keyword here in our practice question what is most likely the shape of our distribution well we can automatically eliminate right and left skewed as we know we only only really have one Peak so we're only left with uniform and bodal well uniform as we discussed doesn't have any Peaks at all right it's uniform it's one straight line across so we can automatically eliminate that and if we're thinking about two distinct Peaks we talked about that bodal two modes that is our two distinct Peaks so out of all the answers that we have available to us the one that makes sense is C bodal our next question States a teacher records the test scores of students in a class and notices that the frequencies of scores is evenly distributed across all positive values which of the following best describes the shape of the distribution of these test scores so remember when we're seeing even distribution we're looking at one thing in particular so let's take a look at our examples well right skewed and left skewed we can automatically eliminate that because as we know we don't have an even distribution fustion it kind of skews off either to the right or skews off a little bit to the left next we have uniform and bodal so as we said before in our previous practice question bodal means we're going to have two peaks right two modes so that is obviously not going to be the right answer based on everything that we have available to us knowing that the data set is evenly distributed the only answer that's going to make sense is a uniform let's take a look at our last question before we move on to probability so the question States in a study of household income in a certain region it was found that most of the data points cluster at the lower end of the income range with fewer households earning significantly higher income this results in a long tail to the right of the distribution how is this distribution best described so if we're talking about this we know that the cluster points for the lower end of the incoming range right is going to be most of the data with fewer households earning a significantly higher so we know that we've got some households right having higher data and then as we start to see income households it's going to start to kind of long tail off to the right based on what the question is saying longtail off to the right so based on what we have available to us we have uniform well that doesn't make any sense right because that would mean we would have no modes it would just be all evenly distributed so we know that that is incorrect we have bodal well we don't have bodal because we know we don't have two peaks we have to have two modes it tells us that our results are going to long tail off to the right of the distribution so we can automatically eliminate that next we have left skewed well look at your toes based on what we just drew is your left foot the way that it Trends down exactly with what we're looking for here no right we would have to have larger numbers to the right right of the graph with the smaller numbers trailing down to the left so we can automatically eliminate that leaving us with our only correct answer which is going to be d a right skewed graph now let's talk about probability in most mathematical scenarios outcomes are going to be certain right if we say 1 + 1 is going to give us two if we multiply two by three is going to give us six however sometimes in math you're often going to be presented with situations that are far from predictable for this we use the probability equation which is the total number of favorable outcomes divided by the total number of possible outcomes let's take a look at the simple Act of flipping a coin we can't be certain if it's going to land on heads or tails and while we can't predict the outcome of each individual coin toss we do understand some basic principles about it for example in a Fair coin toss the likelihood of landing on heads is going to be equal equal to the chance of it landing on Tails since there's only two outcomes heads or tails each is going to be equally as likely we would expect that half the flips would be heads and half the flips would be Tails meaning that the probability of flipping heads is 50% or 1/2 and the probability of flipping tails is also going to be 50% or 1 half let's dive deeper into this concept using the probability line which is essentially a number line ranging from 0 to 1 on this scale a probability of zero indicates the impossibility of an event occurring and the probability of one guarantees that this event will most likely occur marking it as complete certainty this is the reason why the probability line is confined from 0 to one think about it no events can have a probability lower than zero as nothing is less likely than impossible and similarly no events can have a probability higher than one since nothing can be more certain than certainty itself when we talk about a probability of 1/2 such as with the case with our coin toss it means that the event has an equal chance of occurring as it does not occurring if the probability Falls below 1/2 the event is considered unlikely whereas if the probability is above 1/2 it's going to indicate the event is likely to happen in addition the te's will commonly practice expressing probabilities in various formats including fractions decimals and percentages they're all going to be used interchangeably for example a probability of zero can be viewed as 0% chance of an event happening similarly a probability of 1 half can equate to a 50% chance of something happening and a probability of one corresponds to 100% chance of an event from occurring now let's take a look at an example that's a bit more complicated than a coin toss now let's take a look at dice so a standard dice has six faces each numbered one to six when you roll the dice each face has an equal opportunity of Landing face up similar to the 50/50 chance that we see with a coin flip either being heads or tails however that's where the comparison is going to stop because unlike a coin which only has two outcomes the roll of a dice is going to have six possible outcomes the certainty of landing on one of the six faces is represented by a probability of one or 100% however since only one face can be on top after a roll the probability must be distributed equally among all six outcomes so we divide that by six doing the math 1 / 6 is going to give us six which can be translated into 0.167 or 16.7% this value should be positioned in our probability line to indicate the likelihood of rolling a specific number say maybe a three while that suggests that the three is unlikely it is equally probable as to Rolling any other number on our dice I'd like to highlight an important aspect of probability remember that probability of Landing heads when you flip a coin is 1/2 and the chance of get tails is also 1/2 when it comes to Rolling a dice the probability of rolling a one is indeed 16 now that you tally the probabilities of all outcomes of flipping a coin the sum is going to be 2 over 2 which is going to simplify to one likewise summing up all the probabilities for each face when it comes to a dice is going to be 6 over 6 which is also going to equate to one the cumulative probability of all potential outcomes for a given event always is going to equal one or 100% this reflects the certainty that one of the possible outcomes will occur in our last example let's make it a bit more complicated by introducing a spinner if our spinner only had six sections of equal size the probability is going to mirror that of rolling a dice but to make things more interesting let's increase the number of sections now our spinner has 16 equal segments so what's the likelihood of landing on a number say 10 similar to dividing a die 100% chance among the six faces we're going to evenly distribute that percentage across the Spinner's 16 sections therefore the chance that that spinner is going to land on 10 is 116th or 6.25% which again we can plot on our probability line it's now evident that l on a 10 with our spinner is going to be less likely probable than rolling a three when we have a die this is logically given with the number of outcomes based on our spinner and our dice now let's add one more layer to this scenario how do we determine the probability of a spinner landing on a specific color with our spinner Now featuring five blue tiles and 11 gray ones let's delve into the probability of landing on one of those blue tiles drawing parallels to previous examples like the coin flip resulting in 1/2 and the dice resulting in six we notice a common theme the numerator from all of these fractions were one this is because we were focusing on One Singular outcome such as rolling heads on a coin flip or rolling three on a dice however in our current scenario the numerator of this fraction is going to change to five because we are reflecting the five blue tiles that must meet our criteria for a successful outcome the denominator is still going to remain the total number of outcomes which is 16 representing the total number of tiles that we find on our spinner thus the probability of the spinner landing on a blue tile is calculated as 5 over 16 also translated to 31.25% well 31.25% might still be considered not highly probable it's significantly more likely that we're going to land on a blue tile than we were to land on one specific number with our spinner hang tight with me as we're going to work through this probability question I promise we're going to get through it the question States a bag contains five red marbl three blue marbles and two green marbles if two marbles are drawn one at a time from the bag at random with that replacement what is the probability that both marbles drawn are going to be red so we're looking four red marbles we know that we're drawing marbles one at a time from the bag at random and we're not replacing them without replacement so we start off by asking ourselves how many marbles do we have because based on probability our equation is going to be the number of favorable outcomes that's our top number our numerator and underneath that it's going to be the total possible number of outcomes so that's what we're going to figure out first the total possible number of outcomes so we know that we have five red marbles we have three blue marbles and we have two green marbles so if we add all of these together we know that we have a total of 10 marbles so we know that the denominator of our equation is going to be 10 okay so now we need to figure out what we're drawing each time we draw we know that we have five red marbles so the possibility of us getting one of those red marbles is going to be five out of 10 five is our number of favorable outcomes cuz we have five red marbles and 10 is a total number of possibilities that can occur because we have 10 marbles in the bag so say we pulled out a red marble the first time we're not going to replace it right without replacement that's what it states so now how many marbles do we have in the bag well now we have nine right so now our denominator is going to to be nine because we