## Math Workout for the GMAT

## Part IV GMAT: Math Practice Test

### Chapter 12 Answers and Explanations

**ANSWER KEY**

__1__. E

__2__. C

__3__. A

__4__. E

__5__. D

__6__. B

__7__. E

__8__. B

__9__. D

__10__. D

__11__. A

__12__. C

__13__. C

__14__. D

__15__. B

__16__. B

__17__. D

__18__. B

__19__. B

__20__. B

__21__. D

__22__. E

__23__. E

__24__. B

__25__. C

__26__. B

__27__. E

__28__. D

__29__. A

__30__. D

__31__. C

__32__. C

__33__. B

__34__. C

__35__. C

__36__. B

__37__. D

**ANSWERS AND EXPLANATIONS**

__1__.**E**If the area of the uncut square was 100, then each side of the square is 10. Given that the line on each side of the circle is 3, the diameter of the circle is 10 − 3 − 3 = 4, so the radius is 2. That means the area of the circle is π*r*^{2} = 2^{2}π = 4π, or approximately 12.4 (using 3.1 as an approximation for π). Thus, the area of the cut sheet is about 100 − 12.4 = 87.6. Choose (E).

__2__.**C**You can translate Statement (1) into *b* = *a*. However, that’s only 1 equation in 2 variables; you can’t solve that. For example, you could have *a* = 8 and *b* = 2, in which case *a* – *b* = 6, or you could have *a* = 16 and *b* = 4, in which case *a* – *b* = 12. Narrow your choices to (B), (C), and (E). You can translate Statement (2) into *a* + *b* = 100. Once again, that’s only 1 equation in 2 variables; there are many possible solutions. Eliminate (B). With both statements together, you have 2 equations in 2 variables. You can solve that system to find *a* and *b* and then calculate *a* – *b*. Choose (C).

__3__.**A**Try Plugging In The Answers. With (C), *z* = 3, so = = , which is undefined, so eliminate (C). With (A), *z* = 9, so = = = 2. That matches what the question stated, so (A) is the right answer. Alternatively, you could factor the quadratic terms to get = = . Solve = 2 to get *z* = 9. Choose (A).

__4__.**E**Translate Statement (1) into 10 < 2*j* < 32. Divide by 2 to get 5 < *j* < 16. More than one value is possible, so narrow the choices to (B), (C), and (E). Translate Statement (2) into 4 < < 10. Multiply by 2 to get 8 < *j* < 20. Since more than one value is possible, eliminate (B). Combining both statements, you still have more than one possible value for *j*, such as *j* = 9 or *j* = 10. Choose (E).

__5__.**D**With Statement (1), you can find the fee for June; it is 180 × 0.25 = $45. You can answer the question, so narrow the choices to (A) and (D). With Statement (2), you can find the fee for 460 minutes by using the two rates ($0.25 for the first 200 minutes and $0.50 for 260 minutes over 200) and then dividing the result by 4 to get the fee for June. Since you can answer the question, choose (D).

__6__.**B**Try Plugging In The Answers. With (C), the press produces 23 pages per minute or 23 × 60 = 1,380 pages in an hour. Of those, 15% × 1,380 = 207 are unusable, so there are 1,380 − 207 = 1,173 usable pages. That’s too many, so eliminate (C) and try (B). At 20 pages per minute, the press produces 20 × 60 = 1,200 pages in an hour. Of those, 15% × 1,200 = 180 are unusable, leaving 1,200 − 180 = 1,020 usable pages. That’s what the question stated, so choose (B).

__7__.**E**Plug *x* = 6 and *y* = 2 into the equation to get 2(6) − 2*c* = 18. Solve that to get *c* = −3. That means the equation is 2*x* + 3*y* = 18. (Be careful with the negative signs!) Plug *y* = 3 into that to get 2*x* + 3(3) = 18, which you can solve to get *x* = . Choose (E).

__8__.**B**This question is essentially about divisibility. If you increase *p* by 20%, you get 1.2*p*. Since *p* must be in whole cents, change each answer into cents by moving the decimal point 2 places to the right, and then divide each answer by 1.2. You’re looking for the one that isn’t an integer after the division. (B) becomes = . Choose (B).

__9__.**D**Let *p* = price per copy. From the given information, you can set up the equation *n* × *p* = $31.50. To find *p*, you need the value of *n*. You can translate Statement (1) into 2*n* × $1.75 = $31.50. There is only one variable, so you can solve that for *n*, and then plug that into the original equation to find the price. Statement (1) is sufficient, so narrow the choices to (A) and (D). You can translate Statement (2) into (*n* − 4) × (*p* + $2.80) = $31.50. Although there are two variables in this equation, you also have the equation from the initial setup. With two equations for two variables, you can solve for *p*. Statement (2) is sufficient, so choose (D).

