We publish practice problems on our social media pages. Here's where you can find the correct answers—and explanations.

THE CORRECT ANSWER IS C.

There are two ways to go about this problem. One is to use the Bowtie method to compare fractions. 2/3 versus 5/8 yields a 16 versus 15. Pretty close. 3/4 versus 5/8 yields 24 versus 20. Not so close, so eliminate it. 7/11 versus 5/8 yields 56 versus 55; that’s really close on a percentage basis because the numbers are bigger. Eliminate (A). 19/25 versus 5/8 yields 152 versus 115. Get rid of it. Choice (E) yields 18 versus 50. Get rid of it. Alternatively, you could also use long division, but if you do, there is no need to finish out the math for each answer. 5 divided by 8 = 0.625. 2/3 = 0.66. Keep it. When you start to divide 3 by 4, the first number you see is a 7. Don’t continue to divide; just eliminate it because 0.7 is farther from 0.625 than 0.66. Choice (C) yields 0.63, so keep it and eliminate (A). The answer for 19 divided by 23 begins with 0.8, so get rid of it. The answer to 23 divided by 30 begins with 0.7, so get rid of that too.

THE CORRECT ANSWER IS D.

To solve this question Plug In. Since the question deals with percents, try 100. If the F train arrives 100 times per day, then the B will arrive 10% fewer times than 100: *B *=* (*1 – 10/100)*F* = (90/100)100 = 90 times, and the Q will arrive 30% fewer times than 90: *Q* = (1 – 30/100)*B* = (70/100)90 = 63 times Translating the question “the Q train’s frequency is what percentage of the F train’s” gives *Q* = (*x*/100)*F* and thus 63 = (*x*/100)100, which means that the Q’s frequency is 63% of the F’s. The correct answer is (D).

THE CORRECT ANSWER IS 1,800.

The question asks about the sum of the degree measurements of the interior angles of a polygon, which is given by 180(*n *– 2) where *n *is the number of sides of the polygon. Since the question states that the polygon has 12 sides, the sum of the degree measurements of the interior angles is 180(12 – 2) = 180 × 10 = 1,800°. The correct answer is 1,800.

THE CORRECT ANSWER IS B.

This problem has numbers in the answer choices and asks for one specific thing, so Plug In the Answers. Label the answer choices as *z*. To save time, first consider whether there are any answer choices that don’t make sense with what the problem states. The problem states that *z *is a prime number. Choice (D), 10, is not a prime number, so eliminate it. The problem also indicates that *z *is a factor of 150. Choice (E), 13, is not a factor of 150, so eliminate it. Now start in the middle of the remaining options, with (B). If *z *= 3, determine what *x *and *y *must be. The problem indicates that *x, y, *and *z *multiply together to equal 150, so divide 150 by 3 to get 50. The value of *xy *must be equal to 50 in this case. Consider the factor pairs of 50: 1, 50; 2, 25; 5, 10. The problem states that *x *is equal to twice *y*. Thus, *x *must be 10 and *y *must be 5. This works, so the correct answer is (B).

THE CORRECT ANSWER IS C.

The presence of rectangles implies that this is a geometry problem, so first follow the steps for the basic approach to geometry. (Draw and label the figure.) Write down relevant formulas. The area of a rectangle is *A = lw*, and its perimeter is *P *= 2*l *+ 2*w*. To solve this problem directly would require some detailed algebra. There are no variables named in the problem or the answer choices, and the answers represent a defined object (the length). This problem is a good candidate for the Plugging in the Answers (PITA) strategy. Start with (C), plug it into the problem, and work through the question, making columns. If the short side of tile B is 20, the long side is 30 because 20 is 2/3 of 30. This is also the short side of tile A. Since the area of A is 1500 and its shorter side is 30, use the area formula *A* = *lw* to find that the other side is 50. The last remaining information is the perimeter of tile A. If the sides are 30 and 50, the perimeter is 160. This matches the information in the problem, so the correct answer is (C).

THE CORRECT ANSWER IS D.

The diameter of the circle is 12, so the radius is 6, and the area is 36π. The total number of parts in the ratio is 3 + 4 + 5 = 12, so each part covers an area of 36π/12 = 3π. The largest ratio part is 5 times this amount, or 15π.

THE CORRECT ANSWER IS 29,456.