have only nine possible outcomes since we've already drawn one of the red marbles well we've already drawn one of the red marbles how many red marbles do we now have in the bag 5 - 1 is equal to 4 so our next fraction is going to be four out of nine there's four number of favorable outcomes cuz we have four red marbles left and we have nine total number of possible outcomes cuz we only have nine marbles left so base on our our two drawings our first should be 5 out of 10 and our second should be four out of n now we are going to multiply these two possibilities together okay so we know that we have 5 out of 10 we are going to multiply that by 4 over 9 because we have the word and so whenever we are looking for two things together we use the word and that means we are going to multiply our outcomes something very important that you're going to have to know for the t's is when we are looking for two outcomes together we use usually the word and and anytime we have the word and we're going to multiply the outcomes together because we're going to have two outcomes that we're going to need to know together whereas if you start to see the word either or or we are going to add the outcomes together so anytime that you are trying to figure out two things like and in this case we're trying to figure out the probability of both marbles right and two marbles we are going to multiply outcomes whereas if it was asking or whether we were going to get a red or a blue then we would add the outcomes together so very important when you are taking your te's exam so in this particular case we are trying to figure out both marbles right that's our and so we are going to multipli so we are going to multiply our top together 5 * 4 is equal to 20 and we're going to multiply our denominators together so 9 * 10 is equal to 90 but if we take a look at our examples here for our answers we don't have anything that says 20 over 90 right we're going to have to simplify our fraction you can either do the math if you want to but what I do is I just remove the corresponding zeros from the end of each equation it just makes it a lot easier and that is going to give me 2 over 9 so the probability that we are going to pull a red marble both times without replacement is going to be 2 over 9 do we have that yes we absolutely do the correct answer is going to be b 2 over9 the atits is going to test you on different graphical display types and how you use them there are five different types of displays that you're going to encounter when you take your te's one is the cartisian coordinate scatter plot line P or Circle and bar graphs let's break each one of these down starting with cartisian coordinate chart also known as cartisian graph consists of two perpendicular lines or axes the horizontal axis is called the x axis and the vertical one is called the Y AIS their intersection at zero creates An Origin which is the starting point of plotting data as an example let's look at how we would plot five comma 4 so starting at our origin 0 0 right here in the middle if you look look to the left and right of the origin Point you're going to see that you have negative numbers that move towards the left and you're going to have positive numbers that move towards the right it's very similar if you're looking at a number line in our example 54 the first data point is a positive number so we would move to the right five spaces on our x axis when you're looking at our y AIS we can see that the origin point 0 0 has kind of the same thing like we saw with our number lines in our x axis so as we move up our number line it becomes more positive and as we move down our number line we become more negative our next Adda point is a positive number so we're going to move up four spaces to get to four now we're just going to place a DOT at this position of our plot and that is our Point 54 on our cartisian coordinate chart not too bad right next let's look at Scatter Plots so this type of graph is essential when want to examine relationships between two different data sets or numerical variables essentially it's about observing how one variable correlates to the other let's consider the example of our scatter plot found here on the right side of your screen it investigates the amount of ice cream sales dependent on environmental temperatures as you can see over time the hotter the temperature outside becomes the more sales of ice cream increase Scatter Plots do hold significant value in scientific studies they are particularly useful when we need to alter one variable known as our independent variable and observe its impact on the other called our dependent variable when we're setting up our scatter plot it's crucial that we place that independent variable on our x axis this would be our X for instance if you're conducting an experiment where you vary the amount of fertilizer given to a plant to measure its growth over time the amount of fertilizer or independent variable would go on the x- AIS the growth of our plant the dependent variable would be along our y AIS by doing it this way you can effectively analyze the correlation between these two factors as you can see the amount of fertilizer increases the growth of the plant now let's look at a line graph so a line graph is ideal in observing changes over time so take for instance examining the produce sales over 9 months in this scenario the timeline is plotted along our xais ACC showing us the individual months while the financial values are marked on our y AIS showing the sales in the thousands in a scientific context we may be studying the respirations of a nursing student during clinical check off a line graph proves invaluable for this information if you're measuring how much oxygen the nursing student consumes over a period of time from just going into checkin during and then after a line graph will notice in illustrate these changes over time this makes it a perfect tool for visualizing and analyzing data that varies with time this is probably one of the most familiar charts and this is a pi or circle chart they use both names interchangeably as you can see from our pride chart that we have here as our example there was a survey that was completed in 2020 for nurses who were leaving a facility the data collectors collected information based on their reasons for departure and these were the results as you can see from our chart each color corresponds to a different reason so for example people leaving for personal reasons was about 50% those who transition to a competitor was about 20% travel nursing was about 133% no opportunity to advance was 10% and then lastly those individuals who did not provide a reason was 7% a bar graph is particularly useful when we need to compare multiple groups or categories for example in our first set of bar graphs you can see that it illustrates the number of worldwide electric cars per year as you can see as the years progress the number of electric cars also increase going back to our plant experiment that we talked about before imagine that we're in a laboratory setting investigating how various light colors affect the rate of photosynthesis in such a case a bar graph would be an excellent choice for this experiment we could plot the rate of photosynthesis here on our y AIS with each color represented as a distinct bar down here on our xais often in bar graphs each bar signifies the average or mean derived from the collected data this makes bar grph a useful tool and a very powerful tool at that for visualizing representations of comparing average values across different categories or groups here's a quick and useful tool as to why we would use different graphical displays and what specific cases we would use them as it's a little cheat sheet to help you pass your atits when you come across these questions on the exam all right once we've determined the graph to use we can begin the graphing process let's experiment with this concept using hypothetical data from an ice cream sales that we discussed earlier imagine that we have data showing the average temperature in an area and the corresponding ice cream cone sales during different months if we want to illustrate the total ice cream sales in each month as a proportion of the year's total sales a part chart would be ideal each slice of the pie chart would represent a different month showing us its average sales that they share over that year on the other hand if our focus is tracking the average high temperatures in an area over time a line graph would be the way to go this graph would effectively depict temperature Trends throughout the year now let's say that we're interested in comparing ice cream sales across specific months in this scenario a bar graph would be highly effective it would allow us to compare the ice cream sales to the specific months side by side providing a clear visualization of the comparison let's dive into the Practical creation and interpretation of graphical displays this is very important for you to know for the te's and we're going to start by focusing on Scatter Plots using our previous data set here is how we would construct the scatter plot and what each element represents the very first thing that we want to do is we want to establish our axes so on the xaxis we're going to plot the average high temperature and on the Y AIS we're going to plot our ice cream sales it's important that you make sure that you've label each axis clearly including the unit of measures that you're using for instance the xais might be labeled average high temperature and we're using the unit of measure of Fahrenheit and on the Y AIS it is our ice cream sales and we're doing cones per month we also want to make sure that we title our graph a good title should encapsulate the essence of the graph content it's not enough to just have a vague title like ice cream right instead it should be informative such as correlation of ice cream sales and average high temperatures 2023 this title should essentially narrate the story of the data that we're trying to present after labeling our chart we can now plot the data points in a scatter plot each point represents a data pair correlating between temperature with sales the arrangement of these points can reveal patterns or correlations an important feature of our scatter plot is our best fit line this line represents the average Trend through our data points when drawing it freehand we aim to position it so that roughly half the points lie above it and the other half lie below it remember this line should not go beyond our range of data as it represents interpolation within the existing data set avoid putting any kind of arrows or extending them past the outermost data points in addition in terms of scaling we want to ensure that the increments of each axis is uniform and clear for instance if each major grid line of the x axis represents an increment increase of 15° Celsius this should be consistently applied throughout our line in our case with this graph the ice cream sails an average temperature increase in increments of 10 lastly remember that including zero on your graph is