## 1.

The Correct Answer is (C) — Factoring out x gives 0 = x(x + 1). Therefore, x = 0 or -1, for a total of two solutions. You may have chosen (E) if you thought that it was the equation $y=x^2+x$.

## 2.

The Correct Answer is (H) — Distributing the negative 2 gives C + D – 2C – 2D. Collect like terms and simplify to get –CD.

## 3.

The Correct Answer is (B) — Factor out $4x^2$ from the numerator to get $\frac{4x^2 (x^2+2x+4)}{4x^2}$ is in both the numerator and the denominator, so it cancels out, leaving $16^x=4$.

## 4.

The Correct Answer is (J) — First, find the number of tickets that Gabriella bought by dividing the money spent by the cost per ticket: $\frac{42 \text{ dollars}}{6 \text{ dollars/ticket}} = 7 \text{ tickets}$. To find the price of the original tickets, take the total original cost of 7 tickets and divide by 7: $\frac{42 \text{ dollars} + 56 \text{ dollars}}{7 \text{ tickets}} = 14 \text{ dollars}$ for a regular ticket.

## 5.

The Correct Answer is (E) — To find the greatest common factor, find the prime factors of all the numbers and multiply the factors that all the numbers have in common. The prime factors of 51 are: 3, 17. The prime factors of 68 are: 2, 2, 17. The prime factors of 119 are: 7, 17. The only prime factor that all the numbers have in common is 17, so 17 is the greatest common factor.

## 6.

The Correct Answer is (H) — To begin, find the total surface area of the walls that Henry is going to paint. $4\text{ walls }\times\frac{9\text{ feet}}{\text{wall}}\times\frac{13\text{ feet}}{\text{wall}} = 468 \text{ square feet}$ Then, subtract the surface area that Henry is able to paint: 468 square feet – 175 square feet = 293 square feet.

## 7.

The Correct Answer is (D) — First, notice that all the answer choices are fractions, and that the question asks you to solve for the denominator x. Gathering the left side together:

$\frac{2 +3}{x}=40$
Now solve for x:

\begin{align*} 5&=40x \\ \frac{5}{40}&=\frac{1}{8}=x \end{align*}

## 8.

The Correct Answer is (G) — The question tells you that there are two parallel lines intersected by a third line, which means that this third line is a transversal. You can apply the rule of corresponding angles to find that the angle supplementary to q is also 120°. Since q is supplementary to 120°, q = 180° – 120° = 60°.

## 9.

The Correct Answer is (C) — First, find the total number of hours that Alex works: $20 \text{ days}\times\frac{8 \text{ hours}}{\text{ day}} = 160 \text{ hours}$. To find the monetary value of these hours, multiply $160 \text{ hours}\times\frac{\text{\}12}{\text{ hour}} = \1920$.

## 10.

The Correct Answer is (J) — Use the distance formula, and substitute the known values in. Then solve for a. \begin{align*} 4&=\sqrt{(\Delta x)^2+(\Delta y)^2} \\ 4&=\sqrt{(5-1)^2+(a-4)^2} \\ 4&=\sqrt{4^2+(a-4)^2} \\ 4^2&=4^2+(a-4)^2 \\ 0&=(a-4)^2 \\ a&=4 \end{align*}

## 11.

The Correct Answer is (A) — The prime numbers between 80 and 100 are 83, 89, and 97.

## 12.

The Correct Answer is (J) — First calculate all value of the triangle containing the 31 degree angle. 180 – 99 = 81 is the upper angle of the triangle, determined because the 99 degree angle is crossed by L; the other side of this crossing is the upper angle of the triangle. The final angle in the triangle is found by subtracting the two known angles from the sum of the triangle’s angles: 180 – 81 – 31 = 68 degrees. Angle c is supplementary to 68, so c = 180 – 68 = 112 degrees.

## 13.

The Correct Answer is (C) — To solve this question, factor out 4 from the expression: 20x + 8y – 80 = 4(5x + 2y – 20).

