In this video, we're going to focus on solving common ACT math questions. Now, for each question that you see, pause the video, work on the example, and then unpause it to see if you have the right answer. So let's go ahead and begin.
So if 3x over 4 plus 5 is equal to 11, what is the value of 5x over 2 minus 16? If we could find the value of x in the first equation, then we could find the value of this expression. So let's go ahead and solve. So the first thing that we need to do in this problem is subtract both sides by 5. 11 minus 5 is equal to 6. Now, in the next step, we need to multiply both sides by 4. This will get rid of the fraction on the left. 3 fourths times 4 is 3. The 4's will cancel.
And 6 times 4 is 24. Next, let's divide both sides by 3. 24 divided by 3 is 8. Now that we have the value of x, we can plug it into this expression. So let's replace x with 8. So what is 5 halves of 8? 5 times 8 is 40. And 40 divided by 2 is equal to 20. 20 minus 6 is 8. 16 is 4, so 4 represents the value of the expression.
Therefore answer choice D is the right answer. Number 2. What is the sum of the solutions of the equation? x squared plus 4x minus 45 equals 0. So what can we do in this problem?
Well first, we need to find the value of x in this equation. So we have a trinomial with a leading coefficient of 1. So we need to factor the equation. So what two numbers multiply to negative 45 but add to the middle term 4? What are those two numbers?
Well, we know that 5 times 9 is 45, but it has to be negative 45. So we can use negative 5 and positive 9, or positive 5 and negative 9. Now we need to pick the pair that adds up to positive 4. So, negative 5 and plus 9 adds up to positive 4. So, to factor it, it's going to be x minus 5 times x plus 9. So, now what we're going to do is set each factor equal to 0. And then solve. So x is equal to 5 and x is equal to 9. I mean negative 9. But those are the two solutions. Now the problem wants to know what is the sum of the solutions of the equation. so if we were to add 5 and negative 9 this will give us a sum of negative 4 so negative 4 is the final answer of the problem so B is the right answer number three if f of x is equal to 5x cubed minus 2x plus 7 then f of 2 is equal to what? Well all we got to do in this problem to find the answer is replace x with 2 and simply find the value of the expression.
So it's going to be 5 times 2 raised to the third power minus 2 times 2 and and then plus 7. So what's 2 raised to the third power? 2 raised to the third power is basically you're multiplying three 2's together. 2 times 2 times 2 is 8. So this is 5 times 8. Negative 2 times... times 2 is 4. 5 times 8 is 40. Negative 4 plus 7, that's positive 3. And 40 plus 3, well that's 43. And so this is the answer. So E is the right answer.
Number 4. The area of a square is 36. What is the perimeter? Well let's draw a picture. So here's a square and the area is 36. Now what is the formula for the area of a square? Well, if we call this x and x, all sides are the same of a square by the way. The area is simply length times width or x squared.
So 36 is equal to x squared. And if we take the square root of both sides, we can see that x is equal to 6. So each side of the square has a value of 6. So now that we know that, we can calculate the perimeter. The perimeter is basically the sum of all sides.
It's 2 times the left plus 2 times the width. And both the left and the width is equal to 6. So it's going to be 12 plus 12, which is 24. You could also do 6 times 4, because you're adding 6 4 times. And that will also give you the perimeter.
So C is the right answer in this problem. Number 5. If f of x is equal to 3x squared minus 5x plus 8, and f of x is equal to 36, then x is equal to what value? Number 6. So in this problem, we need to replace f of x with 36, and then we could solve and find the value of x. So let's go ahead and do that. So what's the first thing that you would do at this point in order to find the value of x?
The best thing we could do right now is subtract 36 from both sides. So, 0 is equal to 3x squared minus 5x minus 28. At this point, you can use the quadratic formula, or you can factor the equation to find the value of x. Let's begin by factoring.
Let's get rid of this. So we have a trinomial with a leading coefficient that is not 1. In order to factor it, we need to multiply 3 and negative 28. 3 times negative 28 is equal to negative 84. Now we need to find two numbers that multiply to negative 84, but add to the middle coefficient, negative 5. If we make a list, negative 84 divided by 1 is negative 84. If we divide it by 2, negative 4 is negative 4. 42 if we divide it by 3 that's negative 28 if we divide it by 4 negative 21 none of these differ by 5 5 does it go into 84 we could divide it by 6 let's see 84 divided by 6 that's 14 and if we divide it by 7 that's 12 now these two differ by negative 5. If we add 7 and negative 12, that is negative 5. So that's the pair of numbers that we need. Now what we're going to do at this point is we're going to replace negative 5x with negative 12x plus 7x.
and then we could factor by grouping. In the first two terms, take out the GCF. The greatest common factor is 3x.
3x squared divided by 3x is x, and negative 12x divided by 3x is equal to negative 4. Now do the same thing for the last two terms. The greatest common factor is 7. 7x divided by 7 is x. Negative 28 divided by 7, that's negative 4. Now we can factor out the x minus 4. So if we take out x minus 4 from this term, we're going to have 3x left over. And if we remove it from this term, we're going to have a 7 left over.