not always necessary if your data doesn't include or approach zero it does not need to be present on your graph this helps keep the focus on the relevant data ranges and avoids misleading representations one of the things that the teas loves to test you on is bad graphs and misrepresentations what are we missing what could be better about our graphs let's take a look at the example of a bad graph focusing on one that's supposed to show how different amounts of fertilizers affect plant growth when you look at this you immediately notice several errors firstly the title of the graph while it is present isn't descriptive enough it needs to clearly indicate what is being plotted on our X and Y AES such as the relationship of fertilizer to plant growth next there's a major issue with how the axes are set up plant height is actually listed down here on our x axis but since we have varying amounts of fertilizer that actually should be our independent variable and should be placed down here on our xaxis and that plant height should be moved to our y AIS because it is our dependent variable it's also crucial to include units for each variable but as you can see they're missing here plant height should be labeled on the Y AIS with appropriate units and fertilizer amounts should also be labeled on the x- axis with its amounts of units the scaling of these axes is another problem the X axxis shows long linear scaling numbers of 5 12 15 19 23 and 30 in a well-made graph the distance between each point should be consistent uniform and clear additionally only having a single number on our Y axis 13 doesn't help with accurate measurements either lastly our best fit line on this scatter plot extends past our range of data this is also going to be incorrect that best fit line should only Encompass the range of data points on our graph in summary the graph has quite a few significant errors demonstrating what to avoid when creating or interpreting a scatter plot let's take a look at our first practice question which of the following descriptions best matches a scatter plot is it a a graph that uses vertical or horizontal bars to show comparisons among categories b a graph that displays data points along a line typically used to show changes over time c a circular graph divided into slices to illustrate numerical proportions or is it d a graph that displays data points to represent the relationship between two variables and the correct answer is d a graph that displays data points to represent the relationship between two variables as we know a scatter plot is characterized by its use of data points plotted on a cesion coordinate system showing how one variable is affected by the other which is why D is the most correct answer you were tasked with presenting the monthly sales data for three different products over the course of a year which type of graph would be the most effective in displaying this information is it a a pie chart b a line graph c a scatter plot or d a cartisian coordinate and the correct answer is b a line graph as we know a line graph is ideal for showing changes over time making it the best choice for presenting sales data across different months it allows the audience to easily see Trends and compare the performance of the three products throughout the year pie charts are more suited for showing parts of a whole at a single point in time Scatter Plots are used to show relationships between two variables and cesan coordinates would not be used in this scenario a bar graph is used to compare the average test scores of four different classes algebra biology chemistry and English sounds like the tease test right however the bars are not labeled with the names of the classes how does this affect the interpretation of the graph is it a it allows for a straightforward comparison of the test scores across subjects B it does not impact the interpretation as long as scores are visible C it creates ambiguity making it difficult to attribute the correct scores to each class or is it D it improves the visual Simplicity of the graph focusing attention solely on the numerical scores and the correct answer is C it creates ambiguity making it difficult to attribute the scores for each class without these labels indicating which BX represents which classes it becomes impossible to determine the average test scores for each individual class of algebra biology chemistry and English now let's talk about linear exponential and quadratic graphical Trends in order to understand these three different kinds of Trends we're going to put our imagination hats on so let's say that we're a scientist who are observing Turtle populations in a series of islands focusing first on two islands funky Island and get down Island in what we term Turtle Euro zero three turtles each make their way to Funky and get down Island upon visiting each island after one year we note that funky Islands Turtle count has risen to five while get down islands has reached six the following years F's population which is our funky island has grown to seven and our G island which is get down island has impressively doubled to 12 year after year we meticulously record our findings aiming to discern any emerging patterns we Ponder whether the turtle populations expand at a steady Pace a concept known in mathematics as constant difference let's take a closer look at each one of these different kinds of islands and graph Trends so starting with funky Island we started with three turtles if two more arrived each subsequent year then after a decade the calculation for Island F would be the initial three Turtles plus two turtles for each of 10 years totaling 23 as we generalize this if x equals the total number of years elapsed the turtle population on island F would be represented as 2x + 3 which intriguingly forms a linear equation y = mx + b where m is the constant rate of increase and B is the starting number of the turtles upon graphing this relationship of funky Island we discover a linear Trend affirming our hypothesis this linear model is characterized by a constant increase or decrease turning our attention to the turtles on git funky Island G it's another intriguing pattern that emerges their numbers seem to double each year indicating exponential growth rather than linear growth unlike that steady addition that seemed to take place on funky Island get down Islands Turtles increase by a factor of two annually to illustrate the population for the seventh year we would be calculating that by taking the initial three Turtles and doubling them seven times a process conveniently expressed using an exponent this kind of a pattern aligns with exponential function which is y = a * B with the exponent of X where it represents the starting numbers of turtles b as the growth rate in this case two and X is the number of years our analysis is confirmed as depicted in the graph of get down Islands Turtle population as you can see we have a nice curve steadily increasing over time our findings highlight a fundamental principle of growth patterns given enough time any exponential function with a growth multiplier greater than one will surpass a linear function lastly let's focus our efforts on quickstep island initially there was only one Turtle but by year one the count reached 13 by the second year the number stored to 45 and by the third year it stood at 97 these patterns present a curious pattern that seems to correlate with the passing of years the turtle population on quickstep island can be described by the formula y = 10 x^2 + 2x + 1 where X represents the number of years this means that for the 9th year the population would be 10 * 9 years sared + 9 * 2 + 1 1 doing our calculations we have 92 is equal to 81 10 * 9 is = to 810 2 * 9 is equal to 18 and then of course we bring up the number one so if we add all of these numbers together we're going to get a total of 829 turtles by the time that we reach our 9th year this pattern is indicative of a quadratic function which takes the form of Y is equal to a * x^2 + BX + C in this case b is equal to 2 and C is equal to 1 the graph of this function reveals that classic U-shaped curve that's also symmetrical and distinctive when we're looking at quadratic equations when complimenting which Turtle population would outgrow the others over time it might be tempting to say that the quadratic growth of quickstep Island would be the given Choice given its rapid Ascent however it's it's important to remember that exponential growth when it comes to capacity is going to increase indefinitely it will eventually surpass that quadratic growth the quadratics function's growth is tether to the square of X whereas the exponential growth X serves as the exponent itself offering balance potential for increase let's take a look at how this will be displayed on the aits starting with our first practice question Calvin is selling apples for $2 each at a farmers's market he has written down his sales for the last 30 minutes as follows and as you can see over here on the right hand side of our screen this is our data points what type of relationship is depicted between the numbers of the apples bought and the total cost is it a linear B exponential C quadratic or D cubic and the correct answer is a lineer the cost increases by a constant amount of $2 with each additional Apple indicating a constant rate change this consistent increment is characteristic of a linear equation where the graph is going to be a straight line Julie is growing bacteria collected by a bacterial sputum sample for a patient admitted with pneumonia she documents her findings every hour what type of graph best represents the relationship between the time and the number of bacteria is it a linear B exponential C quadratic or D D cubic and the correct answer is B exponential the number of bacteria doubles with each passing hour indicating the bacteria count is not increasing by a fixed amount but rather a fixed proportion this pattern of growth where the rate of increase becomes progressively more rapid is indicative of an exponential relationship next let's take a closer look at directions of Trends in graphs when we're analyzing graphs it's important to understand that different Trends can depict either increasing decreasing or no change trends when we're looking at graph Trends we want to look from left to right where does the line start in the left and what is the trend when we move across to the right let's start with increasing Trends so graphs with increasing Trends show values that rise As you move from left to right as you can see from our two different examples if you look at the starting point on the left you can see that each line increases in value as we start moving to the right so as you can see here we start moving up in value and again on this example we also move up in our value so we're going to get a concave up or concave down increasing kind of distribution with decreasing trending graphs they depict values that fall as they progress from left to right on the graph so as you can see from our example as you look at the starting point on the left you're going