1.

The Correct Answer is (D) — For this question, you have to group like terms. Multiplying the constants gives –18. Multiplying the variables gives the following:


The result is –18x. If you chose (C), you probably multiplied the exponents rather than adding them.

2.

The Correct Answer is (G) — Simply divide the number of particles by 2:


If you chose (F), you probably divided the exponent by 2 as well.

3.

The Correct Answer is (E) — You know that (0.20) x = 15, therefore 20% of x is 15. So:


75% of 120 is (120)(0.75) = 90.

4.

The Correct Answer is (H) — Recall that distance = speed time. To find the distance that Maria traveled, substitute the values from the question into the formula. Remember to convert the extra 15 minutes into hours by multiplying by the conversion factor for minutes to hours:


Add this to the 5 hours to get a total of 5.25 hours driving, then multiply by speed to find distance:



If you chose (G), you probably didn’t convert the minutes into hours.

5.

The Correct Answer is (C) — The original price is $237. The first discount the customer receives is 16% off, which is equivalent to 84% of the original price. Then, at the counter, she receives an additional 10%, which is equivalent to 90% of the first discount price.


If you chose (B), you probably added the discounts rather than applying them sequentially.

6.

The Correct Answer is (F) — To find the lowest common denominator, find the lowest common multiple of all the denominators. The easiest way to do this is to prime factor and to multiply all the factors, but shared factors only once. 17 is prime, so its only factors are 1 and 17; 12 is composed of 2, 2, 3; 4 is composed of 2, 2; 6 is composed of 2, 3. LCM = 2 x 2 x 3 x 17 = 204.

7.

The Correct Answer is (C) — As given in the question, substitute x = –2 into the expression and evaluate:

\begin{align*} &= ((-2)^3+6)((-2)^2-3) \\ &= (-8+6)(4-3) \\ &= (-2)(1) \\ &= -2 \end{align*}

8.

The Correct Answer is (F) — Using the distance formula, the distance between (-1, 2) and (2, 6), the two points that form the line that is the base of the isosceles triangle, is:


The only value of k that gives the same distance from (k, 2) and (2, 6) is k=5, where:


Note that the points (-1,2) and (2,6) must form the base in this question. If one of the other sides were to form the base, then k would have to equal 4, which is not an answer choice.

9.

The Correct Answer is (B) — The variable x is a common factor in the expression. After factoring x out, you are left with x(x — 16) = 0. The only possible values of x that would make the equation true are x = 0 and x = 16.

10.

The Correct Answer is (F) — To find an expression equivalent to the one in the question, expand the expression and add like terms:

\begin{align*} &=3(5x-2y)+8(\frac{1}{2}y-2x)) \\ &=15x-6y+4y-16x \\ &=15x-16x-6y+4y \\ &=-x-2y \\ \end{align*}

11.

The Correct Answer is (C) — The ratio blueprint:structure is 15:45, which reduces to 1:3. Now, you know that the length of the structure is 3 times as large as indicated on the blueprint. (8)(3) = 24 and (10)(3) = 30. Adding all the sides together, you find that 24 + 30 + 45 = 99 meters of metal are needed.

12.

The Correct Answer is (J) — Substitute all the values that are given in the question into the expression, then isolate and solve for a:

\begin{align*} d&=vt+\frac{1}{2}at^2 \\ 61&=(15)(3)+\frac{1}{2}a(3)^2 \\ 61&=45+\frac{9}{2}a \\ a&=\frac{32}{9} \end{align*}

13.

The Correct Answer is (B) — Kristin pays $5 per shirt, so the cost of s shirts is 5s. Similarly, she pays 6j per jacket. To find the total cost of s shirts and j jackets, just add those two terms together. Finally, add the initial fee of $9 to get the final expression 5s + 6j + 9.

14.

The Correct Answer is (F) — The roots are the x coordinates of where the parabola intersects the x-axis. The roots are the values of x that make the function equal to zero. Graphically, this is represented by the x-intercepts of the parabola. From the graph, you can see that the roots are —38 and 10. (—38) + 10 = —28.

15.

The Correct Answer is (E) — Using the formula for finding the slope of a line, the slope is . The equation of the line in slope intercept form is where b is the y-intercept. Substitute either given point to find the value of b. b = 4.

16.

The Correct Answer is (G) — The question tells you to evaluate the difference between 2a and 2a+1. The exponent increases by 1, which means b will increase by a factor of the base, which is 2. So, b doubles in size.

17.

The Correct Answer is (A) — Subtract the number of outfit options that Alex has after losing a shirt and a pair of sandals from the original, complete number of outfit options that Alex had. To find the number of outfit options, multiply the numbers of each article of clothing that Alex has:


If you chose (C), you probably found the number of original options and stopped there. If you chose (B), you probably found the number of new outfit options and stopped there.