So now, if we set x minus 4 equal to 0, and 3x plus 7 equal to 0, we can see that x is equal to 4, and x is equal to negative 7 over 3. Now, negative 7 over 3 is not listed as an answer, but 4 is. So, answer choice D is the right answer. Now, let's get the same answer used in the quadratic equation. Because if you get to a point where you're not sure how to factor it, you may have to use that equation. So you have to make sure the equation is in the form ax squared plus bx plus c.
So we can see that a is equal to 3, b is equal to negative 5, and c is negative 28. So here's the formula. x is equal to negative b plus or minus the square root of b squared minus 4ac divided by 2a. So, b we said is negative 5. Don't forget about this negative sign in front of the b.
So, b squared, negative 5 times negative 5 is positive 25. Minus 4 times a, which is 3. And c is negative 28. Divided by 2a, or 2 times 3. Negative negative 5 is the same as positive 5. And if we multiply negative 4 by 3, that's negative 12. And negative 12 times negative 28, that's going to be positive 336. 2 times 3, that's 6. So next... Let's go ahead and add 336 plus 25. That's going to be 361. And the square root of 361 is 19. So we have 5 plus or minus 19 divided by 6. At this point, you can separate it into two expressions. the first one is going to be 5 plus 19 divided by 6 and the second one 5 minus 19 divided by 6 5 minus 19 is negative 14 and you can reduce negative for by dividing the top and bottom by 2. This will give us negative 7 over 3, which was one of the answers. Now to find the other one, we need to add 5 plus 19, which is 24. And 24 divided by 6 is 4. And this is the answer that we're looking for.
So D is the right answer. So now you have two ways to solve a quadratic equation. You can solve by factoring it, or by using quadratic formula.
Or sometimes you can even complete the square. But I wouldn't do that for this problem. Number 6. Which of the following equations represents the graph shown below?
Is it going to be A, B, C, D, or E? So the equation is in slope-intercept form. Y is equal to MX plus B.
The first thing we should look for is the y-intercept. which you could find it here. The y-intercept is negative 3, so that should be our b-value, which means that we could eliminate answer choice D and C.
Answer choice A, B, and E all have a y-intercept of negative 3. Now the next thing we need to look for is the slope, the number in front of x. So a quick way to calculate the slope is to use the rise over run method. As we travel from this point to this point, we need to go up 3 units and travel 4 units to the right.
So the slope, which is the rise over the run, it's 3 over 4. So therefore, answer choice B is the right answer. It has a slope of 3 over 4. Number 7. In triangle ABC, angle A equals 45 degrees. What is the area of the triangle? So first, let's start with the equation. The area of a triangle is 1 half base times height.
So this is the base, this is the height. Therefore, we need to find the lengths of segment AB and BC. If we could do that, then we could find the area of the triangle. Now we know the angle is 45 degrees, which means that this other angle must be 45, since this is 90. The three angles inside a right triangle must always add up to 1. to 180. 90 plus 45 plus 45 adds up to 180. Now because these two angles are the same, that means that these two sides are equivalent to each other.
So let's go ahead and call this side X and the other side is also X. Now we can use the Pythagorean Theorem to theorem in order to find the value of X. a squared plus b squared is equal to c squared.
a and b represents the legs of the triangle so a and b are both X. C is the hypotenuse which is across the red box. C is 10. X squared plus X squared is 2X squared.
10 times 10 is 100. If we divide both sides by 2, we can see that X squared is equal to 50. To find the value of X, we need to take the square root of both sides. So X is root 50. Now, we can plug in the values that we have. The base is equal to x and so is the height. So both the base and the height is equal to root 50. The square root of 50 times the square root of 50 is equal to the square root of 2500, which is 50. And half of 50 is 25. So therefore, the area of this triangle is 25 square units, which means that answer choice B is the answer.
Number 8. Sally's test scores in chemistry were 86, 93, 79, 82, and 90. What was her average test score? Whenever you wish to calculate the average, it's simply the sum of all the numbers divided by the number of numbers in the list. So let's add up all five numbers.
Let's find the sum. It's going to be 86 plus 93 plus 79 plus 82 plus 90. Now there's a total of five numbers, so we're going to divide the sum by five. 86 plus 93 plus 79 plus 82 plus 90, that's equal to 430. So now we need to divide 430 by 5. So this will give us an average of 86. So that's her average test score, which means that C is the right answer. Number 9. Sarah has taken 6 math tests and has an average score of 85. If she scores a 99 on her next math test, what is her average for these 7 tests? Well, in order to find the average of the 7 tests, we need to calculate the total sum of the scores of the 7 tests and divide it by the number of tests, which in this case is 7. Now how can we find the sum of all seven tests?
If we rearrange the equation, if the average is equal to the sum divided by n, then it turns out that the sum is equal to the average times n. The total sum is basically the sum of the first six exams plus the seventh exam. That's going to equal the total sum. Now, we could find the sum of the first six exams because we have the average of those six tests, that's 85, and we have the number of tests, which is 6. So, 85 times 6 is equal to 510. So now that we have the sum of the first six tests, and we know the score for the seventh test, which is 99, we now have the total sum of all seven tests.
So 510 plus 99, that's equal... to 609. So now we can use this equation to calculate the average of all seven tests. So the average is going to be the total sum, which is 609, divided by 7, which is the number of tests. And this will give us an average score of 87. So therefore, D is the right answer.