## 14.

The Correct Answer is (F) — To find the coordinates of the midpoint of a line segment, take the average of the x and y coordinates. The midpoint of the x coordinate is $\frac{-10 + 6}{2}=\frac{-4}{2}=-2$. The midpoint of the y coordinate is $\frac{-4 + 8}{2}=\frac{4}{2}=2$. So, the midpoint is of the line segment is (-2, 2).

## 15.

The Correct Answer is (B) — You know there are 124 total students enrolled in science courses, but only 84 total students. A student may be enrolled in 1, 2, or 3 classes. However, since the question is asking you for the maximum number of students enrolled in more than 1 science class, assume that no student is taking 3 science classes. The calculation is therefore simply 124 enrollments – 84 students = 40 additional enrollments per student. There can be at most 40 students enrolled in more than one class.

## 16.

The Correct Answer is (H) — In the large triangle, there are a total of 16 smaller triangles. The shaded area is composed of 4 triangles. You can reduce 4:16 to 1:4.

## 17.

The Correct Answer is (C) — To find the sum, write out all the terms and add them together.

15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 + 55 + 60 = 375

If you are familiar with the Gaussian method of adding arithmetic sequences, recognize that the pattern of this sequence is 15+(n-1)5, where n is the $n^{th}$ day: \begin{align*} 60 &= 15 + (n-1)5 \\ 45 &= (n-1)5 \\ 9 &= n-1 \\ n &= 10 \end{align*}

By the 10th day, Herb will have built a total of $\frac{n}{2}(15+60)=\frac{10}{2}(75)=375$ fence posts.

## 18.

The Correct Answer is (J) — A circle is the only shape where all the points on its perimeter are equidistant from its center.

## 19.

The Correct Answer is (B) — Expand and simplify the expression:

\begin{align*} &=(3a^2-4b)(3a^2+4b) \\ &=9a^4+12a^2 b-12a^2 b-16b \\ &=9a^4-16b \end{align*}

## 20.

The Correct Answer is (H) — The best way to solve this question is to derive two equations from the text and solve for the number of suits. Let’s say that suits are, t, and shirts are s. You know that:

\begin{align*} t(200)+s(20)&=8,080 \\ s&=0.10(t) \end{align*}

Now you just need to substitute s into the first equation, to solve the question:

\begin{align*} t(200)+0.1(t)(20)&=8,080 \\ t(202)&=8,080 \\ t&=40 \end{align*}

## 21.

The Correct Answer is (D) — To answer this question, find the height of the bottom half and the top half of the kite separately. Where a is the height of the bottom of the kite: $30=\sqrt{10^2+a^2}, 30^2 = 10^2 + a^2, a = \sqrt{800}$ Where b is the height of the top of the kite: $15=\sqrt{10^2+b^2}, 15^2 = 10^2 + b^2, b = \sqrt{125}. \sqrt{800} + \sqrt{125} \doteq 39.46$ Rounding to the nearest inch, the height of the kite is 39 inches.

## 22.

The Correct Answer is (H) — The difference between the second and fourth terms is 10. Because there is one term between 12 and 22, the difference between each of the terms must be 5 (10 divided by 2). Since 12 + 5 = 17, you can determine that only (H) is correct.

## 23.

The Correct Answer is (B) — Howard has half a bag of flour. $\frac{14\text{ cups}}{\text{bag}}\times\frac{1}{2}\text{ bags} = 7 \text{ cups}$. To find how many cakes he can make with 7 cups divide 7 cups by 1 and $\frac{2}{3}$ cups of flour per cake (coverts to $\frac{5}{3}$):

$\frac{7}{\frac{5}{3}}=4.2 \text{ cakes}$

Howard also has 1 liter of milk, or 4 cups of milk. If you similarly convert the 4 total cups of milk to the possible number of cakes, you will get:

$\frac{4}{\frac{3}{4}}=5.3 \text{ cakes}$

Realize that Howard can only make as many cakes as his ingredients will allow. He cannot make 5 cakes, since he does not have enough flour for the fifth cake. Since he cannot make a fraction of a cake, he can only make 4 cakes.