18.

The Correct Answer is (F) — Isolate and solve for x:

$$23 \geq -(17+5x)$$

$$-23 \leq 17+5x$$

$$-40 \leq 5x$$

$$-8 \leq x$$

If you chose (G), you probably forgot to flip the direction of the inequality after multiplying through by a negative.

19.

The Correct Answer is (D) — You could add up all the sides individually, but recognize that since all of the lines intersect at right angles, the perimeter is equivalent to calculating that of a rectangle. Adding gives the lengths of the top or bottom of the figure:

20.

The Correct Answer is (G) — To solve this question, set up the ratios. You know that the two ratios given in the question are equivalent to each other. Equate the two ratios and solve for the unknown value:



21.

The Correct Answer is (D) — Recognize that the triangle formed by the rope and the ground forms a triangle. You know that one of the angles is 40ᣞ, and that there is a right angle formed by the piano and the ground. A triangle has 180ᣞ, hence the missing angle must equal 180ᣞ – 40ᣞ – 90ᣞ = 50ᣞ.

22.

The Correct Answer is (K) — You can compare fractions by converting them into equivalent fractions with the same denominators, or by converting the fractions into decimals. By using either method, is the smallest fraction, and K is the only answer choice that begins with . is the second smallest, the third smallest, and the largest.

23.

The Correct Answer is (C) — To solve for x, rearrange the equations such that y is expressed in terms of x and equate the two equations to each other and solve:

\begin{align*} 2y&=1-x \\ y&=1-x \\ x^2-1&=1-x \\ x^2+x-2&=0 \\ (x+2)(x-1)&=0 \end{align*}

The possible values of x are -2 and 1. The only possible value that is listed in the answer choices is 1.


If you chose (D), you probably forgot the negative sign on 2.

24.

The Correct Answer is (K) — Recognize that , according to the graph. The question asks you to find , or . So, .

25.

The Correct Answer is (D) — Notice that 81 is 3 to a power, so you can reduce the expression 817 to have a base of 3. Using exponent laws, reduce 817 to have a base of 3:



If you chose (B), you probably added the exponents instead of multiplying them.

26.

The Correct Answer is (J) — Recall that the ratio of arc length to circumference is equal to the ratio of sector angle to 360 degrees. Therefore:
\begin{align*} \frac{\frac{\pi}{2}}{2\pi}&=\frac{x}{360} \\ \frac{1}{4}&=\frac{x}{360} \\ x&=90 \end{align*}
The arc length of a sector is calculated by multiplying the radius by the sector angle. For this circle, the arc length is , where x is the sector angle in radians. From the equation, you can determine that is equivalent to 90°.

27.

The Correct Answer is (D) — Note that the question does not ask on what day a book reaches a fine of $4.00, but the maximum number of days before a fine of $4.00 is applied. On the first day, the fine is $0.25; on the second, $0.50; on the third, $1.00; on the fourth, $2.00; on the fifth, $4.00. Hence, a book can be borrowed for four days past its due date, or for 25 days in total.
If you chose (E), you probably gave the number of days that would give a fine of $4.00.

28.

The Correct Answer is (H) — To solve problems involving absolute value, evaluate the expressions inside the absolute value bars first:

\begin{align*} &=|9-21|-|8+3-16| \\ &=|-12|-|-5| \\ &=12-5 \\ &=7 \end{align*}

29.

The Correct Answer is (D) and . So, .
If you are familiar with trigonometric identities, you may recognize that .

30.

The Correct Answer is (H) — The question tells you that n is doubled, giving . The number is then tripled, giving . The number is then divided by 4, giving .

31.

The Correct Answer is (D) — When working with logarithms, rewrite them into their exponent form to make them easier to work with. is equal to and is equal to . Now, you can see that the question is asking you for the value of . , and , therefore .
If you are familiar with logarithm rules, you may recognize that this equation can be simplified to .

32.

The Correct Answer is (J) — Let x represent the number of soda bottles that John bought.
\begin{align*} 130&=3x+4(40-x) \\ 130&=3x+160-4x \\ -30&=-x \\ x&=30 \end{align*}

33.

The Correct Answer is (B) — The shape of the graph indicates that it represents an absolute value expression. So, you can rule out (A). (D) and (E) may be tempting, but note that none of the slopes in the answer choices match what is shown on the graph. Your two remaining choices are (B) and (C). (C) indicates that the graph was moved 2 units up from the x-axis, which does not correspond to the graph. (B) is the correct answer.

34.

The Correct Answer is (K) — Matrices can only be added (or subtracted) when they have the same dimensions. The matrices in the question are of different dimensions, therefore they cannot be added and the resulting matrix is undefined.