to see that each line is is going to decrease in value as we move to the right so up here we have a concave up decreasing graph meaning that we start high and then we come down low and then over here on the right we have a concave down decreasing graph again starting high and as we move to the right we lower the number lastly we have a no change Trends kind of graph so these graphs will show a flatline representation of a situation where there's no change over time the value remains constant regard of the changes that are happening along the horizontal axis as you can see here we just have this flat line on our graph which shows that as we move left to right there is no change making it a no change graph outliers on graphs are data points that stand apart from the crowd catching your eye because they don't quite fit in with the rest imagine that you're looking at a graph and while most of the data points are in a cozy cluster like we see in the graph here on our screen there's one point or maybe even two points that are going to kind of miss the memo with the rest of our data points they're going to be sitting off far to the side above or below our best fit line as you can see here in our example sometimes these outliers are a result of a simple slip up maybe a typo or data that was entered incorrectly or even a glitch in the measurement equipment that we're using other times these are real deal genuine deviations that tell a story of their own like a sudden spike in sales after a viral marketing campaign or an unexpected drop in temp temperatures on an otherwise warm month if any of you live in Florida you know exactly what I mean these types of questions on the te's tend to be easier because we can visually see the data points and can easily denote that something is wrong we introduced dependent and independent variables a little bit earlier but now we're going to dive a little bit deeper into understanding what they mean the independent variable is often seen as the cause is the variable that we manipulate or change to observe how it affects another quantity on the other hand the dependent variable is considered the effect it's the value that depends on or is determined by the changes of our independent variable for instance in an experiment measuring the growth of plants based on the varying amount of water the amount of water given that is going to be your independent variable is going to directly influence the growth or dependent variable so let's start with the basics when it comes to graphing these variables the independent variable which is often plotted on our X AIS as we know is the variable that we change in order to control an experiment think of it as the input of our cause and effect relationship on the flip side we have our dependent variable which is represented here on our y AIS and this is the outcome are the effect that we are going to observe as a result of the changes to our independent variable so essentially the value of the dependent variable is going to hinge on the influence of our independent variable if you're trying to figure out how independent and dependent variables affect an equation it'll look something similar to this we have the equation y = 3x + 5 based on our equation we know that Y is going to equal the outcome of the value because it is the lone variable on the other side of our equal sign that means that X is going to be our independent variable we have control over what we're going to put in X which is ultimately going to influence our y let's take a look at another example imagine a hospital system is conducting an experiment to understand the relationship between nurse to Patient ratios and average recovery time for patients on an orthopedic unit the task is to represent the given data on a graph this scenario poses an interesting question between the nurse to Patient ratio and the average recovery time which is our dependent variable and which is our independent variable let's recall that our dependent variable is is ultimately going to be influenced by your independent variable so in our case does the nurse to Patient ratio depend on the recovery time or is it the other way around given that in this experiment we can choose the nurse the patient ratio and implies that the ratio is the variable within our control hence nurse to Patient ratio is going to be our independent variable which we are going to plot down here on our X AIS on the other hand the average recovery time time which we aim to measure naturally is going to become our dependent variable which we're going to plot over here on our y AIS as you can see down here on our graph the higher the nurse the patient ratio is is ultimately going to increase the average recovery time for our patients lastly to test our knowledge from before what kind of relationship do we see when we're looking at these two variables is it linear exponential or quadratic that's right you've guessed it it's linear because we have a straight line good job in a study examining the effects of study hours on test scores researchers track the number of hours spent studying X and the results testing scores y given the relationship described in the equation of y = 2x + 70 or Y represents the test scores and X represents study hours determine which variable is independent and which is dependent is it a x is dependent variable because it relies Li on the test scores b y is the independent variable because it is determined by the number of study hours is it c x is the independent variable because it represents the controlled amount of study time influencing the test scores or is it D why is the dependent variable because it dictates the number of study hours needed to achieve a certain score and our correct answer is C in this context the number of study hours which is X is the variable that the researchers control or manipulate to observe the effect on another variable making it our independent variable the test scores which would be our y on the other hand are affected by the changes in our study hours making them the dependent variable the equation y = 2x + 70 clearly shows that Y which is our test scores is going to change in response to X which is our study hours underscoring the dependency of test scores in the amount of time spent studying in a clinical trial to assess the effect iess of a new dietary supplement on improving blood pressure levels participants Baseline blood pressure readings are taken before starting a supplemental regimen and their daily supplemental dosage varies among participants based on the information provided which variable is independent and which is dependent is it a medication dosing is the dependent variable because it is influenced by the changes in blood pressure readings is it B blood pressure readings is the independent VAR variable because they determine the dosage of dietary supplement is it C medication dosing is the dependent variable because it represents the dosage of the supplement which is manipulated to observe its effect on blood pressure or is it D blood pressure readings are the dependent variable because it depends on the dosage of the dietary supplement and the correct answer is D in this clinical trial the variable the researchers actively manipulate is the dosage of that dietary supplement given to the participants making that medication dosing our independent variable the purpose of varying that dosage is to observe its impact on a specific outcome which is in this case is going to be our blood pressure readings making that our dependent variable next let's talk about correlation and co-variance so correlation and covariance is a statistical measure that describes the relationship between two variables like in all of our examples for mathematics we call these variables X and Y to illustrate this let's imagine that we plot points on a graph using X and Y AIS and the pattern emerging somewhat like we see in our examples the way that it's depicted in our first example here suggests that positive correlation between X and Y that is that when X goes up Y is also going to Trend up as well now let's also consider a scenario where the inverse relationship is true like we see in our second example with negative correlation as X increases Y is actually going to decrease causing an inverse relationship in our last example we have no correlation meaning that the points are plotted on the X and Y axes but they don't follow the same Trend or have any correlation as you can see from our graph we're unable to draw any kind of line through any of these plotted points to show a relationship among them this shows that there is no relationship between these points after as they're scattered all throughout our graph a study was conducted to explore the relationship between a number of hours spent studying per week and the resulting GPA of college students the data collected from a sample of students is presented in our table that we have right here on the right hand side of our screen based on the data provided what kind of correlation exists between the number of hours studied per week and the GPA is it a positive correlation a b a negative correlation C no no correlation or D cannot be determined from the data provided and the correct answer is a we have a positive correlation the table shows that the number of hours studied per week which is our X increases and the GPA which is our y also increases this upward Trend suggests that we have a positive correlation indicating that students who spend more time studying tend to have higher GPA so this next question can tend to be a little bit tricky so hang tight with me a fitness coach is analyzing the relationship between the number of calories consumed daily that's our X and the weight change Y in kilograms observed over a month in clients the following data is recorded in a table what does the data suggest about the co-variance between daily caloric intake and weight change do we have a positive covariance a a negative covariance B C zero covariance or D the data is insufficient to determine covariance and the correct answer is B we have a negative co-variance so the table illustrates that as the daily calorie intake X increases the number change y shift decreases this pattern implies that the negative covariance between caloric intake and weight change indicates that a lower calorie intake is associated with less weight gain and more weight loss lastly we're going to highlight the concept of proportionality which fundamentally comes in two flavors direct and inverse proportionality let's start with direct proportionality in this scenario we use variables Y and X like we have always so far to describe a situation where y increases or decreases in tandem with X mathematically we express this relationship as Y is equal to KX where K represents proportionality constant this constant of K is crucial as determines the specific relationship between Y and X ensuring that any change in y is going to be directly linked to the proportional change in X so what does this mean in real world scenarios so if you consider the relationship between the Distance by a