__10__.**D**Since the number of homework assignments is never defined, you can plug in your own number, such as 10 assignments. To work with averages, you should set up average circles, as shown below. For the homework already submitted, Torry has × 10 = 4 assignments, with an average grade of 75. So, he has a total of 4 × 75 = 300 points on these assignments. Since he has 10 assignments total, to get an average grade of 90, Torry needs 10 × 90 = 900 points. That means he needs to get 900 − 300 = 600 total points on his remaining assignments. Since there are 6 assignments remaining, that works out to an average grade of = 100 per assignment. The formula for percent change is . In this case, that is = = 33%. Choose (D).

__11__.**A**This one is a bit tricky, because the order of *m* and *n* is reversed in the question stem. You need to determine the value of *n* @ *m*, which becomes 4 @ 3 when you plug in the values provided. Take those numbers and plug them into the formula *m* @ *n* = (2*m* – *n*)(*m* + *n*); remember that *m* = 4 and *n* = 3 for this formula, because the 4 is first and the 3 is second. Thus, 4 @ 3 = [(2)(4) − 3] × (4 + 3) = 5 × 7 = 35. Choose (A). If you chose (B), you forgot to switch the order of *m* and *n*.

__12__.**C**With 150 students in each of 3 grades, there are 3 × 150 = 450 students altogether. During the blizzard, 10% × 150 = 15 fourth-graders, × 150 = 25 fifth-graders, and 60 sixth-graders miss school. That is 15 + 25 + 60 = 100 students absent. To find percent change, divide the difference by the starting number. In this case, = = %. Choose (C).

__13__.**C**Set up an average circle, as shown below, to find Laura’s average figurines per box before the addition. 207 figurines in 9 boxes is = 23 figurines per box. After the addition, she has = approximately 32 figurines per box. Feel free to round off numbers, because the question says “approximate.” The percent change is = ≈ 39%. The closest answer is 40%, so choose (C).

__14__.**D**Try Plugging In The Answers. You can eliminate (A) and (B) right away, because 2 and 3 can’t be the largest of 3 consecutive primes. With (D), the three consecutive primes are 3, 5, and 7. Their sum is 3 + 5 + 7 = 15. The largest integer smaller than 3 is 2. The product is 15 × 2 = 30, which matches what the question says. Choose (D).

__15__.**B**An obvious thing to do would be to calculate the volume of the box, 12 × 18 × 10 = 2,160 cubic inches, and the volume of the cylinder, π*r*^{2}*h* = 3^{2} × 5 × π = 45π cubic inches. If you divide the volumes, you get = ≈ 15.3, which seems to imply that 15 cans would fit. However, that trap answer ignores the dead space that comes from putting round objects in a rectangular box. If you stack the cans side by side in rows, each one takes up 6 inches (the diameter) in each horizontal direction. So you can fit = 6 cans in each horizontal layer. Since the box is 10 inches tall and the cans are 5 inches tall, you can fit 2 layers, or 2 × 6 = 12 cans. Choose (B).

__16__.**B**Plug in some numbers for *a*, *b*, and *c* that match the equations. Suppose *a* = 2, *b* = 4, and *c* = 10. Take those values and fill in the lengths on the perimeter of the rectangle. That makes the rectangle a 12 × 20 rectangle, so the area is 240. Since each corner is a right angle, you can find the area of each of the four triangles at the corners. The area of the upper-left triangle is × 4 × 14 = 28. The area of the upper-right triangle is × 6 × 10 = 30. The area of the lower-left triangle is × 8 × 10 = 40. The area of the lower-right triangle is × 2 × 10 = 10. The total area of the triangles is 28 + 30 + 40 + 10 = 108, so the shaded region is 240 − 108 = 132. As a fraction of the whole rectangle, the shaded region is = . Choose (B).

__17__.**D**From the initial setup, you can write the equation *g* + *r* = 225. With Statement (1), you can substitute *r* = 75 to find the number of green pencils. Statement (1) is sufficient, so narrow your choices to (A) and (D). With Statement (2), you can write the equation *g* = 2*r*. With two equations and two variables, you can solve for the number of green pencils. Statement (2) works, so choose (D).

__18__.**B**Substitute *x*^{2} for *y* in the equation *xy* = 125. That gives you *x*^{3} = 125, so *x* = 5. Now plug that into the equation *x*^{2} = *y* to get *y* = 5^{2} = 25. So *x* – *y* = 5 − 25 = −20. Choose (B).

