## 1.

The Correct Answer is (B) — In scientific notation, the decimal place comes after the first digit of the number, which is multiplied by a power of 10. The only answer choice that gives the number stated and follows this rule properly is B.

## 2.

The Correct Answer is (H) — To answer this question, it may be helpful to work backwards. To take the average of 2 numbers, you add them together and divide by 2. To find one of the original numbers, multiply the average by 2 and subtract the known number: (184 × 2) — 170 = 198.

## 3.

The Correct Answer is (E) — To find the GCF, multiply together all of the prime factors that the numbers have in common. $72=2\times2\times2\times3\times3$ and $88=2\times\2\times2\times11$ and $120=2\times2\times2\times3\times5$. For these three numbers, the common factors multiplied together give a product of $2\times2\times2=8$. In this instance, however, it may be faster to guess and check.

## 4.

The Correct Answer is (F) — The questions equates two expressions that are both in terms of z. First, isolate for z. Then, collect like terms and solve.

\begin{align*} 6z+2&=5-11z \\ 6z+11z&=5 \\ 17z&=3 \\ z&=\frac{3}{17} \end{align*}

## 5.

The Correct Answer is (E) — You know that the area of a rectangle is length multiplied by width, so to find the missing length, divide the area by the known width: $\frac{320000}{400}=800$.

To find the perimeter, add all the lengths together: $400+400+800+800=2400$.

If you chose (A), you probably found the missing length and stopped there.

## 6.

The Correct Answer is (J) — To answer this question, it may be easier to convert the ratio into a decimal: $\frac{\frac{2}{5}}{\frac{1}{8}}=3.2$. The only ratio from the answer choices that gives this same ratio is J.

You can also multiply both sides of the ratio by $\frac{5}{8}$ to get 16:5, or J.

## 7.

The Correct Answer is (D) — Since this is a function, simply substitute x=2 and solve. \begin{align*} f(x)&=(a^4b^3)^{2x+1} \\ f(x)&=(a^4b^3)^{2(2)+1} \\ f(x)&=(a^4b^3)^5 \\ f(x)&=a^{20}b^{15} \end{align*}

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

## 8.

The Correct Answer is (K) — First, determine how many times the recipe can be completed with the given quantities of ingredients. $\frac{7.5}{1.25}=6$ iterations of the recipe. This means $6\times5=30$ cupcakes.

If you chose (G), you probably found the number of times the recipe could be done and stopped there.

## 9.

The Correct Answer is (D) — The number can be odd or even, so (A) and (B) are incorrect. Doubling a number makes the number even, so (C) is incorrect. Tripling a number can make the product odd or even depending on the original number, so (E) is incorrect. (D) is the correct answer: doubling the number will always result in an even number, and adding 1 to an even number will always result in an odd number.

## 10.

The Correct Answer is (H) — To find the angle, first find the proportion of the circle graph that homework occupies on the graph. $\frac{24-8-8-1-3}{24}= \frac{4}{24}= \frac{1}{6}$. Multiply this proportion by 360° to get the angle measure, or 60°.

## 11.

The Correct Answer is (B) — Expand and simplify the expression using the FOIL technique: \begin{align*} &=(x-5)(x+2) \\ &=x^2-5x+2x-10 \\ &=x^2-3x-10 \end{align*}

## 12.

The Correct Answer is (H) — Translating the question from words into equations, $a= \frac{1}{2}b$, and $3a=c$, . Rewrite the latter as $a= \frac{1}{3}c$ and substitute this into the first expression:

$\frac{1}{3}c= \frac{1}{2}b$ and $c= \frac{3}{2}b$

b is greater than c by a factor of $\frac{3}{2}$.

## 13.

The Correct Answer is (B) — Working through |x – 5| = 3, gives you: x – 5 = 3, which simplifies to x = 8, and –(x – 5) = 3, which simplifies to x = 2. Possible values of x are therefore 8 and 2. Substituting x = 8 into the second equation yields 33, which is not an answer choice. Substituting x = 2 into the second expression yields 9.