35.

The Correct Answer is (B). This can either be solved algebraically:


where x is the smallest number. In the interest of time, it may be easier to substitute answer choices and select the correct answer through process of elimination.

36.

The Correct Answer is (J)

If you are familiar with the Gaussian method of adding arithmetic sequences, recognize that the pattern of this sequence is , where n is the nth day. On the 10th day, Fred will have deposited . Sum of deposits .

37.

The Correct Answer is (B) — On June 5th, Fred is still depositing more than Angela. On the 6th, Fred will have deposited $50 and Angela $32. On the 7th, Fred will have deposited $55 and Angela $64. Therefore, Angela deposits more than Fred on the 7th.

38.

The Correct Answer is (H) — Use the Pythagorean Theorem and set the height as one of the legs of the triangle, with the cable length acting as the hypotenuse:

\begin{align*} c^2&=a^2+b^2 \\ (13)^2&=5^2+b^2 \\ (13)^3-5^2&=b^2 \\ 144&=b^2 \\ 12&=b \end{align*}

Since a height of -12 ft. is impossible, the height of the tent is 12 ft.
You may have recognized that 13-12-5 is a Pythagorean Triple. If you did, solving this question would have been much easier and quicker.

39.

The Correct Answer is (C) — Let x represent Corwin’s favorite number:

\begin{align*} 5&=\sqrt{x+2} \\ 5^2&=x+2 \\ 25-3&=x \\ 23&=x \end{align*}

40.

The Correct Answer is (H) — A quick sketch of the circle may help you to solve this question:

Recall the equation of a circle, , where h and k are the center.

You know from the equation that the center is at (15, 4) and that the radius is 9. The circle will not cross the y-axis, since the center is further from the y-axis than the length of the radius. The circle will, however, cross the x-axis because the distance from the center of the circle to the x-axis is smaller than the length of the radius. Since the x-axis is not tangent to the circle, it will intersect the circle twice.

If you chose (G), you may have missed that a circle will intersect the x-axis more than once. If you chose (F), you may have miscalculated the radius or center of the circle and misplaced it on the coordinate plane.

41.

The Correct Answer is (B) — When vector is broken up into its components, it is -5 units in the direction of the i unit vector (parallel to the x-axis) and 3 units in the direction of the j unit vector (parallel to the y-axis). Hence, the component form is .

42.

The Correct Answer is (F) — In order for to be true, . Therefore, and . Remember that you are not finished yet: .
If you chose (G) or (J), you may have found just the value of x or y without multiplying them together.

43.

The Correct Answer is (C) — Since the radius is tripled, sub 3r in for r.
\begin{align*} A&=\frac{4}{3}\pi(r)^3 \\ A&=\frac{4}{3}\pi(3r)^3 \\ A&=\frac{4}{3}\pi(r)^3\times27 \\ \end{align*}

The volume will increase by a factor of 27.

44.

The Correct Answer is (G) — Start by finding . Then, substitute into .

45.

The Correct Answer is (B) — In order for four points to form a square, all points must be an equal distance from its adjacent points. A quick sketch may help you with this problem:

Answer choice (A) is from (2, -1) and (2, -1) is from (-2, 3). Answer choice (C) is 4 units away from both provided points. Answer choice (D) is 4 units away from both provided points. Answer choice E is from (2, -1) and (2, -1) is from (-2, 3). Answer choice (B) is from (-2, 3) but (-2, 3) is from (2, -1).

46.

The Correct Answer is (F) — Recall that the sum of any quadrilateral’s interior angles is always 360ᣞ. To find the value of x: 360ᣞ — 85ᣞ — 105ᣞ — 70ᣞ = 100ᣞ.

47.

The Correct Answer is (A) — To find the length of , subtract the length of from . Then, collect like terms and simplify.

\begin{align*} &=3x+5a-(x-2a) \\ &=3x+5a-x+2a \\ &=2x+7a \end{align*}

48.

The Correct Answer is (H) — The question tells you that the triangle is an isosceles right triangle, so you know that both of the legs of the triangle are the same size and that you can use the Pythagorean Theorem.

\begin{align*} (\frac{10}{\sqrt{2}})^2+(\frac{10}{\sqrt{2}})^2&=c^2 \\ \frac{100}{2}+\frac{100}{2}&=c^2 \\ 100&=c^2 \\ c&=10 \end{align*}

49.

The Correct Answer is (C) — By looking at the diagram, you can see that the area of the garden with pansies is equivalent to 2/4 of the garden, minus the 2 small squares of lilies.

\begin{align*} &=2(\frac{100}{4})-2(\frac{\frac{100}{4}}{4}) \\ &=2(25)-2(\frac{25}{4}) \\ &=50-12.5 \\ &=37.5 \\ &\approx38 \end{align*}

50.