Number 10. Triangle ABC is inscribed in a circle below. Segment AB passes through the center of the circle at point O. What is the area of the shaded region?
Well first, we need to realize that the area of the shaded region is the difference between the area of the circle and the area of the triangle. The area of a circle is pi r squared, so we've got to find the radius of the circle. The area of a triangle is 1 half base times height.
Now we already have the base. The base is equal to 6, and the height of the triangle is 8. So B is 6, H is 8. So we can find the area of the triangle at any time we wish. What we need to do is calculate the radius of the circle. Now you need to be familiar with certain special triangles.
Special right triangles include the 3-4-5 triangle, the 5-12-13 triangle, the 8-15-17 triangle, and also... the 7, 24, 25 triangle and any ratio of those numbers. Now let's focus on the 3, 4, 5 triangle.
If we multiply it by 2, this will give us the numbers 6, 8, 10. Notice that we have two of these numbers, 6 and 8, which means that the missing side of the triangle has to be 10. And we can prove it using the Pythagorean theorem. So, a squared plus b squared is equal to c squared. The legs of the triangle are 6 and 8, and we're looking for c.
6 squared is 36, 8 squared, or 8 times 8, that's 64. 36 plus 64 is 100. And to find the... value of C when you take the square root of both sides, the square root of 100 is 10. So C is 10. Now segment AB, which passes through the center, represents the diameter. So if the diameter is 10, that means that the radius of the circle is half of that.
The radius is 5. So this is equal to 10. The entire segment AB. But segment AO or even BO, that's the radius that's equal to 5. Now that we have the radius of the circle, we can now calculate the area of the circle. And we can also find the area of the shaded region.
So, r is 5, the base is 6, and the height is 8. So, the area of the circle is going to be 25 pi. And the area of the triangle, 6 times 8 is 48, half of 48 is 20. So that's the area of the triangle. Therefore, the area of the shaded region is the difference between those two values. Now looking at the answer, all of them are in decimal values.
So we got to plug in pi in this equation. So use the exact value of pi. If you don't know what the exact value is, you can use this. 3.14159.
It's still a rounded value, but this should give you an accurate answer. So this is about 54.53975. So we can round that to 54.54, which means C is the right answer.
Number 11. If sine z is equal to negative 7 over 25, and tangent z is less than 0, what is cosine z? Well, before we begin this problem, let's go over some basics of trigonometry. So you need to know something called SOHCAHTOA. Perhaps you've heard of this expression before.
Let's start with the first three letters. S stands for sine. Sine, let's say sine of an angle theta, is equal to the opposite side divided by the hypotenuse.
That's what the so part represents. Cosine is equal to a over h, or the adjacent side, divided by the hypotenuse. And tangent. is opposite divided by the adjacent. So what does this mean?
How do we know which side is opposite, which one is the hypotenuse, and which one is the adjacent side? Now let's say if we have a triangle. right triangle and we need an angle. Relative to this angle, this side is opposite to the angle.
This side is next to it so it's adjacent. The hypotenuse is always across the 90 degree angle so this is the hypotenuse. Now let's say if we have the 3, 4, 5 right triangle.
Let's say this side is 3, this is 4. and that's five. What is sine theta and what is cosine and also tangent? Sine theta based on SOHCAHTOA is equal to the opposite side which is 4 divided by the hypotenuse which is 5. So sine of theta is 4 over 5. Now cosine theta we know it's adjacent over hypotenuse.
So relative to this angle, 3 is the adjacent side, 5 is the hypotenuse. So cosine theta is 3 over 5. tangent theta is opposite over adjacent opposite is for adjacent is 3 so tangent theta is 4 over 3 so that's how you can find a sine cosine and tangent ratios given a right triangle so now what about the question that we have because sometimes sine or cosine could be positive or negative Now there's something else that you need to know. You need to know that this is quadrant 1, 2, 3, and 4. Sine is associated with the y-values, cosine is associated with the x-values. X is positive on the right side, so cosine is going to be positive in quadrants 1 and 4. Cosine is negative on the left side, in quadrants 2 and 3. Sine is associated with the y values, and y is positive above the x-axis, that is in quadrants 1 and 2. Sine is negative below the x-axis, that is, in quadrants 3 and 4. Now, tangent is basically sine over cosine. It's y over x.
Tangent is positive when sine and cosine have the same sine. So, it's positive in 1 and 3. Tangent is negative. 2 and 4 when sine and cosine have opposite signs So these are some things to keep in mind So now let's go ahead and work on this problem.
The first thing we need to do is we need to know in which quadrant we should draw the right triangle. Is it going to be in quadrant 1, 2, 3, or 4? Now we know that sine is negative based on this information here.
And sine is negative in quadrants 3 and 4. Now tangent is less than 0. That means tangent is negative. And tangent is going to be negative in quadrants 2 and 4. So therefore, these two statements are true in quadrant 4. That's where we're going to have to draw the right triangle. And here's going to be the angle theta, which in this case we're going to use the angle z instead. Now, sine of z is negative 7 over 25. And we know sine is opposite over hypotenuse.
Opposite to z is this side. This is going to be negative 7. It's negative because it's below the x-axis. y is negative in quadrant 4. The hypotenuse is 25. Now we've got to find the missing side.