car and the time it takes to travel that distance is a constant speed as the time increases the distance traveled also is going to increase if a car is traveling at a steady rate of 60 MPH the distance cover is directly proportional to the time traveled now let's pin a little bit to talk about inverse proportionality in contrast to direct proportionality when we say that Y is inversely proportional to X we're describing a situation whereas y increases X is going to decrease and vice versa the mathematical representation for indirect or inverse proportionality relationships is y is equal to K overx where K remains our proportionality constant however in this Formula K is positioned above X and a fraction signifying the inverse relationship the setup implies that X as it becomes larger Y is going to get smaller and the opposite of true when X decreases a real ball example of this is a relationship between the amount of time taken to complete a task and the amount of workers assigned to that task if more workers are assigned to that task the time taken to complete the task is going to decrease and if fewer workers are assigned to that task then the time taken is going to increase this is a great example of inverse variation where the increase in the number of workers tends to decrease the amount of time that it takes to complete a task graphs representing direct and inverse relationships exhibit distinct characteristics based on the nature of the relationship between the two variables as we discuss in a direct relationship as one variable increases the the other variable is going to increase at a constant rate this is reflected in our graph as a straight line if you plot y or dependent variable against X or independent variable for a direct relationship Y is equal to K * X the line will pass through the origin 0 0 the slope of this line is determined by that constant K and the line will Ascend to the right if K is positive indicating that y increases and X is also going to increase in an inverse relationship as one variable decreases the other variable is going to increase this type of relationship is captured by our equation like we talked about before is y is equal to K overx and the graph is going to look significantly different from a direct relationship graph instead of seeing that straight line that we've become accustomed to the graph is actually going to feature a curve line that approaches the axes but it's actually never going to touch them as X increases the Y value is going to decrease approaching zero but never actually reaching it reflecting that inverse relationship between the two variables a Biology experiment investigates the growth of a plant species under different light intensities the table below shows the height of the plants y after 30 days of various light intensities X based on the data provided which statement best describes the relationship between light intensity and plant height is it a as light intens density increases plant height decreases indicating an inverse relationship is it B there is no clear relationship between light intensity in plant height is it c as light intensity increases plant height increases at a constant rate suggesting a direct relationship or is it D plant height changes unpredictably with changes of light intensity and the correct answer is C the table shows a consistent pattern where we double the light intensity is going to result in doubling that plant's height this proportional increase indicates a direct relationship where the plant height is directly proportional to our light intensity a physics study examines the cooling time of a hot beverage as a function of the amount of cream that was added the table shows the cooling time why for the beverage to reach a drinkable temperature for various amounts of cream added x what does the data suggests the relationship between the amount of cream added and the cooling time of the beverage is it a adding more cream significantly increases the cooling time showing a direct relationship is it B the cooling time decreases as more cream is added illustrating an inverse relationship is it C there's no significant change in cooling time with varying amounts of cream indicating no relationship or is it D the relationship between the amount of cream added and the cooling time is inconsistent and the correct answer is B the data indicates that as the volume of cream increases the cooling time of the beverage to reach drinkable temperature decreases this temperature suggests that the cooling time is inversely related to the amount of cream that's added as an increase in one leads to a decrease in the other let's start off by exploring the concepts of perimeter area and volume as a whole to quantify a one-dimensional object we use a one-dimensional measure known as length for instance the length of a given line might be precisely 1 cmet a standard unit of measure when it comes to length if we take that same line and we extend it perpendicularly by 1 cm this is going to transform into a two-dimensional shape specifically a square two-dimensional shapes are Quantified by a two-dimensional measure referred to as area since our initial line of 1 cm in length was extended by 1 cm to the right the resulting Square covers an area of 1 square cm a typical unit of measure when we are trying to figure out area now let's consider extending that two-dimensional shape up from its plane of 1 cm this new action creates a three-dimensional shape known as a cube to quantify a three-dimensional object such as this we're going to utilize the three-dimensional measure known as volume when we talk about volume we're quantifying the amount of three-dimensional space inside our object in which it fills so what is the volume of a cube given that our volume of cube was formed by extending a square cm up into a third dimensional shape by 1 cm the volume of this cube is defined as 1 cubic cm a standard unit of measure when we are measuring volume to make it easier area is measured as square units and volume in cubic units Square units results from multiplying two one-dimensional units cimer * cimer often expressed in exponent notation as CM squar or CM raised to the second power conversely cubic units arise from the product of three one-dimensional units cimer * cimer * cimer this is abbreviated in its exponent notation as cm cubed or CM raised to the Third let's break each one of these three concepts down further and apply them based on the guidelines of the te's so let's start by diving into the fundamentals of geometry when it comes to Perimeter it refers to the total distance or length around a shape now you might wonder what does it mean to measure the distance around a shape well distance or length is the concept that exists in one dimension and is quantifiable in units such as centimet inches or miles this means that perimeter is also a one-dimensional measure Quantified with units of length so when we talk about a shape's perimeter we're not just saying 10 we're saying 10 cm or it's not just 25 it's 25 miles the specifity of units is crucial when we're discussing perimeter but then what do you mean by around the shape what we mean by this is It's the absolute shortest path around a shape this is the distance that you cover if you were to trace a line around the shape border or its Edge a helpful way to understand perimeter is to imagine yourself walking along the edge of a shape such as this Pentagon visualize yourself beginning at one end of the Pentagon and walking all along the edge of the shape until you reach back to your starting point that complete distance that you cover is equal to the perimeter of the shape in our example we know that every side of the Pentagon measures 10 m transversing all five sides of this shape is going to equal a total distance of 50 m another insightful method to grasp the concept of perimeter is to imagine taking the shape like our Pentagon and separating it from its Corners you can unfold that shape into a straight line that length from the start of your line to the end of your line is your perimeter calculating the perimeter of polygons which are shapes that only have straight edges is pretty straightforward you simply sum the lengths of all the sides and the result of the polygon is the perimeter let's practice some examples of how this actually works we're going to start with a right triangle the triangle sides are 3 cm 4 cm and 5 cm to determine the triangle's perimeter we need to add the lengths of each individual side so we calculate 3 + 4 + 5 total gives us 12 CM and that is our right triangle's perimeter in our next example we have a hexagon a polygon with six equal sides this hexagon is described is regular meaning that every single side is the exact same length this actually simplifies our task because only one size length is actually given to us which is 4 cm we can infer that every single side along this hexagon is going to be 4 cm long instead of adding each side individually as we did in our previous example we can actually take a shortcut and use multiplication since all the sides are of equal length that's because multiplication is really just rep reped addition to find the total perimeter we would simply multiply the number of sides that would be six by our length of the one side that we know which is 4 cm 4 * 6 is going to give us 24 cm which is the overall perimeter of our hexagon let's do a more complicated example this time we're looking at a polygon with six sides again but this one isn't regular that means that the length of each side is going to vary this example is a bit more challenging because the DI only reveals the lengths of four sides leaving us with two sides that are unknown encountering incomplete information is very common in mathematics and also on the te's the strategy here is to leverage the information that we do have to deduce the information that we're missing here is how we're going to approach this pay attention to the two vertical sides that we have here on our screen 4 in and 6 in now imagine that you could slide these two vertical sides across the opposite side and both of those sides are going to equal the length of that missing side so as we know we have four and we have six so when you add these together we actually get a total of 10 we can apply this similar logic to our horizontal sides as well by shifting these lengths that we have available to us of 10 in and 5 in so if we slide those down we have 10 and we have five we're going to add those two totals together in order to get the horizontal side that we're missing so our missing side is 15 in by utilizing the lengths we knew we managed to deduce the lengths that were unknown to us with the knowledge in Hand of every s's length now we can actually calculate our perimeter so 10 which is our number up here + 4 + 5 + 6 + 15 + 10 gives us a total of 50 in which is the perimeter of our irregular polygon now let's step it up a notch by talking about how we calculate circumference and area of circles it's it's crucial that you familiarize yourself with the formulas for both of these as they are going to be