First, set up two equations with two variables. Let *m *= the pre-tax cost of Mabel’s car, and let *r *= the pre-tax cost of Rose’s car. Equation one: Mabel’s car cost 12% more than Rose’s car: *m *= (1.12)*r*. Equation two: The combined price of their cars is $53,000: *m + r *= 53,000. Next, solve for the pre-tax price of each car by substituting (1.12)*r *from equation one for *m *in equation two such that (1.12)*r *+ *r *= 53,000. Now solve for *r: *2.12*r *= 53,000. Therefore, *r *= 25,000. Now substitute that back into equation two to calculate that *m *= 28,000. Lastly, since you’re looking for the price of Mabel’s car after sales tax, multiply 28,000 by 1.052 to find that 28,000 × 1.052 = 29,456.

THE CORRECT ANSWER IS E.

To solve this question, Plug In the answers as the number of seniors to determine whether the total number of students, seniors (*S*) + juniors (*J*) + freshmen (*F*), adds up to at least 150. Start with (C). If *S* = 24, then 24 = (60/100)*J*; thus, there are 41.67 juniors, which is incorrect since it is impossible to have a fraction of a student. For (E), if *S* = 27, then 27 = (60/100)*J*; thus, *J* = 45; since 45 = 50% of *F*, then *F *= 90; since 27 + 45 + 90 = 162, which is at least 150, (E) is the only correct answer.

THE CORRECT ANSWER IS 8.

Use brute force to solve this one. Write down the 2 given terms, find half the sum of the previous 2 terms, and repeat the process until you have a non-integer. When you work it out, Sequence *S *should begin 64, 32, 48, 40, 44, 42, 43, 42.5; the first non-integer term is the 8th term, so *n *= 8.

THE CORRECT ANSWER IS E.

List out the times until you figure out the pattern. The vent releases steam at 6:25 p.m. and then 20 minutes later at 6:45 p.m., then 7:05 p.m., then 7:25 p.m. So the pattern is that steam is released at 5, 25, and 45 minutes after the hour. Only (E) fits the pattern.

THE CORRECT ANSWER IS D.

There are variables in the quantities, so Plug In. Begin with something that affects exponents in a unique way, such as 1. When *a *= 1, Quantity A is 3 and Quantity B is something greater than 3. Eliminate (A) and (C). Now Plug In again, and try *a *= 0. In this case, both quantities are equal. Eliminate (B). The correct answer is (D).

THE CORRECT ANSWER IS D.

There is a variable in the question stem and in the quantities, so Plug In. The variable is any number between 130 and 150. The question asks about the greatest factors, so start by Plugging In the greatest integer which is 149. If *x *is 149, then the greatest odd factor of *x* is 149. However, there is no even factor of 149, so Quantity A is greater. Eliminate (B) and (C). Plug In again. If *x* = 148, then the greatest even factor is 148 and there is no odd factor, so Quantity B is greater. Eliminate (A). The correct answer is (D).

THE CORRECT ANSWER IS D.

Simplify this question by using Average Pies to deal with the two steps of the problem. First, figure out how many total inches of rain fell in the 7 months from April to October. This results in 11.9, which can be rounded to 12 since the problem said “approximately.” Next, subtract that total from 37 to find out how many remaining inches fell during the other 5 months. Finally, take those remaining 25 inches and divide by the remaining five months. The correct answer is (D).

THE CORRECT ANSWER IS A.

The question asks for a specific value, and that value is represented by the answer choices, so Plug In the Answers. The answer choices represent the price of a bagel. Start with (A). If one bagel costs $1.00, then 56 bagels costs $56.00. Because there is a discount of $1.40 per dozen and there are 4 complete dozens in the order, the total discount is $5.60. Therefore, the actual price paid is $50.40, or an average of $0.90 per bagel. This matches the information in the problem. The correct answer is (A).

THE CORRECT ANSWERS ARE A AND C.

The correct answers are A and C.

There are variables in the question stem and answer choices, so Plug In a value for *a. *If *a *= –42, then it is divisible by 6 and 21 but is not positive or equal to 42, so eliminate (B) and (D). If *a *= 84, it is still divisible by 3 and 14, as well as by 21 and 6. In fact, *a *is always divisible by 6 and 21, because the prime factors of 3 and 14 are 2, 3, and 7 and the distinct prime factors of 6 and 21 are also 2, 3, and 7. The correct answers are (A) and (C).