__19__.**B**The circumference of the hoop will tell you how far the hoop travels in one revolution. The formula for circumference is *C* = 2π*r*. In this case, the circumference of the hoop is (2)(π)(12.5) = 25π = approximately 75 inches. Convert this measurement to feet (because the problem wants to know how many minutes it takes the hoop to roll 75 feet). To make this conversion, divide 75 inches by 12 (the number of inches in a foot): 75 ÷ 12 = 6.25. Now you can set up the following ratio:

Cross multiply to obtain the following equation:

6.25*x* = 750

*x* = 120 seconds

Divide 120 seconds by 60 (the number of seconds in a minute). The result is 120 ÷ 60 = 2 minutes. Choose (B).

__20__.**B**You know that 63% of all votes cast were for the winner. To find the actual number of votes for the winner, you need to know the total number of votes cast. Statement (1) tells you the number of eligible voters, but the actual number of votes cast might be smaller. That’s insufficient, so narrow the choices to (B), (C), and (E). With Statement (2), you know that 37% of votes cast equals 55,500, since 63% for the winner implies 37% for the others. You can use that equation, 0.37*x* = 55,500, to find the total number of votes, which allows you to answer the question. Statement (2) is sufficient, so choose (B).

__21__.**D**You know that *AB* = 5 and *AC* = 12. What you don’t know is whether they are the base and height of the triangle. Base and height must be perpendicular, so you need to know whether *y* is a right angle. With Statement (1), you know that *BC* is 13. That means the triangle is a right triangle, and you can plug 12 and 5 in for the base and height to find the area of the triangle. Narrow the choices to (A) and (D). With Statement (2), *y* is a right angle, so you can use 12 and 5 as the base and height. Statement (2) also works, so choose (D).

__22__.**E**With Statement (1), you know that two of the sides are equal. The triangle could be equilateral, if all three are equal, but that’s not necessarily the case. Imagine that *NO* is very small, while the other sides are equal. Narrow the choices to (B), (C), and (E). With Statement (2), knowing that *h* = 5 may make you think that the triangle is equilateral. However, that’s true only if the middle line is perpendicular to *MO*, so that it becomes the height of the triangle. Imagine that the line leans to the right instead of pointing straight up. Eliminate (B). With both statements together, you still can’t conclude that the middle line is the height of the triangle. See the following diagram for a counterexample. Choose (E).

__23__.**E**With Statement (1), you know that Brand *X* was chosen by an overall majority of participants, but you don’t know whether that translates into a majority in three of the contests. It’s possible that Brand *X* had received 51 votes in two of the taste tests, winning those, but only 17 or 18 votes in each of the others, losing those, so that Brand *Y* was the overall winner. Narrow the choices to (B), (C), and (E). With Statement (2), you know that the winner won three of the first four contests, so that the last one was moot, but you don’t know which brand that was. Eliminate (B). With both statements, Brand *X* still could be the overall winner or the loser, and knowing the order of the wins and losses wouldn’t change them. Choose (E).

__24__.**B**With Statement (1), primes 2, 3, 5, 7, 11, 13, and 17 are definitely less than *k*, but you don’t know whether 19 and 23 are less than *k*. Narrow the choices to (B), (C), and (E). With Statement (2), primes 2 through 23 are less than *k*. You don’t know whether 29 is greater than *k* or equal, but in either case, it’s not less than *k*. That’s enough to answer the question, so choose (B).

__25__.**C**You’re looking for the value of *k* + *p* + *s*. With Statement (1), you can write the equation *k* + *p* = . But that’s only one equation for three variables, which is insufficient to solve the system and answer the question. Narrow the choices to (B), (C), and (E). With Statement (2), you can write the equations *k* + *s* = 109 and *p* + *s* = 126. However, two equations are not enough to solve for all three variables. Eliminate (B). With both statements together, you have three variables for three equations, which is enough to find all three variables and answer the question. Choose (C).

__26__.**B**With Statement (1), you don’t know whether the distance from the origin to *A* is the same as the distance from the origin to *D*. If it were the same, you could apply the Pythagorean theorem to find the length of the side of the square and then its area. Since you don’t know if it’s the same, then you can’t find the side of the square. That’s not sufficient, so narrow the choices to (B), (C), and (E). With Statement (2), you know that the distance between *B* and *D* is 8. That’s the diagonal of the square. You could use the 45-45-90 triangle ratio to find the sides of the square and then its area. Choose (B).