fundamentally covered when it comes to the t's the formula to calculate circumference of a circle is going to be circumference is equal to pi multiplied by the diameter and Shand we have C is equal to Pi * D where C is our circumference and D is our diameter so you might be asking yourself what the heck is a diameter right so if we were to take a circle and we were to draw a line through the Center of that Circle that line from one end of the circle to the other end of the circle that measurement is considered our diameter an easy sentence to remember this equation is cherry pies delicious where C in our Cherry is equal to our circumference Pi is of course equal to Pi and the D and delicious is equal to diameter now how do we calculate a circle's area so the area which is the region enclosed within our Circle boundaries is area = piun ultied radius squared so what exactly is a radius so our radius is actually half of our diameter so as we know we drew a line down the center of our Circle in order to get a diameter we're just going to take the diameter and divide it by the number two in order to get our radius which is half the diameter an easy sentence to remember this equation is apple pies r two where the A and apple is our area the pi is of course our PI the r and our R is our radius and the two is going to be the square of our equation a crucial point to remember is that R squar does not mean 2 * R this is frequently misunderstood among te test takers who first learned to calculate area of a circle when we're examining both formulas closely their similarities become apparent each formula involves multiplying Pi by a specific Circle measurement to determine either circumference or area for circumference the calculation involves pi multiplied by the diameter while for Circle it's pi multiplied the radius squared how do we distinguish between a diameter and a radius well as we know the diameter is the line that we draw across the center of a circle that divides a circle into two equal halves that line measurement from one end of the circle to the other end of the circle is our diameter sometimes the te's is going to give you questions that's only going to give you the radius well in order for us to figure out the diameter base on the radius is we're going to take the radius and multiply it by two because 2 times the radius is going to give us the full length of our diameter so when we're talking about area we're going to do something a little bit different when it comes to our radius we're going to actually Square our radius and when we talk about squaring it is not the same thing as 2 * R when we Square we actually multiply the number by itself so if our radius was 2 2 * 2 is going to equal 4 if our radius was 5 5 * * 5 is going to equal 25 so that is the big key differences between diameter and radius when we're trying to figure out circumference and area let's take a look at an example of how this will actually be applied on the t's so starting with our first example we're given the exact same Circle and our circle is given us a radius of eight so when it comes to considering the circumference of a circle we're actually going to multiply the radius by two like we discussed before so in this case 2 * 8 is equal to 16 so we're going to plug that into our equation Pi is just a fancy way of saying 3.14 when it comes to your t's so you can automatically manipulate the pi to be 3.14 so we're going to multiply that by 16 and that is going to give us our circumference of this circle as being 50.2 4 M next we're going to figure out the area of the exact same Circle so again we know that the radius is 8 but instead of multiplying it by two like we did in circumference this time we're going to square it so 8 * 8 gives us 64 M squared we're going to multiply that by pi which is 3.14 and that is going to give us our total area of 200.96 M squared now that we have a better understanding of one-dimensional values of perimeter and two-dimensional values of area when it comes to a circle let's explore other shapes when it comes to polygons and area so to grasp this concept of area we imagine that that one meter line and length is going to be dragged across perpendicularly to create another 1 meter line when we do this we create a two-dimensional shape in this case it's a square so when we talk about area we're trying to figure out the measurement within this shaded region when it comes to our different polygons so we're going to start by trying to figure out the measurement for three shapes squares rectangles and triangles to determine the area a square or a rectangle we simply multiply the dimensions of its side typically we're only measuring two sides which would be our length times our width so if we have in our first example we have a square that has two sides of 2 cm each so we're going to multiply length * width 2 * 2 gives us 4 cm squar so the area of this square is equal to 4 cm squ just like with our Square we're going to use the same formula when it comes to our rectangle so in this case we have a length of 4 cm and we have a width of 2 cm so we're just going to multiply length time width 4 * 2 is going to give us 8 cm squared which is going to give us the overall area when we're trying to consider our squares and our rectangles now how do we figure out the area of a triangle so we're going to begin with a rectangle once more and this time we have a dimension of 3 cm by 4 cm using our previous discussed formula the area of a rectangle is calculated as 3 * 4 is equal to 12 square cm now let's imagine that we're slicing this rectangle in half diagonally from one corner to the opposite corner this action creates two triangles each occupying half of the area of our original rectangle given that the rectangle's area is 12 square cm the area of each triangle would be half of the overall area that we saw in our rectangle meaning that each triangle is 6 square cm this Revelation leads us to the formula for the area of a triangle being essentially half of the area of a rectangle so instead of area equals length time width the formula of triangle modifies to area equal 12 of length * width however there's a slight adjustment that we need to note for our triangle dimensions we actually refer to them as base time he height instead of length time width here's the reason the labels length time width are suitable for a right triangle as a right triangle forms precisely 1/2 of our rectangle however these terms don't fit as well with other triangle types like we see with acute and obtuse triangles where it's less clear which side should be labeled what in the case of these triangles we adopt a different approach we select one of the three sides as our base the choice of the base is really up to you and often the base is predetermined in mathematical problems anyways and once we select That Base we envision placing that triangle on the ground with its base being horizontal now we have to figure out what is going to be our height we do this by finding our triangle's Peak also known as the Apex of our triangle once we find that we're going to highlight that and draw a little line down until we hit the base that is going to be the height of our triangle in certain cases such as with acute triangles the line is going to fall within our triangle boundaries but for OB triangles this height line extends outside of our triangle's perimeter regardless of its position the formula to calculate the area of a triangle is going to remain the same consistently area equals 1 12 of our base multiplied by our height let's take a look at a couple different examples when it comes to triangles we're going to start off with our acute triangle so as we know our acute triangle formula is going to be 1 12 time our base time our height because as we know any triangle is basically 1/2 of a rectangle so we have our base is equal to 5 m and our height is equal to 8 m we're just going to go ahead and plug in play we've got 5 * 8 is going to be equal to 40 40 m squar of course because we're dealing with area and then we are just going to divide that overall number by two so the area of of our acute triangle is going to be 20 M squared now let's take a look at our obtuse triangle so again using the same formula we have a base of four and we have a height of seven so we're going to plug those numbers into our equation 4 * 7 is equal to 28 in squared and we're just going to take that number and divide it by two so the area of our obtuse triangle is equal to 14 in squared now let's talk about parallelograms and trapezoids because they are most commonly tested on your te's starting with a parallelogram it kind of resembles a rectangle but it lacks those four right angles typically that you would see when it comes to our rectangles to determine a rectangle's area we typically use the formula area equals length times width we apply a similar approach when we're looking at parallelograms so let's imagine taking a segment of our parallelogram and sliding it across to the other side of our shape fitting it in like the perfect puzzle piece so if we take down here one part of our parallelogram which is shaped like a triangle and then add our triangle here down to the end of our parallelogram this rearrangement is going to transform our parallelogram into a rectangle without altering its Dimensions therefore we can still utilize the formula area equals base time height to calculate the area of our parallelogram following those same principles that we see with our rectangle so in this example we have a base of five and a height of two we're just going to go ahead and plug those numbers into our equation and that is going to give us the overall area of our parallelogram is 5 * 2 is equal to 10 ft squared now finishing off this area section with a trapezoid you might have noticed that the formula for calculating this trapezoid Bears a remarkable resemblance when it comes to triangles the area of a triangle can be expressed as base time height / 2 however for a trapezoid the formula adapts to include both bases base one plus base 2 * our height is all divided by 2 let's explore why that is a trapezoid can effectively be segmented into two triangles by drawing a diagonal line from the bottom of one end to the top of the other end depicted by the red line that I just drew for you this division splits the trapezoid into two triangles the bases of these triangles align with the length of the trapezoids top and bottom edges and their assured height is identical to the trapezoid vertical Heights in order to find the area of a trapezoid we could calculate the area of each triangle separately and then sum them up together but to save ourselves from that frustration this step can be streamlined by simply adding both bases together base one and base two multiplying that outcome by its height and then dividing it by two so taking a look at our example that we have here we know that we have a base of eight a base of four and our height overall is six so we're just going to add our two bases together multiply it by its height and divide it by two so 8 + 4 ultip by 6 gives us 72 and then