## 14.

The Correct Answer is (H) — The ratio of the longer side of the floor to the longer side of the blueprint is 15:10, or 3:2. This means that the floor is 1.5 times as large as the blueprint. Cross multiply to find the length of the floor: $\frac{6}{10}=\frac{x}{15}, x=\frac{6\times15}{10}=\frac{90}{10}=9$.

## 15.

The Correct Answer is (C) — Since you know two of the angles in triangle CDE, you can find angle DCE, which is 55°. Since CDE and BCE are supplementary angles, you can find the value of BCE by subtracting 55° from 180°, which yields 125°. Since there are 180° in a triangle, subtracting 125° from 180° gives the value of the remaining two angles. x + y = 55°.

## 16.

The Correct Answer is (K) — Solving for x given the first equation gives x = 4. Substituting this into the second expression gives $4^2+4^3=16+64=80$.

## 17.

The Correct Answer is (D) — To solve this question, drawing a diagram may help.

Since the distance from the Sun to Jupiter is 65 cm and from the Sun to Mars is 20 cm, the distance from Mars to Jupiter must be 45 cm. To find the distance from Mercury to Mars, subtract the distance from Mercury to Jupiter, 60 cm, from the distance from Mars to Jupiter, 45 cm. 60 — 45 = 15 cm.

## 18.

The Correct Answer is (H) — From the question, if y is the larger number, then the smaller number is equal to 2y — 6. Subtracting this from the larger number gives y — (2y — 6). The question tells you this expression is equal to 1, therefore the equation is y — (2y — 6) = 1.

## 19.

The Correct Answer is (D) — You can factor $x^4$ out of the expression under the exponent, leaving $x^2 - x$ inside the brackets. Since the entire expression is to the exponent $\frac{1}{2}$, when $x^4$ is factored out from the exponent $\frac{1}{2}$ (or the square root), it becomes $x^2$. The final expression is $x^2\sqrt{x^2-x}$.

## 20.

The Correct Answer is (F) — To answer this question, substitute the known values using $a^2 + b^2 = c^2$ from the Pythagorean Theorem, and solve for the unknown height. \begin{align*} 6^2&=4^2+x^2 \\ 36&=16+x^2 \\ x^2&=20 \\ x&=\sqrt{20}=2\sqrt{5} \end{align*}

## 23.

The Correct Answer is (B) — This question is most easily solved by the process of elimination. You know that the line slopes downwards, which means the slope must be negative. This eliminates answer choices (C), (D), and (E). Of the two remaining answer choices, the slopes are $-2$ and $\frac{-1}{2}$. The line is not steep enough for a slope of $-2$ , therefore $\frac{-1}{2}$ must be the slope and (B) is the answer. This can be verified by solving for the slope: $\frac{0-2}{3-(-1)}=\frac{-2}{4}=\frac{-1}{2}$.

## 24.

The Correct Answer is (F) — The chimpanzee eats up to 63 bananas per week, or an average of $\frac{63}{7} = 9$ bananas per day. If the chimpanzee eats up to than 63 bananas a week, the average daily consumption must be less than or equal to 9 bananas. The only answer choice that represents this inequality is choice (F).

## 25.

The Correct Answer is (E) — To solve this question, simplify the expression by factoring the bottom and top separately, then cancelling out like terms when possible: \begin{align*} &=-(\frac{z^2-4}{16z^3-64z}) \\ &=-(\frac{z^2-4}{16z(z^2-4)}) \\ &=-\frac{-1}{16z} \end{align*}

## 26.

The Correct Answer is (J) — Set up a ratio between the height and the length of the door stopper: $\frac{2}{5}=\frac{3}{x}$. Cross multiply and solve for the length of the door stopper: $x = \frac{3\times5}{2}=\frac{15}{2}=7.5$

## 27.

The Correct Answer is (E) — Remember to apply the exponents first, according to the order of operations:

\begin{align*} &=(8x^2 y^3 )^2\times(-2x)^3 \\ &=64x^4 y^6×-8x^3 \\ &=-512x^7 y^6 \end{align*}

All of the answer choices other than (E) have a + or – y term at the end, which is incorrect.