Now this is a 7, 24, 25 triangle. If you're not sure about that, you can use the Pythagorean theorem to find the missing side. So let's say we're looking for A. B is going to be 7, or negative 7, and C is 25. 25 squared is 625 minus 49 that gives you 576. So the square root of 576 is 24 which is the missing side and it's positive 24 because x is positive on the right side. but y is negative as you go down.
So that's why we have a negative 7. Now that we have all three sides of the triangle, we can now find the value of cosine z. So cosine is adjacent over hypotenuse. The adjacent side is 24, the hypotenuse is 25. So cosine is positive 24 over 25. which means that D is the right answer. Number 12. If the sum of a number and 6 times its reciprocal is 5, what is the value of 5 more than 4 times the number?
So the number, we're going to call it x. So we're dealing with the sum, which means addition. The reciprocal of the number is 1 over x.
So 6 times its reciprocal is 6 over x, or 6 times 1 over x. So the sum of that number and 6 times its reciprocal is 5, or meaning it's equal to 5. Once we find the value of x, then we can answer the second part of the problem. What is the value of 5 more than 4 times the number? So that's 5 plus 4 times x.
So let's focus on the first equation. How can we find the value of x in that equation? What I would recommend is to multiply both sides by x.
The purpose is to clear away the fraction. So let's distribute it. x times x is x squared and then x times 6 over x, we can cancel the x variable so it's just going to be 6. And finally, 5 times x is 5x.
Now, let's subtract both sides by 5x. So let's take the 5x from the right side and move it to the left side. So the sign is going to change from positive 5x, and it's going to become negative 5x. So now what we can do is we can factor in order to solve this quadratic equation. So what two numbers multiply to 6 but add to the middle coefficient negative 5?
We know that 2 times 3 is 6, and it adds up to positive 5, so we're going to use negative 2 and negative 3, which adds up to negative 5. So it's x minus 2 times x minus 3. So now at this point, we're going to set each factor equal to 0. So x minus 2 is equal to 0 and x minus 3 is equal to 0, which means that x is equal to 2 and 3. So we have two possible answers. So first, let's find the value of this expression if x is 3. So it's going to be 5 plus 4 times 3, and 4 times 3 is 12, 5 plus 12 is 17. But none of the answers... are listed at 17. So let's try the other one. Let's replace x with 2. 4 times 2 is 8, and 5 plus 8 is 13. Now this answer matches answer choice D. So D is the right answer.
Number 13, if 3x minus 4y is equal to 3, and 2x plus 3y is equal to 19, then what is the value of x squared minus y squared? So how can we figure this out? Well, if we could find the value of x and y, we could simply plug it into this expression and get the value of x squared minus y squared. So we're given two equations, and we have two variables, which means we can solve it using a system of equations. You can use the elimination method or the substitution method.
I'm going to use the elimination method. So first, I need to get rid of either x or y. I'm going to try to cancel the y variable. So a common multiple between 3 and 4 is simply 12. You could just multiply the two numbers.
4 times 3 is 12. So I'm going to multiply the first equation by 3, so I can get negative 12y. And I'm going to multiply the second equation by 4 to get positive 12y. So when I add the two equations, the y variable will cancel. So let's multiply everything by 3. 3x times 3 is 9x, and then 3 times negative 4y, that's negative 12y, and then 3 times 3 is 9. 4 times 2x is 8x, 4 times 3y, that's positive 12y, and 4 times 19. 4 times 20. is 80. If you take away 4, that's 76. So 4 times 19 is 76. Now let's go ahead and add these two equations. So the y variables will cancel.
9 plus 8 is 17, and 76 plus 9 is 85. So now let's divide both sides by 17. So what is 85 divided by 17? 85 divided by 17 is 5. So that's the value of x. Now let's go ahead and calculate the value of y.
So we can use any one of the equations that we need to. Let's use the first equation. 3x minus 4y is equal to 3. Let's replace x with 5, and let's find the value of y.
3 times 5 is 15, and if we subtract both sides by 15... 3 minus 15 is negative 12 so now we could divide both sides by negative 4 and negative 12 divided by negative 4 is 3 so y is equal to 3 So now that we have the value of x and y, we can now find the value of x squared minus y squared. All we need to do is plug it in. So let's replace x with 5 and y with 3. 5 squared, or 5 times 5, that's 25. 3 squared is 9, and 25 minus 9 is 16. So this is the final answer, which means that c is the correct answer.
So let's write an equation for the first part. 7 apples plus 8 bananas, that's going to be 8B, is equal to $13.95. And for the second equation, 9 apples and 6 bananas cost $15.15. Using these two equations, we could find the value of a and b. Once we have it, then we need to find the value of 11a plus 12b.
So this is the goal that we have to attain. So now let's use elimination again to find the value of a and b. Let's cancel b. A common multiple between 6 and a is 24. To get to 24, I'm going to multiply the first equation by 3, because 8 times 3 is 24. The second equation I'm going to multiply by negative 4. 6 times negative 4 is negative 24. them to be positive and the other to be negative if I'm going to cancel these two equations I need to cancel the B variable so let's begin by distributing the three so three times seven a is 21 a 3 times 8 B that's going to be 24 B and 3 times 1395 so 13.95 times 3 that's going to be 41.85 negative 4 times 9a that's negative 36a negative 4 times 6b negative 24b and negative 4 times 15.15 That's going to be negative 60.6. Now let's add the two equations.