we're just going to divide that by two to give us the overall area of our trapezoid as being area is equal to 36 in squared so let's take a look at one more example of a complex shape so here we have a heart and we're trying to figure out what the overall area of that heart is so Step One is we want to identify the shapes and formulas that we're going to use in order to find the area of our heart so we have a couple different shapes we have a square which is this area that's right here and then we have a circle formula which are these areas right here step two is we're going to start with our easier shape we're going to start with our square and figure out the area for that so we know that the square area is length times width so we have a length of four and we also have a width of four so we're going to multiply 4 * 4 and that's going to give us the overall area of our Square as being 16 cm squared so step three is we want to calculate the next shapes formula so as we know we have two semicircles along the edges of our Square so we need to figure out what is going to be the area of our semicircles and all we do is we take our equation when it comes to area of a circle and then we divide it by two in order to get the overall area of a semicircle so starting with our equation we have apple Pi's R2 which is a is equal to Pi R 2 that's how we're going to figure out the area so in this particular case we know we have a diameter of four and we know that in order for us to determine the radius is we have to divide that diameter by two so we're going to divide by two and that is going to give us two as our radius so now we just need to plug it in so a is equal to < * 2^ 2 we know that 2^ 2 is equal to 4 4 * 3.14 which is the equation of a pi is equal to 12.56 now we need to divide this number by two because we don't have a full circle we only need half the circle so as we divide by two that is going to give us overall area of our semicircle as being 6.28 cm squared and finally in step four we can add all of our areas together we have our two semicircles so each one was 6.28 this is semicircle 1 semicircle 2 and then our last shape number three which we know was our square is equal to 16 if we add 6.28 + 6.28 + 16 is going to give us the overall area of our irregular shape as being 28.57 cm squared now let's talk about volume as we discussed before volume quantifies the amount of three-dimensional space an object fills we create three-dimensional objects by extending a two-dimensional shape up in height from its plane to create a three-dimensional object starting with square and rectangle projecting these shapes into the third dimension yields a three-dimensional figure known as a square prism or cube and a rectangular prism similarly if we extend a triangle into its Third Dimension it's going to produce a triangular prism to determine the volume you can simply calculate the area of an initial two-dimensional shape and then multiply that by the height in which you projected it so if you can understand the area when it comes to squares and rectangles we know that volume's going to be really easy because the only additional thing that we are measuring here is the height so in this case we have length * width time height will give us the volume of square and rectangles in our example we have 4 * 3 * 10 is going to give us the overall volume of 120 cm cubed next let's calculate the volume of a triangular prism just like with the area of the triangle we multiply half of our base times our height but again we've added an additional unit of measure which is our length in this situation so if you can remember the area of a triangle it's going to be really easy for you to remember the area when it comes to a triangular prism because the only thing that we are adding is that additional unit of measurement which is our length so in this case we have our base that is 10 and our height that is eight length that is 50 so we are going to multiply 10 by 8 by 50 and we're going to divide that by 2 just like we would do with area but this time our volume using our new equation is equal to 2,000 in cubed another easy three-dimensional shape to remember the formula for is going to be our cylinder so again the base of our cylinder is just a circle so how do we find the area of a circle well Apple Pi's R2 right so area is equal to Pi R 2 just like we see here at the beginning of our formula but because we made this into a three-dimensional shape we're adding or I should say multiplying one additional unit of measurement and that is our height so we're going to take the formula that we have for an area of a circle and we're going to multiply that by its height so in our example here we have a radius of two so we just plug that into our equation and we have a height of 10 10 m so 2 2 is equal to 4 M 4 * 10 is equal to 40 * pi which is again 3.14 is going to give us the overall volume of our cylinder as being 125.638 's volume is precisely 1ir that of the cylinder to visualize this imagine placing a cone inside a cylinder where both the radius and the height are the same this setup reveals that the cone occupies oneir of the cylinder's volume this phenomenon is why when determining the volume of a cone we apply that same formula that we have for a cylinder but we are going to divide the result by three this adjustment to the formula accounts for the cone's proportional volume relative to that of a cylinder so taking a look at our example we have volume is equal to < * R 2 * height / 3 so we know that our radius is five and our height is 12 so we're just going to plug in our numbers so this is going to give us > * 5^ 2 * 12 is ided 3 5^ 2 is equal to 25 and we're going to multiply 25 by 12 * pi and that is going going to give us 300 Pi all we're going to do now is we're going to divide that total number by three and that is either going to give us 100 Pi or 314 in cubed remember the t's might test you on both of these examples as with cones the same concept is also going to apply when it comes to rectangular pyramid shapes the base of our shape is just a rectangle and the volume of a rectangle pyramid is precisely 13 of a rectangular prism when we're trying to determine the volume we apply that same formula for a rectangular prism and we're going to divide the total outcome by three so taking a look at our equation we have volume is equal to length * width time height / 3 we have a length of 8 a width of seven and a height of 11 so we're just going to plug in our numbers 8 * 7 is equal to 56 * 11 is equal to 66 16 and then we're just going to divide that overall outcome by three which is going to give us a total volume of 2533 in cubed now finally we have the volume of a sphere it's difficult to explain the volume when it comes to a sphere without getting into a whole bunch of calculus and how this formula is determined to save ourselves some time and frustration I use a sentence to remember this equation four spheres we find the space inside with 4/3 piun * R cubed applied so looking at our formula we have volume is equal to 4/3 * < * R Cub remember anytime that we're dealing with volume when it comes to Circle we're going to cube the radius any times we're dealing with area when it comes to circles we're going to square the radius so again we know that we have a radius of 10 in so we're just going to go ahead and start plugging in so we have 4/3 * pi * 10 cubed we know that 10 cubed is 1,000 we are going to times that by 4/3 and that is going to give us either two answers depending on how the t's list it it's either going to be 13333 Pi or 4,186 point6 throughout the world various units of measurements are used across many different countries however the most widely adopted system globally is known as the metric system in fact there's only three countries around the world that don't use the metric system and guess what the United States is one of them starting with the standard system also known as the United States customary system includes units such as length you're going to see inches feet yards and Miles we've got weight where we use ounces and pounds and we have capacity where we use teaspoon tablespoon cups pints quarts and gallons in comparison we have the metric system also known as the International System of unit SI the system is actually decimal based meaning that it's based on powers of 10 which simplifies calculations and conversions between units in the metric system you're going to see length in which they use the unit of meters then we have weight which is the unit of grams and we have capacity which uses the unit of liters overall the metric system is much simpler than the standard system understanding both systems though is going to be beneficial when you're taking your teas especially in the field of science as you're going to be using these conversions quite a bit when you enter Healthcare we're going to begin by converting customary units of length specifically working on feet inches yards and Miles so we have about eight conversions in front of us that we're going to tackle together aiming to solidify our understanding of these measurements so let's get started on our first example you're going to notice up here at the top of your screen I gave you like a little cheat sheet of the break sound of the most commonly seen units of measure conversions when it comes to the standard system that the tease is going to test you on so the first one as we know 1T is equal to 12 in 1 yard is equal to 3T or 36 in and 1 mile is equal to 1,760 yard or 5,280 ft or 63,360 in so let's break down our examples to see what that'll look like on your test so for this first problem we need to consider that 1T is equal to 12 in like we have up here at the top of our screen so when you think about the length of a ruler right a ruler is 12 in it's about 1T when we have 2 feet it's logic that we would follow by just multiplying that 2 feet by 12 in so 2T * 12 in is going to give us 24 so the correct answer to this question is 24 in so this is a really important concept for you to know whenever we're converting from a larger unit like feet to a smaller unit like inches the operation that we're going to use is multiplication and if we're going the other way around where we have a smaller unit going to a bigger unit we're going to use division you have to remember this is going to be crucial when you're taking your aits and you're navigating between units because even in the metric system you're going to use these same principles so our next example is 48 in to blank feet so we're going from a smaller unit to a bigger unit so we are going to use division so we know that 1T is equal to 12 in so we're just going to divide our 48 in by 12 so 48 / by 12 gives us 4T which is the correct answer to problem two our next example is three yards to how many feet so again we're going from a larger unit to a smaller unit so we're going to multiply we know that 1 yard is equal to 3 ft so we are we going to multiply by three and that is going to give us our correct answer of 3 yards is equal to 9 ft so how many inches are in 2 miles so as we can see up here