## 28.

The Correct Answer is (F) — Set the two expressions on the right-hand side of both equations equal to each other, and then solve: \begin{align*} x+1&=-3x-3 \\ 4x&=-4 \\ x&=-1 \end{align*}

Substitute $x=-1$ into either of the original equations: $y=-1+1=0$.

The solution is the point that lies on both lines: (—1,0).

## 29.

The Correct Answer is (A) — The difference in votes, ab, must be greater than or equal to 30 votes. The only answer choice that represents this inequality is (A).

## 30.

The Correct Answer is (G) — You can rewrite 32 as $2^5$, so the expression becomes $(2^{5})^{\frac{2}{5}}$. Using exponent laws: $(2^{5})^{\frac{2}{5}}=2^2$.

## 31.

The Correct Answer is (C) — You can see from the right triangle formed at the right of the parallelogram that the height of the parallelogram is 4. This is because the right triangle is a 3-4-5 triangle. Alternatively, you can use the Pythagorean Theorem to solve for the missing length. It may, however, be faster to memorize some Pythagorean triples. To find the area of a parallelogram, multiply the height of the parallelogram by the base. (5+3)(4)=32.

If you chose (A), you probably forgot to add 3 to 5 when finding the length of the base.

## 32.

The Correct Answer is (H) — This is an arithmetic sequence because each new row has two more than the row before it. You can write out the number of circles for the first 9 rows and sum them together:

$5+7+9+11+13+15+17+19+21=117$

Or, recognize that each "pair" found by starting from the outside edges and working in, such as 5 and 21, 7 and 19, and so on sums to 26. There are four pairs that sum to 26 plus the remaining row of 13, so 4×26 + 13 = 117. If you are familiar with the Gaussian method of adding arithmetic sequences, recognize that the pattern of this sequence is $5+(n-1)\times2$, where n is the 9th row. 9th row $= 5+(n-1)2=5+(9-1)2=5+16=21$. The $9^{th}$ row will have a total of $\frac{n}{2}(5+21)=\frac{9}{2}(26)=117$ circles.

## 33.

The Correct Answer is (E) — Remember that $\sin a = \frac{\text{opposite}}{\text{hypotenuse}}$. You can sub in the relevant values from the question to get $\sin a=\frac{6}{12}=\frac{1}{2}$.

## 34.

The Correct Answer is (G)$\frac{60}{4}=15$ students participate in musical theater. Of those 15 students, $\frac{15}{3}=5$ students do not play an instrument. Therefore, 10 students participate in musical theater and play an instrument.

## 35.

The Correct Answer is (E) — To average $100 per day for a week, the fundraiser needs to raise a total of$700. Sum up the money they have made in the first 6 days: 52+84+106+66+94+98=500. $200 more needs to be raised. Each cookie is$2, therefore 100 cookies need to be sold.

## 36.

The Correct Answer is (H) — The outputs of the function increase at regular intervals of 3. For each increase by 1 in the x column, the output of the function increases by 3 times as much. Therefore, the slope of the function, or the value m is 3.

Alternatively, you could plug any two points into the slope formula to get an m value of $\frac{7-4}{1-0}=3$.

## 37.

The Correct Answer is (C) — You know that angle BFC is 90°, and that the triangle BCF is an isosceles triangle. Therefore, the other two angles, including angle FBC, of the triangle must each be 45°.

## 38.

The Correct Answer is (H) — The equation of the first line is $y = \frac{1}{2}x$. Since it passes through the origin, the y intercept must be 0. Using the given point of intersection and the origin, you can deduce that the slope is $\frac{1-0}{2-0}=\frac{1}{2}$.

The slope of the second line is perpendicular to the first line, so it is the negative reciprocal: —2. To find the y-intercept, substitute the known point of (2,1) into the equation of the second line and solve. \begin{align*} 1 &= (—2)(2) + b \\ 1 &= —4 + b \\ 5 &= b \end{align*}

The y intercept is 5, therefore the line intersects the y axis at (0,5).