So these two will cancel. 21a plus negative 36a, that's going to be negative 15a. And 41.85 minus 60.6, that's negative 18.75. So now let's divide both sides by negative 15. So we can get the value of a. Thank you.
So negative 18.75 divided by negative 15 is 1.25. So the cost of each apple is $1.25. So now that we have that, let's calculate the cost of each banana. And let's use the first equation to do that. So 7a, or 7 times 1.25, plus 8b is equal to 13.95.
So 7 apples cost $8.75. Now let's subtract both sides by 8.75. So 13.95 minus 8.75 is 5.2. Now to find the value of b, let's divide by 8. So 5.2 divided by 8 is 65..65 So the cost of each banana is 65 cents. Now that we have the cost of a single apple and a single banana, we could find the cost of 11 apples and 12 bananas.
So it's 11 times 1.25 plus 12 times.65. so this is going to be twenty one dollars and fifty five cents so therefore answer choice D is the right answer Number 15. Which of the following is equivalent to the expression shown below? Now, before we work on this problem, let's go over some basic properties of exponents.
For example, when you multiply by a common base, you need to add the exponents. x squared times x cubed is x to the fifth power, because 2 plus 3 is 5. When you divide, you need to subtract. So 9 minus 3 is 6, so this is going to be x to the 6th. And also, when you raise one exponent to another, you need to multiply.
For example, let's say if you have x to the 4th raised to the 7th power. This is x raised to the 28th. 4 times 7 is 28. Now let's understand why.
So why is it that x squared times x cubed is x to the 5th? x squared is basically two x variables multiplied to each other. x cubed represents three x variables.
Together, we have a total of five x variables being multiplied. That's why it's x to the fifth. Now, what about this one?
Why is it that x squared raised to the third power is x to the sixth? Why do we multiply 2 times 3 to get to 6? x squared raised to the third power means that you have 3 x squared values multiplied to each other. And each x squared represents 2 x values.
So therefore, we have a total of 6 x values being multiplied, so it's x to the 6. In terms of division, we need to subtract. 5 minus 3 is 2. x to the 5th is basically 5x variables. x to the 3rd is 3x variables. x divided by x is simply 1. So if you cancel 3x variables, you're going to have 2 left over on top. That's why it's x squared.
Now the last thing you need to be aware of are negative exponents. For example, let's say if we have x to the negative 3. This is the same as 1 over x cubed. So if you move the x variable from the top to the bottom, the exponent changes sign. Likewise, if we have 1 over x to the minus 5, you can move it to the top, and the negative 5 will become positive.
So now let's work on this example. So y to the fourth raised to the fifth. Whenever you raise one exponent to another, you need to multiply.
4 times 5 is 20. Now, 3 times negative 8 is negative 24. And now that we're multiplying two common bases, we need to add the exponents. So, 20 plus negative 24, that's equal to negative 4. And since we have a negative exponent, we need to move the y variable to the bottom. So, therefore, the final answer is 1 divided by y to the 4th. So, d is the right answer.
Number 16. What is the slope of the line in the equation shown below? So we have an equation, a linear equation, in standard form. That is in ax plus by equals c format. And what we need to do is we need to rewrite the equation in slope and in set form.
That is in y. equals mx plus b form. The slope is simply the number in front of x. So once we put it in that form we can easily identify the slope.
So let's get y by itself on one side of the equation. So let's begin by subtracting both sides by 3x. So negative 5y is equal to negative 3x plus 8. Now in our next step, we're going to divide everything by negative 5. So y is equal to negative 3 divided by negative 5 is just positive 3 over 5. And this is going to be minus 8 over 5. So negative 8 over 5 is the y-intercept, which is b.
But the slope... is the number in front of X so the slope is positive 3 over 5 which means answer choice C is the right answer in this problem 17 if R is an integer that satisfies the inequality shown below what is the sum of the largest possible value of R and the smallest possible value of R Well, let's go ahead and find out. So let's focus on the first part of the inequality. That is, 3 is less than the square root of r. So in order to find the value of r, we need to square both sides.
3 squared is 9, and the square root of r squared is simply r. Notice that r is greater than 9, but not equal to 9, and r has to be an integer. So the lowest possible value of r is 10. Because r has to be greater than 9. And the next number greater than 9 that's an integer is 10. r can't be 9.5 or 8.2. It cannot be a decimal value. Now let's focus on the second part of the inequality.
The square root of r is less than or equal to 10. So once again, we're going to square both sides. 10 squared is 100, so r is less than or equal to 100, but not greater than 100. So the highest value of r is 100. Now, we want to find the sum of the largest possible value of r and the smallest possible value of r. So that's going to be 100 plus 10, which is 110. So therefore, D is the right answer. Number 18. Which of the following represents the equation of the circle shown below?
So the first thing that we need to know is, we need to know the standard form of the equation of a circle, which is x minus h squared plus y minus k squared, and that's equal to r squared, where r is the radius of the circle and the center has the points. So first, let's find the center of the circle. Now we have four points of interest.