__27__.**E**Try Plugging In. With Statement (1), suppose *x* = 5, *y* = 3, and *z* = 10; the median would be *x*. However, if *x* = 5, *y* = −20, and *z* = 0, the median would be *z*. That’s two possible answers, so narrow the choices to (B), (C), and (E). With Statement (2) alone, you don’t know anything about *z* and whether it’s bigger or smaller than *x* or *y*. Eliminate (B). With both statements together, you could still use the two sets of numbers plugged in above. With two possible answers, that’s still insufficient. Choose (E).

__28__.**D**Plug In. If *m* = −2 and *n* = −3, then you can eliminate (A) and (E). Try weird numbers such as 1, or −1 in this case. If *m* = *n* = −1, then you can eliminate (B) and (C). Choose (D).

__29__.**A**Plug In for the amount of water. Suppose the large bottle has a capacity of 36 gallons. (The number 36 works nicely with the 3s and the 4 in the fractions.) It currently contains × 36 = 12 gallons of water. The small bottle has a capacity of × 36 = 24 gallons and currently holds × 24 = 18 gallons of water. If you combine the water, you get 12 + 18 = 30 gallons of water, which is = of the capacity of the larger bottle. Choose (A).

__30__.**D**Try Plugging In The Answers and the ratio box. The ratio of men to women is 3 to 4 at the start, so the ratio total is 7 people. With (C), you have an actual total of 28, so the multiplier is = 4, giving you 3 × 4 = 12 men and 4 × 4 = 16 women to start. Adding 10 men gives you 22 men and 16 women, or a = ratio. That’s not correct. Eliminate (C) and try another answer. With (D), you have an actual total of 42, so the multiplier is = 6, giving you 3 × 6 = 18 men and 4 × 6 = 24 women to start. Adding 10 men gives you 28 men and 24 women, or a = ratio. That matches the information in the question, so choose (D).

__31__.**C**Try Plugging In. Suppose the cube has an edge of 4 units. That means the height of the cylinder is 4 and its radius is 2. The cube has a volume of 4^{3} = 64. The volume of the cylinder is π*r*^{2}*h* = 2^{2} × 4 × π = 16π. The ratio of the cube to the cylinder is = . Choose (C).

__32__.**C**Try Plugging In. You’ll probably need to try several sets of numbers, and you need one of the answers to be true at least once. The key is to try *y* = 2. If *x* = 3 and *y* = 2, then (C) becomes = 2, which is an even integer. None of the other answers are ever true if *x* and *y* are different from each other and primes. (*Remember:* 1 is not prime.) Choose (C).

__33__.**B**The best way to solve this problem is first to find the probability that there are *not* at least 2 consecutive men’s speeches and then subtract that probability from 1. The only way to avoid having consecutive men’s speeches is to alternate man-woman-man-woman-man. For the first spot, there are 3 men out of 5 people. For the second spot, there are 2 women out of 4 people; and so on. That probability is × × × × = = . That’s the probability of what you *don’t* want to occur, so the probability of what you *do* want is 1− = . Choose (B).

__34__.**C**The best way to think about this question is to focus on the final round, the one in which two children choose one number and one child chooses the other, so that there is a winner. It doesn’t really matter whether this is the first round or the hundredth, the probabilities will come out the same. In this final round, there are three possibilities: Ringo chooses 2 while both John and Paul choose 1; Ringo and John choose 2 while Paul chooses 1; and Ringo and Paul choose 2 while John chooses 1. Each of the three possibilities is equally likely, and Ringo wins in only one of them, so the probability that he wins is . Choose (C).

__35__.**C**With Yes/No questions, try Plugging In. For Statement (1), you can plug in both an integer, such as *q* = 2, and a non-integer, such as *q* = . Since you can get both “Yes” and “No” answers, narrow the choices to (B), (C), and (E). For Statement (2), you can plug in both an integer, such as *q* = 2, and a non-integer, such as *q* = . That’s insufficient, so eliminate (B). With both statements together, you can plug in an integer such as *q* = 2, but there are no non-integers that fit both statements. Since you can get only the “Yes” answer, that is sufficient. Choose (C).

__36__.**B**The easiest way to solve this problem is Plugging In The Answers. The correct answer must have the right slope for both line *k* and the line through the origin. Using the formula for slope, you need both = − and = −2. All of the answers work for the line through the origin, but only answer (B) also works for line *k*. With (B), = = − and = = −2. Choose (B).

__37__.**D**There are three orders in which Will can set up his password: letter-digit-digit, digit-letter-digit, and digit-digit-letter. The number of permutations for letter-digit-digit is 26 × 10 × 9 = 2,340. Since the digits must be distinct, there are only nine options left for the second digit after the first is chosen. Digit-letter-digit and digit-digit-letter each have 2,340 permutations as well, so Will has 2,340 + 2,340 + 2,340 = 7,020 possibilities from which to choose. Choose (D).