## 39.

The Correct Answer is (B) — This problem requires you to work backwards from what you are given. You know there are 5 times as many footballs sold as baseballs; if the store sold 720 footballs, it sold $\frac{720}{5}=144$ baseballs. Additionally, there are 3 times more baseballs sold than hockey sticks: $\frac{144}{3}=48$ hockey sticks sold.

## 40.

The Correct Answer is (J) — The area of the larger square is $(2+3)^2=25$. To find the side length of the smaller square, use $a^2 + b^2 = c^2$ from the Pythagorean Theorem. $2^2+3^2=c^2,c^2=13, c=\sqrt{13}$. The area of the smaller triangle is $(\sqrt{13})^2=13$. Now, divide the area of the smaller square by the larger square and multiply by 100 to get the percentage of the larger square that is occupied by the smaller square: $\frac{13}{25}\times100 = 52\%$.

## 41.

The Correct Answer is (B) — Simply plug in the given expressions for c and d, and reduce.

\begin{align*} \frac{c}{d}&=\frac{2x^2-10x-28}{4x+8} \\ \frac{c}{d}&=\frac{2(x^2-5x-14)}{4(x+2)} \\ \frac{c}{d}&=\frac{2(x-7)(x+2)}{4(x+2)} \\ \frac{c}{d}&=\frac{x-7}{2} \end{align*}

## 42.

The Correct Answer is (F) — The shaded area is above the line $y=\frac{1}{2}x-\frac{1}{2}$, indicating greater than $y=\frac{1}{2}x-\frac{1}{2}$. The shaded area is also above $3x+2$, indicating greater than $3x+2$. The only answer choice that combines both of these inequalities is (F).

## 43.

The Correct Answer is (C) — Expanding and simplifying this expression gives:

\begin{align*} &=i^2-2i+2i-4 \\ &=i^2-4 \\ &=-1-4=-5 \end{align*}

## 44.

The Correct Answer is (G) — For this question, treat the vectors as equations, where i and j are variables. Substitute the expressions u and v and solve. \begin{align*} &=2u — 3v \\ &=2(7i + 4j) \\ &=14i + 8j — 9i + 6j \\ &=5i + 14j \end{align*}

## 45.

The Correct Answer is (B) — This question requires knowledge of special triangles. The easiest way to solve this question is to be familiar with the trigonometric ratios of special triangles. Alternatively, you can draw out the triangle to help you deduce the value of θ.

The sides of the triangle are $1-2-\sqrt{3}$. The longest side of the triangle must be opposite the greatest angle. Since 2 is opposite the right angle, 60° must be opposite $\sqrt{3}$ and 30° must be opposite 1. Therefore, the value of theta is 30°.

## 46.

The Correct Answer is (J) — Since the perimeter of the square is 16 inches, you know that $\frac{16}{4}=4$ is the length of one of the sides. From the diagram, you can see that a single slant height of the trapezoid (the side lying along the square) is 2 inches long. To find the two bases of the trapezoid, use the Pythagorean Theorem with the legs of each triangle being formed by the square. The total slant lengths of the trapezoid of colored glass is $2+2=4$. The long base of the trapezoid measures $4^2+4^2=c^2,c^2=32,c=4\sqrt{2}$. The short base of the trapezoid measures $2^2+2^2=c^2,c^2=8,c=2\sqrt{2}$. Adding all of these lengths together gives $4+4\sqrt{2}+2\sqrt{2}=4+6\sqrt{2}$.

## 47.

The Correct Answer is (A) — If you rewrite the equation as a logarithm, you get $\log_x a = log_x \text{?}$. From this equation, you can deduce that ? = a.

## 48.

The Correct Answer is (H) — Since the company wants to cover just the top and bottom of each bar, you only need to sum the area of the two rectangular bases of the gold bar: (7)(20)+(6)(20)=140+120=260.

## 49.

The Correct Answer is (C) — To find the number of gold bars that the company can make, divide the total amount of gold that they have by the volume of one gold bar. $\frac{100,000}{1040}=96.15$, or 96 bars can be made – round down, since you cannot make a fraction of a gold bar. Now, find the amount of gold left over by subtracting the total volume of all the gold bars from the total amount of gold. 100,000-(1,040×96)=100,000-99,840=160.