This point has an x value of 2 and a y value of negative 2. The point on the left has an x value of negative 1 and a y value of positive 1. The point on top has an x value of 2 and a y value of 4. And the point on the right has an x value of 5 and a y value of 1. So looking at this, you can see... the coordinates of the center. Notice that the point on the bottom and the top share the same x-coordinate, which is 2. That means that the center has an x-coordinate of 2. And now the point on the left and on the right share the same y-coordinate of 1. which means the center has a y-coordinate of 1. So, that's center 2, 1, which means h is equal to 2, and k is equal to 1. So, if we use this equation and replace h and k with those numbers, it's going to be...
I'm going to need some space, so let's erase this. Instead of x minus h squared, it's going to be x minus 2 squared, plus, instead of y minus k squared, it's going to be y minus 1 squared. So we can get rid of answer choice B, because it has x plus 2, and we can get rid of D, and we can also get rid of E, because it has y plus 1, instead of y minus 1. Now, we've got to find the radius of the circle. And to do that, it's pretty much straightforward. Look at the center.
How far away is the center from the next point? Notice that it's three units apart. You can travel in any direction. It's also three units from the point on the left, three units from the point on top, and three units from the point below. So we can clearly see that the radius Is equal to 3 if you take the difference between 2 and 5?
It's 3 if you take the difference between 4 and 1 it's 3 The same is true from negative 1 and 2 they differ by 3 so the radius Now, the equation of the circle is r squared, and r squared is 3 squared, which is 9. So whenever you see this value here, it's not r, but it represents r squared. So clearly we can see that answer choice A is the answer. R squared doesn't equal 3. R is equal to 3. R squared is equal to 9. Number 19 Which of the following is equivalent to the trigonometric expression shown below?
So go ahead and take a minute and work on this example if you feel that you need to review trigonometric identities Check out my video on YouTube entitled Verifying Triglometric Identities. I have two of them. There's an older version that's like 90 minutes, and there's like a longer version that's an hour.
The newer version is the longer version. It has more examples. So if you want to master this topic, go ahead and check that out when you get a chance.
So, secant x minus secant x times sine squared x, is that equal to a sine x, b cosine x, or is it equal to tangent secant or cosecant? Which one is it? Well, the first thing that we should do is factor out the GCF, the greatest common factor, which is secant x. secant x divided by itself is equal to 1. secant sine squared divided by secant will leave behind sine squared. Now secant...
is equal to 1 over cosine, and cosine is 1 over secant. So let's replace secant with 1 over cosine. 1 minus sine squared is equal to an identity.
If you recall, sine squared plus cosine squared is equal to 1. So if you move the sine squared to the right side, you'll see that cosine squared is 1 minus sine squared. So this is cosine squared. And now we can cancel cosine. So this is going to leave behind cosine. Cosine squared divided by cosine is cosine.
And so that's going to be the answer. Answer choice B is the right answer. 2 minus 1 is equal to 1. Whenever you divide, you need to subtract the exponents. So that's it for number 19. Number 20. Which of the following is equivalent to the trigonometric expression shown below? So here's another example that you can work on.
Go ahead and try it. So what do you think we need to do in this problem? Well, whenever you see 1 plus cosine, 1 minus sine, it's a good indication that you need to multiply by the conjugate.
So, since we have a 1 plus cosine, multiply the top and the bottom of this fraction by 1 minus cosine. So we're going to have sine squared times 1 minus cosine. I'm not going to distribute it there. On the bottom, I will FOIL it.
1 times 1 is 1. And 1 times negative cosine, that's simply negative cosine x. Cosine times 1 is positive cosine. And then we have cosine times negative cosine, which is negative cosine squared.
So as we can see the middle two terms will cancel. Negative cosine plus cosine is 0. So therefore we're left with 1 minus cosine squared on the bottom. And 1 minus cosine squared is sine squared.
We know that sine squared plus cosine squared is equal to 1. So if you move the cosine squared to the right side, we can see that sine squared is 1 minus cosine squared. So we're going to replace these two with sine squared. So now notice that sine squared cancels.
Sine squared divided by sine squared is 1. So this is going to leave us with 1 minus 1 minus cosine. Now let's distribute the negative sign. So it's 1 minus 1, but positive cosine. 1 minus 1 is 0, which leaves us with cosine x. So once again, answer choice B is the right answer.
21 the area of triangle ABC is 70 ad is 10 and BD is 6 what is the left of DC so let's write down what we have this is 80 and BD is 6 we're looking for DC so let's call it X How can we find it? Well, we know that the area of a triangle is 1 half base times height. Now the base of the triangle is the length of segment BC and the height we can clearly see that the height is 10. The area is 70 that's equal to 1 half the base is 6 plus X.
and the height is 10. So now we just got to do algebra. So half of 10 is 5. So what we have is 70 is equal to 5 times 6 plus x. Now let's divide both sides by 5. 70 divided by 5 is 14. So 14 equals 6 plus x.
And now let's subtract both sides by 6. So 14 minus 6 is 8, and that's equal to the value of x. So segment DC is 8 units long, which means C is the right answer. 22. The selling price of a pair of shoes is $70. What is the final price after an 8% sales tax is applied? Now we're going to do this two ways.
First, let's find out how much 8% of $70 is. So we need to convert 8% into its decimal form. To do that, simply divide by 100. So 8% is equivalent to.08. So define 8% of 70, multiplying 70 by.08.