THE CORRECT ANSWER IS B.

There are variables in the question stem and answer choices, so Plug In making sure to follow the restrictions in the question stem. Plug In something simple, such as *x *= 2 and *y *= 3. Therefore, the average is 4, which is equal to the average of *y *+ 2*z*. Plug In for *y *to solve that *z *is 2.5. The question asks for the average of *x *and *y, *which is also 2.5. The target number is 2.5. Plug In for the variables in the answer choices, looking for one that equals 2.5. The correct answer is (B).

THE CORRECT ANSWER IS B.

This question asks for a specific value and that value is represented by the answer choices, so Plug In the Answers. Start with (C). If Reservoir *A* contains 700 million gallons of water, then Reservoir *B* has 450 million gallons less, or 250 million gallons. When 100 million gallons are drained from Reservoir *A* to Reservoir *B*, then the reservoirs hold 600 million and 350 million gallons of water, respectively. The problem states that Reservoir *A* has twice the water that Reservoir *B* has, so this choice is incorrect. Eliminate (C). It is hard to tell if the number needs to be larger or smaller, so pick a direction. Try (B). If Reservoir *A* contains 600 million gallons of water, then Reservoir* B* has 450 million gallons less, or 150 million gallons. When 100 million gallons are drained from Reservoir *A* to Reservoir *B*, then the reservoirs hold 500 million and 250 million gallons of water, respectively. Because Reservoir *A* has twice the amount of water as Reservoir *B*, this is the appropriate relationship. The correct answer is (B).

THE CORRECT ANSWERS ARE A, D, G.

There are variables in the question stem and answer choices, so Plug In values for *x* and *y*. This question involves number lines, so draw a number line. Plug In *x* = 4 and *y* = 10. In this case, point A is on the number line between 4 and 10. Choice (A) is 4 + 1, which equals 5. Since 5 is on the number line between 4 and 10, this works, so select (A). Choice (B) is 4 – 1, which equals 3. This is not between 4 and 10. Eliminate (B). Choice (C) is 10 + 1. Since 11 is not on the number line between 4 and 10, eliminate (C). Choice (D) is 10 – 1. Since 9 is on the number line between 4 and 10, this works, so select (D). Choice (E) is 4 + 10. Since 14 is not on the number line between 4 and 10, eliminate (E). Choice (F) is 4 – 10. Since –6 is not on the number line between 4 and 10, eliminate (F). Choice (G) is 10 – 4. Since 6 is on the number line between 4 and 10, it works. Select (G). The correct answers are (A), (D), and (G).

THE CORRECT ANSWER IS E.

There are variables in the question stem and answer choices, so Plug In. Try numbers that are easy to work with such as *q* = 10, *r* = 5, and *s* = 2. So, *A* = 10 – 5 = 5,* B* = 5 – 2 = 3, and *C* = 10 – 2 = 8. In this case, *A* – (*B – C*) = 5 – (3 – 8) = 5 – (–5) = 10. This is the target number, so Plug In for the answer choices to see if any match the target number. The correct answer is (E).

THE CORRECT ANSWER IS C.

The single-digit primes are 2, 3, 5, and 7. Be systematic in listing the results. Start with 2, adding it to the other numbers, then move to 3, and so forth: 2 + 3 = 5; 2 + 5 = 7; 2 + 7 = 9; 3 + 5 = 8; 3 + 7 = 10; 5 + 7 = 12. Out of these six results, 5 and 7 are prime, but the other four results are not, so the probability is 4/6 = 2/3, and the correct answer is (C).

THE CORRECT ANSWER IS E.

First, determine the value of *x*, which is 3^{2}, or 9. The problem asks for the value of *x ^{x}*, so substitute 9 for

THE CORRECT ANSWER IS A.

Plug in a three-digit integer, such as 341. Interchanging the 1 and the 3 yields 143. Subtracting 143 from 341 is 198 (which is already positive, so its absolute value is also 198). Since 198 is not divisible by 7, 5, or 4, eliminate (B), (C), and (D). Plug in another number, such as 546. Its hash is 645. Subtracting 546 from 645 is 99, which is not divisible by 2, so eliminate (E). Even if the hundreds digit or the units digit is zero, the absolute value of the difference between a three-digit integer and its hash is divisible by 9. The correct answer is (A).