## 24.

The Correct Answer is (F) — To calculate a ratio from a ratio, you can simply divide one by the other. In this case the ratio of pizza that Martin ate was $\frac{1}{3}$ and the amount of pizza that Robert ate was $\frac{2}{7}$:

$\frac{\frac{1}{3}}{\frac{2}{7}} = (\frac{1}{3})(\frac{7}{2}) = \frac{7}{6}$

## 25.

The Correct Answer is (E) — Notice that x is one and a half corners of a square. A square’s corners are all right angles. Since the right angles within angle x are each split into half, this means that $x = (3)(\frac{90}{2}) = (3)(45) =135$.

## 26.

The Correct Answer is (H) — During month A, the total average is 10,000 crates per week. During month B, the total average is 6,000 crates per week. To find percent decrease: $\frac{new-original}{original}=\frac{6,000-10,000}{10,000}=\frac{-4,000}{10,000}=-0.4$ or 40% decrease.

## 27.

The Correct Answer is (B) — Options I and III will change a negative integer into a positive integer, an option II will maintain a negative integer.

## 28.

The Correct Answer is (H) — Substitute g = 4 into the expression to solve the question:

\begin{align*} &=\frac{4^2}{4+1}+\frac{4+1}{4^2} \\ &=\frac{16}{5}+\frac{5}{16} \\ &=\frac{256+25}{80}=\frac{281}{80} \end{align*}

## 29.

The Correct Answer is (D) — To answer this question, first determine the mean, median, and mode. Looking at the graph, you can assume that the median will be between 4 and 5. The median is the $\frac{20 + 60 + 80 + 40}{2} = 100^{th} \text{ data point}$, or 4 items. The mode is also 4 items. Knowing these values, you can eliminate (A), (B), (C), and (E).

## 30.

The Correct Answer is (K) — (F) can be eliminated, since it is possible that a < b and therefore $\frac{a}{b} < 1$. (G) can be eliminated, since it is possible that 0 < a, b < 1 and therefore ab < 2. (H) can be eliminated, since it is possible that a < b, and therefore $a^2 < b^2$. (J) can be eliminated since b is a negative integer to the 3rd power. This means that it keeps the negative sign and therefore $a^2 b^3$ is negative (or less than 0).

## 31.

The Correct Answer is (C) — There are 10 horses in total, Cleo and Jordan plus 8 other horses. If Cleo’s placing is already determined, that means that Joran is racing in a total pool of 9 horses for the second place spot. Jordan has a $\frac{1}{9},$ or 11% chance of finishing in second place.

## 32.

The Correct Answer is (G) — To solve this question, substitute h = 0 into the expression and solve. 0=-t(t-4). The possible values of t are 0 or 4. Since the question asks for the next time after t = 0 when the ball will hit the ground, t must be equal to 4.

## 33.

The Correct Answer is (C) — First, find g(8) through substitution into g(x):

$g(8)=\sqrt{(2)(8) }=\sqrt{16}=4$
Next, substitute this value into f(g(8)):

$(4)^2+2(4)=16+8=24$

## 34.

The Correct Answer is (H) — To solve this question, it may be helpful to construct an inequality based on the number line and then see which of the answer options fit your inequality. You can see that it covers all values in the range -3 ≤ x < 2. (H) is the only choice that matches this number line

## 35.

The Correct Answer is (C) — To find the volume of the box, find the area of the base and then multiply by the height of the box. Using s as one side of the square, the area of the base is:

\begin{align*} &=s^2+\pi(1/2 s)^2 \\ &=6^2+\pi3^2 \\ &=36+9\pi \end{align*} Now, multiply this by the height of the box: (36+9π)(9)≈578 cubic centimeters.