The Correct Answer is (H) — To answer this question, a diagram may help:

Using the Pythagorean Theorem, you can find the diagonal of the garden:


\begin{align*} (10)^2+(10)^2&=d^2 \\ 200&=d^2 \\ d&=\sqrt{200} \\ \end{align*} Now, you can find the length of the clothesline.

\begin{align*} (\sqrt{200})^2+5^2&=c^2 \\ 200+25&=c^2 \\ 225&=c^2 \\ c&=15 \end{align*}

51.

The Correct Answer is (E) — Start by finding the total area covered by daffodils and lilies. You can do this either by subtracting the area of pansies, or by adding the area of daffodils and lilies together. This answer explanation will work through the former method.
\begin{align*} &=100-(2(\frac{100}{4})-2(\frac{\frac{100}{4}}{4})) \\ &=100-(2(25)-2(\frac{25}{4})) \\ &=100-(50-12.5) \\ &=100-37.5 \\ &=62.5 \end{align*}
Now, find the area that one wooden plank will cover. The question gives you the length, width, and height of the planks – realize that you only need the length and the width to answer this question. . Divide the total area that needs to be covered by the area of one planks.

52.

The Correct Answer is (K) — You can find the sum of the interior angles of a polygon by multiplying 180 by the number of triangles formed by connecting lines to each of the vertices from one point, as shown below:

You can see that there are 6 triangles formed, hence the sum of all the interior angles is (180ᣞ)(6)=1080ᣞ. Since the polygon is a regular hexagon, you know that all the interior angles are of equal size. Therefore, divide 1080ᣞ by 6 to get 120ᣞ.

53.

The Correct Answer is (A) — If , then and . Therefore:

\begin{align*} ab&=(x-\frac{1}{2})(x+\frac{1}{3}) \\ ab&=x^2-\frac{1}{2}x+\frac{1}{3}x-\frac{1}{6} \\ &=x^2-\frac{1}{6}x+-\frac{1}{6} \end{align*}

54.

The Correct Answer is (G) — Find the areas of each section. Remember to subtract the area of the inner circles when calculating the area of the outer rings. Building a table like the one below can help to keep all of your calculations organized:


Section A, with radius 5:
Section B, with radius 6:
Section C, with radius 7:
Section D, with radius 8:
Keep your answers in terms of π to avoid messy calculations. Section B has the smallest area.

55.

The Correct Answer is (A) — To answer this question, a diagram might help:

On the diagram, label all the information that you know. By doing this, it is easy to see that the distance from D to B is 100 m. If his total trip was 900 m, subtract the distances that you know he travelled to find the distance between points A and D:

\begin{align*} 900&=(\overline{AD}+100)+300+400 \\ \overline{AD}&=100 \end{align*}

56.

The Correct Answer is (F) — To answer this question, it may help to draw a diagram:

You know that tan⁡θ=√3, therefore you know that the triangle will be a 30-60-90 triangle. From this, you can find that .

57.

The Correct Answer is (C) — Since the area of the square backyard is 64, the side length of the backyard must be . To answer this question, it may help to draw a diagram:

If the side lengths of the square is 8 and the circular pool is perfectly inscribed in the square to create the maximum area, then the radius must be half the length of the square, or 4. Therefore, the area of a circle with radius 4 is .

58.

The Correct Answer is (H) — The area of the sticker is equal to the area of the cassette tape, minus the two circular holes. From the diagram, you can see that the combined diameters of the two holes is 1 in, therefore the radius of one hole must be 0.25 in. The area of both of these holes is . To find the area of the sticker, subtract the area of the holes from the area of the cassette tape.
\begin{align*} A&=(4.0)(2.5)-0.125\pi \\ A&=10-0.125\pi \\ A&\approx9.6 \end{align*}

59.

The Correct Answer is (A) — Any set that contains 0 can be eliminated, since that means the fraction will be undefined. Now, you are left with answer choices (A), (B), and (D). If x equals a negative number, then and the inequality is no longer true; you can eliminate answer choice (D). Realize that when x is equal to any number that is equal to or greater than 2, and the inequality is no longer true; you can eliminate answer choice (B). Therefore, (A) is the correct answer.

60.

The Correct Answer is (F) — From looking at the graph, you can see that there was no horizontal or vertical translations from f(x) to g(x), therefore answer choices (H) and (K) can be eliminated. The period of has not changed from f(x) to g(x), therefore answer choice J can be eliminated. The amplitude of the function has changed from f(x) to g(x), and it has increased, therefore a increased.