we know that 1 mile is equal to 63,360 in so 1 mile is equal to 63,360 in so we know that we have 2 miles in our example so we're just going to multiply that two by our 63,360 and that is going to give us 1,260 720 in so how many feet are in 26.2 miles so you might recognize this because this is actually what we run when we run a marathon right so how many feet are we actually running so as we know 1 mile is equal to 5,280 ft so what we're going to do is we're going from a larger number to a smaller number so we're going to multiply so we're going to multiply that 26.2 miles by our 5,280 ft and that is going to give us a total of 138,00 336 and lastly we're going from feet to yards so we have 900 ft is equal to how many yards so we know that 1 yard is equal to 3 ft so now we just need to do our conversion remember we're going from a smaller unit to a bigger unit right so we're going to divide so we're going to divide our 900 ft by 3 and that is going to give us our answer of 900 ft is equal to 300 y so next up we're going to tackle weight when it comes to the standard system so the number one thing that the teas always loves to test you on is pounds to ounces so as we know one pound is equal to 16 oz so this is going to be the key one for you to remember when you're taking that math portion so our first example says 3 lb is equal to how many ounces so as we know 1 lb is equal to 16 we're going from a larger number to a smaller number which means we're going to multiply so we're going to multiply 3 lb by our 16 and that is going to give us our answer of 48 oz our next example says we have a 5 lb bag of flour and we need to convert that into ounces so again we're going to do the exact same thing as we did in example one and we're going to multiply our 5 by 16 5 * 16 is equal to 80 and that gives us our correct answer of 80 O So for our last two problems we're going from a smaller unit to a larger unit so when we go from small to large we actually divide so in this case we're going to divide each one by 16 in order to get our answer from ounces to pounds so our first one is 64 o is equal to how many pound well 64 / 16 is equal to 4 and our last one 96 oz is how many pound again 96 / 16 gives us 6 so for each answer we know 64 4 oz is equal to 4 lb and 96 oz is equal to 6 lb let's explore a pneumonic to recall the US customary units used for measuring liquid volume or capacity I like to refer to this as Big G or superg to start with we're going to draw a large capital letter G this is going to symbolize our gallon this G serves as the foundation for understanding the hierarchy of liquid volume measured when we use us customary systems next up we're going to add four Q's inside of our big G symbolizing that gallons equals four quarts for each quart represented by Q we're going to introduce two PS indicating that each quart is going to consist of two pints and lastly each pint is comprised of two cups so we're just going to put our little C's inside of our big P's to represent that we know that there's two cups in each pint this means that 1 gallon is equal to four quarts is equal to 8 pints and is equal to 16 cups that is the easiest way that you're going to remember how to break down liquid volume when you're taking the teas another important concept that doesn't quite fit inside of this is tablespoons to teaspoons so this one you're just going to have to memorize a little bit but as we know 1 tablespoon is equal to three teaspoon so let's do some practice questions to figure out how we're going to solve these capacity problems all right let's start our conversion so as we know our first question says three gallons to how many quarts so we know that 1 gallon is equal to four quarts so we're going from a larger number to a smaller number we're going to multiply so we're going to multiply our three gallons by four and that is going to give us our correct answer of 12 quarts our next question says eight cups is equal to how many pintes so as we know between cups and Pints we have for every eight pints we have 16 cups so that's basically just to make it a little bit easier is 2 cups per one pint so what are we going to do whenever we're going from a smaller number to a larger number we're going to divide right so we're going to divide our eight cups by two and that is going to give us our correct answer of four pints so how many teaspoons are in four tablespoons so as we know 1 tablespoon is equal to three tpoon so again we're going from a larger number to a smaller number we are going to multiply so we are going to multiply our 4 tablespoons by three and that is going to give us our correct answer of 4 tablespoon is equal to 12 tpoon and then lastly we have five pints is equal to how many cups so as we talked about before in example two we know that one pint is equal to two cups so what are we going to do when we go from a larger number to a smaller number we're going to multiply so we are going to multiply our five pints by two and that is going to give us our correct answer of 10 cups so I put a little cheat sheet up here on your screen because these are the most common conversions that you're going to see when it comes to the standard measurements of length weight and capacity so just become very accustomed to these sometimes the tease is actually going to give you the formulas sometimes they won't so it's good for you to have a general understanding of how to calculate between them in case you encounter these questions on the te's let's talk about my favorite system the metric system metric conversion are going to be key for you to understand when it comes not only to the aits but also healthc care in general and the pneumonic that I want you to use to remember this is King Henry doesn't usually drink cold milk by using this phrase it's going to help you remember the order of metric prefixes when it comes to kilo all the way to Millie relative to the unit base which could be grams meters and liters so at the heart of our pneumonic we symbolize the word usually with units this is where your general units are going to lie whether you were doing grams lers or meters and around this core are prefixes that indicate whether we're dealing with units that are larger or smaller than our base units so as we move to the right towards cold milk we're going to encounter smaller units Desi CTI Millie each representing a factor of 10 smaller than our base conversely as we move leftward from our units we're going to encounter the high numbers our King Henry's right our Deca hecto and our kilo each representing a tenfold increase over our base unit so a kilo therefore signifies a quantity of a thousand times larger than our base units so this framework is going to categorize and quantify the relationships between different metric units just like we did in our decimal placeholders video by the way if you haven't done so already make sure you go back and watch that to understand what we're about to do right now we're going going to move the decimal either left or right depending on the value that we are converting from so here's a couple of examples to help us with this concept so we start with our first example which is 2,500 millit 2 kilm so how do we do this so what we're going to do is we're going to move our decimal place so it's kind of not really given here but we can just assume that the decimal place is right after our whole number which is 2,500 so what do we do we know we're going from millimet to kilometers right so we're going to move our decimal place 1 2 3 4 5 6 six times to the left so that's ultimately going to give us the correct answer of 0.25 cuz if we did this just so that everybody can see here's our decimal we're going 1 2 3 4 5 6 decimal goes here we add our zeros where we didn't have them before and that gives gives us our correct answer our next question we're going from liters to kiloliters so again here is our base unit where we find liters and we're going to move our decimal place 1 2 three times to the left so I'm going to write my number out here at the bottom our decimal places right here we're going to go 1 2 three times to the left so as we know 750,000 lers is equal to 750 Kil so in our next question we're going in the opposite direction so we're going from kilometers to ctim so we are moving from here all the way down here so we're going to move our decimal to the right 1 2 3 4 five times so writing out our number 42.195 we're going to move it to the right 1 2 3 4 five times and that is going to give us our correct answer of 4,219 500 cm and for our last last example we're going back the other way so we're going from milligram to kilogram so we're going to move our decimal 1 2 3 4 5 6 so writing out our number 250 migr we're going to the left 1 2 3 4 five 6 we're going to add our zeros that we didn't have before and that is going to give us our correct answer of 0. to5 kilog so metric conversions really are quite simple when you you think about it it's just moving the decimal place either left or right but now we've got to figure out how we going to convert between the two systems that is one of the most commonly missed types of questions that we see on the t's is taking measurements from the standard system and converting them to the metric system so over here on the right side of your screen again I gave you a little cheat sheet of the most common ones that you're going to see on the atits these are the good ones to remember in order to pass these types of questions so our first question is 50 lb is equal to how many kilog so this is going to be huge when it comes to healthcare cuz we are constantly converting between pounds and kilogram so make sure you commit this one to memory so 1 kilogram is actually equal to 2.2 lbs so we're going from pounds to kilogram we're going to divide by 2.2 and that is going to give us our correct answer of 22.73 kg next up we're moving from inches to CM so as we know 1 in is equal to 2.54 CM so again we want to multiply this time right we're going from inches to centim so 20 * 2.54 is going to give us our correct answer of 50.8 CM our next conversion is 5 miles to kilom so 1 mile is actually equal to 1.6 km so we're going from miles to kilm so we're going to multiply and we're going to multiply by 1.6 which is going to give us our correct answer of 5 miles is equal to 8 km so how many meters is in 30 ft so 1 M isal to 3.28 Ft so again we're going from feet to meter so we're going to divide we're going to divide by a 3.28 and that is going to give us our correct answer of 9 .15 M and our last question is 100 G is equal to how many ounces so 1 o is equal to 28.35 G so again we're going from gr to ounces we're going to divide we're going to divide by 28.35 and that is going to give us our correct answer of 3.53 Oz that was a lot that's everything you're going to need to know in order to Ace that math portion of the exam if you have any questions make sure that you leave them down below I love answering your questions head over to nurse chunk store.com where there is a ton of additional resources available to you and as always I'll catch you in the next video bye