## 36.

The Correct Answer is (K) — This question involves simplifying both the denominator and the numerator, and then factoring the simplified forms. Starting with the denominator:

\begin{align*} &=\frac{\sqrt{x^2+6x+9}}{x^2+9} \\ &=\frac{\sqrt{x^2+6x+9}}{(x+3)(x-3)} \end{align*} Next simplify the numerator:

\begin{align*} &=\frac{\sqrt{x^2+6x+9}}{(x+3)(x-3)} \\ &=\frac{\sqrt{(x+3)^2}}{(x+3)(x-3)} \\ &=\frac{x+3}{(x+3)(x-3)} \\ &= \frac{1}{x-3} \end{align*}

## 37.

The Correct Answer is (C) — (A) and (E) are true, since the lines l and m are parallel. You can tell (B) is true, because line m is above the point of origin when it intersects with the y-axis. (D) is true, since the lines l and m extend into infinity.

## 38.

The Correct Answer is (K) — To solve this question, substitute t = 3 into the expression: $P(3)=20(10)^3=20(1000)=20,000$.

## 39.

The Correct Answer is (A) — The question tells you that the initial population is 50 and that it increases by a factor of 2, so you can eliminate (D) and (E). The bacteria doubles every half hour, which means that it doubles twice in one hour. (C) does not provide the population per hour, since there is no constant. (B) assumes that the population doubles once every two hours. This leave (A) as the correct answer.

## 40.

The Correct Answer is (G) — To answer this question, substitute what you know into the formula given at the beginning of the question set. You know that the final population is 1000 and that the initial population is 40. You also know that the population grows by a factor of 5, and the growth took place over 5 hours. So, $1000=40(5)^5x$, therefore $25=5^5x,\text{ and } x=\frac{2}{5}$.

## 41.

The Correct Answer is (D) — To find the area of the circle, find the diagonal of the square:

$\sqrt{3^2+3^2} = \sqrt{18}$

This diagonal is the diameter of the circle, so divide it by 2 to get the radius. Then, to find the area of the circle use the formula, substituting the diameter of the circle and dividing it by two:

\begin{align*} A&=\pi(\frac{\sqrt{18}}{2})^2 \\ A&=\frac{18}{4}\pi \\ A&=\frac{9}{2}\pi \end{align*}

## 42.

The Correct Answer is (J) — The outer perimeter is the length of the longest side multiplied by four: (4)(10) = 40. The inner length of the square is the difference of the total length of one side and two ‘walls’ of the square, or 10 – 6 = 4. It’s perimeter is therefore:

$4\times(10-2\times3)=4\times4=16$
Adding the two perimeters together gives 56.

## 43.

The Correct Answer is (D) — To solve this question, guess and check is the most efficient method. You know that the final answer is a whole number, so any answer choices where the larger number is odd can immediately be eliminated: (A), (C), and (E). Substituting 11 and 12 gives 27; substituting 13 and 14 gives 33.

## 44.

The Correct Answer is (J) — The question asks you for the line perpendicular to the given line, which has a slope of 5. So, you know that the slope of the new line must be $\frac{-1}{5}$; this leaves (H) and (J). To find the y-intercepts, substitute the given point into the equation of the new line: $4=(\frac{-1}{5})(-4)+b,4=\frac{4}{5}+b,\frac{16}{5}=b$.

## 45.

The Correct Answer is (B) — Scalar matrix multiplication multiplies every value in the matrix by a constant. Looking at matrix A and B, each of the values in matrix B has changed by a factor of -3.

## 46.

The Correct Answer is (J) — After triangle BFG is reflected across the y-axis, it resembles answer choice F. However, remember that you still have to translate the triangle 4 units to the right, giving answer choice J.

## 47.

The Correct Answer is (D) — The formula for a permutation when choosing from n objects from r positions is $\frac{n!}{(n-r)!}$ The answer for this questions is therefore: $\frac{120!}{(120-9)!}$.

## 48.

The Correct Answer is (K) — Factoring the expression: \begin{align*} y&=\frac{x^2-7x-8}{x+1} \\ y&=\frac{(x+1)(x-8)}{x+1} \\ y&=x-8 \end{align*}

where x ≠ -1. Note that the function is undefined and -1 because the denominator will equal 0 at this point. (K) is the only graph that shows this expression.