## 50.

The Correct Answer is (K) — Substitute the given values into the equation given for the purity of gold and solve: $\frac{12.52}{12.55}\times100=99.76\doteq99.8$

## 51.

The Correct Answer is (B) — All of the other answer choices are infinite, except for a line segment, or (B).

## 52.

The Correct Answer is (F) — If only v can change, v is raised to the power of 2, and F is doubled, the only factor of multiplication that will give 2F is $\sqrt{2}$. Choices (G) and (H ) would increase F by too much, and choices (J) and (K) would decrease F.

## 53.

The Correct Answer is (A) — Isolate the first equation for v in terms of u, and isolate the second equation for w in terms of v.

$v=\frac{5u}{3}, w=\frac{4}{v}$

Substitute the value of v into the second equation and solve:

\begin{align*} w&=\frac{4}{\frac{5u}{3}} \\ w&=4\times\frac{3}{5u} \\ w&=\frac{12}{5u} \end{align*}

## 54.

The Correct Answer is (K) — The length of the base is 2(1-0)=2, as seen by the top length of the triangle on the left. Therefore, a=2. Since (a,b) lies on the x axis, b = 0. $\frac{0}{2}+2=0+2=2$.

## 55.

The Correct Answer is (B) — Recognize from the diagram that the lengths form a right triangle. To find the value of d, you can use the Pythagorean Theorem:

\begin{align*} 60^2+50^2&=d^2 \\ d^2&=6100 \\ d&=\sqrt{6100} \\ d&\doteq78 \end{align*}

You could also use the cosine law, which is stated in the note below the diagram, by substituting the given values and calculating the value of c. However, there is a lot of potential for error when using the cosine law, and it is a much more time consuming.

## 56.

The Correct Answer is (K) — Choices (F) and (J) directly contradict the statement in the question. Choices (G) and (H) are not necessarily true: the statement original statement does not dispute that, for example, a yellow wagon can be painted. The only choice that must be true is choice (K).

## 57.

The Correct Answer is (E) — The minimum volume displaced by the 3 people is, at least, the volume required to make the tank full. As you can see from the diagram, the empty volume of the tank before the 3 people enter is: (30)(15)(0.5)=225 cubic feet.

## 58.

The Correct Answer is (J) — The easiest way to answer this question is by process of elimination. The domain of the three functions is given in the question. Since you know that all the functions are less than or equal to 2, the filled dots must be on the right side of the graph; this eliminates (H) and (K). Of the remaining choices, the parabola is in the same place for all of them, meaning that you have to focus on functions g(x) and f(x), the linear graphs. At x = -1, g(x) = 0, eliminating (G). At x = 1, f(x) = 0, eliminating (F). The answer is (J). You may have incorrectly checked the endpoints of the functions if you didn't notice that the domain is -2 < 2x ≤ 2, which simplifies to -1 < x ≤ 1.Of the remaining 3 choices, (J) is the only one that correctly graphs g(x) and h(x).

## 59.

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

Drawing a perpendicular line from the center of the table to the wall and a line from the center of the table to the point where the table meets the wall gives a right triangle with a hypotenuse of 40 centimeters (since the radius is 40 centimeters) and a height of 32 centimeters. Solve for the missing edge by using the Pythagorean Theorem: $40^2=32^2+c^2,c^2=576,c=24$. Don’t forget to double this value, since the question asks for the entire length of the flat edge of the table. Doubling 24 gives 48.

If you chose (A), you probably found half of the length of the flat edge and stopped there.

## 60.

The Correct Answer is (G) — Changing a changes the height of the tides, or the amplitude. Changing c changes the base height of the tides around which they oscillate, or translates the function up or down. Changing x changes the time, or the input value. Changing y changes the height of the tide at any given time, or the output value. Changing k changes the time it takes for a tide to complete its interval, or the period. Therefore, changing k would change the time between high tides.