This will give you $5.60. So that's going to be the sales tax. So to find the final price, it's going to be the selling price of the pair of shoes plus the sales tax of $5.60, which is $75.60.
So therefore, answer choice C is the right answer. Now sometimes you may need to be able to get to this answer directly. And here's the formula that you want to be familiar with.
This formula will be useful when trying to solve problems like this backwards. The new price is equal to the original price multiplied by 1 plus or minus r, where r is the sales tax or percent increase or decrease in decimal form. So the original price is 70. And the sales tax is going to increase the total price.
So instead of being 1 minus R, it's 1 plus R. And R is 0.08. 1 plus 0.08 is 1.08. So if you multiply 70 by 1.08, this will give you the final answer of $75.60. So that's another way in which you can get the same answer.
The new price... That is $131.43. We're looking for the original price, so let's call it X.
The sales tax is going to increase the price, which means the original price is less. So we have to use 1 plus R instead of 1 minus R. Now, 6%, if we convert that into a decimal, by dividing by 100, that's 0.06. So this is going to be 1 plus 0.06.
So 131.43 is equal to 1.06 times x. So to find the original price, we need to divide 131.43. by 1.06.
So therefore, x is equal to 123.99. So that's the original price of the calculus textbook. It's $123.99.
so D is the answer. 24, the final price of an item sells for $196 after a 20% discount is applied. What was the original selling price of the item?
So go ahead and take a minute and work on this example. So let's start with the equation. The new price is equal to the original price multiplied by 1 plus or minus r. We're looking for the original price, which means that we're looking for x. The new price must be 196. Now we need to convert 20% into a decimal, so let's divide it by 100. so that's going to be.20 now because we're dealing with a discount it's not going to be one plus R it's going to be one minus R so this is one minus.20 and one minus.20 is.80 so to find the original price we need to divide by.8 So 196 divided by 0.8, that's going to be $245.
So that is the original price. And let's confirm our answer. So what is 20% of $245? So if we multiply this by 0.20, we can see that 20% of $245 is $49.
So that means the discount is $49. So to find the final price is $245 minus $49. And this will give us the original price, I mean the final selling price of $196.
Which means the original price is $245 before a 20% discount is applied. So this is the answer. 25. Susan purchases a $300 laptop. The sales clerk applies a 15% discount, followed by a 7% sales tax.
If Susan hands 14 $20 bills to the sales clerk, how much money will Susan receive back from the sales clerk? So, the original price of the laptop is $300. Now we need to see what the value is after a 15% discount has been applied.
So let's use the equation the new price is equal to the original price multiplied by 1 plus or minus r. So the original price is 300 and 15% as a decimal if you divide it by 100 is 0.15. And because it's a discount, it's 1 minus r as opposed to 1 plus r.
So if you take off 15% from the original price, you're going to have 85% left over. The original price is 100%. So we need to multiply 300 by 0.85.
So 85% of 300 is 255. So that's the price of the laptop after a 50% discount. Next, we need to apply a 7% sales tax. So we're going to use the same equation to calculate. So this time, the original price now is going to be $255.
Seven percent is point zero seven and because of the tax is going to increase the total price It's going to be one plus R as opposed to one minus R so We need to find the value of 107% of 255. So 255 times 1.07, and that's going to be $272.85. Now, Susan hands 14 $20 bills. So how much money does Susan gives to the sales clerk?
What is the value of 14 $20 bills? So what's 14 times 20? We know that 5 20s is 100. 10 20s is 200 and 4 20s is 80. So this is going to be 280. 14 times 2 is 28, so you just add the 0, you get 280. So she gives the sales clerk 280, and the price is 272.85. So the amount that she's going to receive back is the difference between these two numbers. So the sales clerk is going to give Susan $7.15 in change.
So therefore, answer choice D is the right answer. 26. If the volume of a cylinder is 80 pi, what is the height if the radius of the cylinder is 4? So this is the shape of a cylinder. This is just a rough sketch.
And so this is the height of a cylinder. and this portion is the radius. The volume of the cylinder is pi r squared times the height. It's basically the area of the circle times the height.
Now the volume is 80 pi. And the radius is 4. And we just got to find the value of h. So we can cancel pi.
We can divide both sides by pi. 4 squared, or 4 times 4, that's 16. So now we just got to divide both sides by 16. So h is 80 divided by 16, which is equal to 5. So the height is 5 units. Therefore, B is the right answer. 27. What is the slope of the line that passes through the points shown below?
To find the slope between two points, we need to use this equation. It's going to be y2 minus y1 divided by x2 minus x1. So the first point is x.
the second point is y. So if that's x1 and y1, this is x2 and y2. So we can see that y2 is 1 over 4, and y1 is negative 2 over 3. x2 is negative 2 over 5, and x1 is 1 half.
Now whenever you have two negative signs next to each other, it's going to change into a positive sign. So we have 1 fourth plus 2 over 3 times, I mean divided by negative 2 over 5 minus 1 over 2. So we need to simplify the complex fraction. So how can we go about doing that?
The best way to simplify the complex fraction is to eliminate the smaller fractions within the larger fraction. To do that, We need to multiply every fraction by a common denominator. It can be the least common denominator or any common denominator. To find the common denominator, you can multiply all of the denominators together.