## 49.

The Correct Answer is (D) — According to log rules, if $log_cx = a$ and $log_cy = b$, then $log_cxy = a + b$. Since both of the logarithms in the question have the same base, this rule applies.

## 50.

The Correct Answer is (H) — To find the width of the wall in bricks, divide the total length, by the length of one brick: $\frac{126}{18} = 7$. To find the height of the wall in bricks, divide the total height by the height of one brick: $\frac{42}{7} = 6$. To find the total number of bricks needed to build the wall, multiply these two numbers together: $(6)(7)=42$

## 51.

The Correct Answer is (C) — To find the point of intersection, equate the two expressions in terms of one variable. The easiest variable to isolate for is x, so the second expression becomes x = 6 - y. Equating the two expressions gives:

\begin{align*} 6 - y &= \frac{2y - 7}{5} \\ 30-5y&=2y-7 \\ 37 &= 7y \\ y &= \frac{37}{7} \end{align*}

(C) is the only choice with this y-coordinate. To double check, you can substitute $y = \frac{37}{7}$ and then solve for x.

## 52.

The Correct Answer is (G) — Subtract the volume of the triangular prism from the volume of the box:

\begin{align*} A&=(14)(14)(6)-(\frac{1}{2})(14)(6)(14) \\ A&=1176-588 \\ A&=588 \end{align*}

## 53.

The Correct Answer is (B) — The vertex begins in quadrant I, then is rotated 180ᣞ to land in quadrant III. When it is reflected across the x-axis, it lands in quadrant II. It may help to draw in the vertex into the graph given in the question in order to help visualize the translations.

## 54.

The Correct Answer is (H) — The only trigonometric equation that can be formed using the information given in the question and diagram is that matches the answer options is:

\begin{align*} \sin(25) &= \frac{12}{\text{length of ladder}} \\ \text{length of ladder} &= \frac{12}{\sin(25)} \end{align*}

## 55.

The Correct Answer is (C) — Find the point that is directly opposite point (7, -1) on the circle that, when connected, would form the diameter of the circle. The slope of the line formed by the given point and the center of the circle is $\frac{1}{3}$, which you can figure out because this line would pass through the center (4,-2). The slope of the line passing through (7,-1) and (4,-2) is $\frac{1}{3}$.

Using this slope, you can determined that the point (1, -3) is the correct choice.

## 56.

The Correct Answer is (K) — You know that the actual length of a is 16. The question gives you a in a ratio, where it is 4. The ratio of the actual length to the trigonometric ratio that the question gives you is 16:4, or 4:1. Because of this, you know that the length of c must be 4 times that of the number given to you in the trigonometric ratio, or 4×5=20.

## 57.

The Correct Answer is (C) — The question deals with angles that are in quadrant I. According to the CAST rule, all trigonometric ratios in this quadrant are positive. Therefore, only statement III is true.

## 58.

The Correct Answer is (G) — Recognize the trigonometric identity, $\sin^2(x) + \cos^2(x) = 1$. Substituting this into the expression gives $\frac{2(1)}{4} = \frac{1}{2}$.

## 59.

The Correct Answer is (D) — The formula for the number of diagonals formed by a polygon is 3 × number of sides. For a nonagon, which has 9 sides, the number of diagonals that it can form is 3 × 9 = 27.

Note that if the question asked for distinct diagonals, you would have to divide this number by 2.

## 60.

The Correct Answer is (K) — Recognize the right triangle formed by the height, the slant height, and the base of the triangle. This gives sides 3 and 5, two sides of a 3-4-5 Pythagorean triple. Knowing this, you also know that the height of the pyramid is 4. Substituting this, along with the other information provided in the diagram, into the formula gives you:

\begin{align*} &=\frac{l^2 h}{3} \\ &=\frac{(6^2)(4)}{3} \\ &=\frac{144}{3} = 48 \end{align*}