So we need to cancel the 3 and the 5. So we need a number that contains 3 and 5. And we need to get rid of the 2 and the 4. So we just need a 4. The 4 will get rid of 2 and 4 because 2 and 4 go into 4. So the least common denominator is going to be 3 times 5 times 4. 5 times 4 is 20. 20 times 3 is 60. So what we're going to do is we're going to multiply the top and the bottom by 60. So let's distribute 60. 60 times 1 fourth, which is 60 divided by 4, that's 15. Now what is 2 thirds of 60? 60 times 2 is 120. 120 divided by 3, that's 40. now what is negative two-fifths of 60 60 divided by 5 is 12 12 times negative 2 is negative 24 and half of 60 is 30 15 plus 40 is 55 negative 24 minus 30 is negative 54 therefore e is the right answer it's negative 55 over 54 28 if John can eat 7 apples in 18 minutes how many apples can he eat in 27 minutes you can solve this problem two ways you can set up a proportion or you can use conversions let's set up a proportion so on one side we're gonna have the information for the number of apples and on the other side the time. So let's put two fractions separated by an equal sign.
So John can eat seven apples in 18 minutes. How many apples, which is x, can you eat in 27 minutes? So whenever you have two fractions separated by an equal sign, you need to cross multiply.
x times 18 is 18x. and here we have 7 times 27. 18 is basically 9 times 2, and 27 is 9 times 3. So if we divide both sides by 9, those two numbers will cancel. So 2x is equal to 7 times 3, which is 21. Now let's divide both sides by 2. So x is 21 over 2, which is 10.5 apples. So if in 18 minutes he can eat 7 apples, in 27 minutes he can eat more, he can eat 10.5 apples, or 21 over 2, which means A is the answer.
Now, let's get the same answer using a conversion. The conversion is this statement here. 7 apples is equal to or corresponds to 18 minutes. So, that's the conversion factor. Now, here's what we're trying to convert.
We need to convert the number of minutes into the number of apples. So, let's start with what we have, which is 27 minutes over 1. Now, in the next fraction, we're going to use the conversion factor. Because we have minutes on top, we need to put it on the bottom.
So that the unit minutes will cancel. So the 7 apples go on top. So it's going to be 27 times 7, which is 189, over 18. Now we can reduce this fraction. We could divide the top and the bottom by 9. 189 divided by 9 is 21. 18 divided by 9 is 2. So you get the same answer. 21 over 2, or 10.5 apples.
So with these problems, you can solve it using a proportion or using a conversion factor. 29. A car can travel 35 miles in 40 minutes. At this rate, how many hours will it take for the car to travel 245 miles?
Now let's solve this problem using two methods, proportions and conversions. So we're dealing with distance in miles and time in minutes. So let's start with a proportion. So the car can travel 35 miles in 40 minutes.
How many minutes can it travel in 245 miles? So let's cross multiply. So we're going to have 35x is equal to 40 times 245. which is 9,800 now let's divide both sides by 35 so X is 280 minutes now the answer is in hours so we got to convert hours I mean minutes into hours 1 hour is equal to 60 minutes.
So we get to divide by 60. We can cancel the 0, so this becomes 28 over 6. 28 over 6 reduces to 14 over 3 if you divide both numbers by 2, which means that answer choice D is the right answer. But now let's get the same answer using conversions instead of proportions. So the conversion factor is that information.
So 35 miles is equal to 40 minutes. That's our conversion factor. Now in the second part of the problem, it tells us what we need to convert.
We need to convert 245 miles into hours. So let's start with 245 miles. Using this conversion factor, let's convert miles into minutes.
So for every 35 miles, well the car travels 35 miles in 40 minutes. So the unit miles cancel, and the last thing we need to do is convert minutes into hours. One hour is equal to 60 minutes.
And so these units will cancel. And now we can get the answer. So it's going to be 245 times 40, which is 9,800.
And 35 times 60, that's 2,100. So we can cancel the two zeros. so becomes 98 over 21 98 is basically seven times 14 21 is seven times three so we can cancel the sevens and so the final answer is 14 over three hours Number 30. One box can hold 27 muffins and 5 cups of flour are required to make 9 muffins.
How many cups of flour are needed to make enough muffins to fill 8 boxes? So this is a multi-step problem, and it's easier to get the answer using conversions instead of proportions. So that's what we're going to do.
So let's write out a list of conversion factors that we have. The first conversion factor... relates number of boxes to muffins so basically one box is equal to 27 muffins that's the first conversion factor now the second conversion factor is given to us in this part of the sentence 5 cups of flour is equal to 9 muffins The objective is listed at the end of the problem. How many cups of flour are needed to make enough muffins to fill 8 boxes?
So we need to convert 8 boxes into cups of flour. So let's go ahead and do that. So let's start with 8 boxes over 1. Now we're going to convert boxes into muffins first using the first conversion factor. And then muffins into cups of flour using the second one.
So one box can hold 27 muffins. So the unit boxes will cancel. Now there's 5 cups of flour that's needed to make 9 muffins. So the unit muffins will cancel as well.
So now all we need to do is do the math. 27 divided by 9 is 3. So we have 8 times 3 times 5. Now 8 times 5 is 40. And 40 times 3, well we know 4 times 3 is 12, so 40 times 3 is 120, which means D is the right answer.