THE CORRECT ANSWER IS 9.

This problem provides a ratio, so draw a ratio box. With a ratio of 5 : 7 : 3, the total number of singers is at least 15. If you double the number, and keep the ratio, there are 30 singers. To have at least 40 singers with the same ratio, the actual total is 45, or 3 times 15, which means there are three times the number of basses (3) in the ratio, or 9.

*a* = 60 and *b* = 100. Since *b* is 40% of *c*, 100 = (40/100)*c*, and *c* = 250. Because *c* is 20% of *d*, 250 = (20/100)*d*, and *d* = 1,250. Now use the values for the variables to translate the last part of the question into the equation 6(1,250) = (*x*/100)(20)(60), which is 7,500 = (*x*/100)(1,200). Therefore, 7,500 = 12*x*, and *x* = 625.

*A* rode for 3.5 hours at 20 miles per hour, so she traveled 20 × 3.5 = 70 miles. Cyclist *B*, then, must have traveled 145 – 70 = 75 miles. Since cyclist B left at 2:00, she rode for 3 hours, giving her a speed of 75 ÷ 3 = 25 miles per hour.

Use the Rate Pie. Ali is traveling 50 feet per second faster than Jeff is traveling. Therefore, that is

the rate at which she is effectively gaining ground on him. Put that in the lower-right segment of

the Rate Pie. We want to know how long it will take her to gain 3,000 feet on him. Put 3,000 in

the top section of the Rate Pie. Now you can see that dividing 3,000/50 will fill in the last segment

of the Rate Pie, telling you how long it takes Ali to do so is 60 seconds. Be aware that (E) is an incorrect partial answer. Now you need to find out how many feet Ali will travel in 60 seconds, by multiplying 60 seconds × 300 feet per second, which equals 18,000 feet. Divide 18,000 feet by the length of one lap, or 3,000 feet, and you’ll find that it will take Ali 6 laps to overtake Jeff.

Start by translating the words into math: 0.25*a* = b, and 0.2*b* = c. Since we're looking for the relationship between *a* and *c*, we'll want to find out what each of those variables equals in terms of *b*. (We'll then be able to get rid of *b*.) Solving for *b* in terms of *c*, we get *b* = 5*c*. So, now we know that 0.25*a* = 5*c*. Solving for *a*, we get *a* = 20*c*. But we're not done yet! If we were to choose choice (C) and move on, we'd get this one wrong. Let's go back and reread the question, which asks: "*a *is what percent of *c*?" We then see that *a* is 2,000% of *c*, and the correct answer is choice (E).

Come up with your own word or phrase for the blank that describes the effect on *the soil*'s relationship to grains brought about by *excessive irrigation and salt accumulation*—an effect that in turn would have caused *ancient Mesopotamians* to switch *from grain production to barley*. Excessive irrigation and salt accumulation must have made the soil bad for grains in some way. So, use the phrase "bad for," and check for answer choices that convey that meaning. Both *inhospitable to* and *unsuitable for* fit the bill, and those are the correct answers.

The transition word that begins this sentence, *despite*, tells us that the *state *in which *many aristocrats...lived* is the opposite of *their noble status*. Because noble status is a positive idea, the word in the blank should be negative. This is reinforced by the additional clues that *many aristocrats were virtually penniless*. Evaluate the answer choices one at a time, eliminating positive words and holding onto negative words. The correct answers are *indigence* and *penury*.

The clue is *the vast amount of time Francis dedicated to learning six...languages*. The opposite direction transition word *despite* indicates that the word in the blank describing Francis as a communicator is at odds with his dedication to learning languages. This idea is continued after the transitional semicolon with an additional clue regarding *his inability to construct cogent prose*. Thus, the word in the blank modifying *communicator *should be something like "poor" or "ineffective." *Inept* and *maladroit* are the correct answers. *Astute* has the opposite meaning of what's expected, and *morose* is out of place because it means gloomy. Though it's possible Francis is *florid* and *prolific*, the clues don't directly support those ideas.

The clue *due to the increased aerodynamic drag* suggests a negative impact on *fuel efficiency...at speeds greater than 50 miles per hour. *Thus, the verb in the blank should be something like "decreases." Both *diminishes* and *wanes* work here.

There is the variable *x* in both the question stem and the answer choices. So, **Plug In** a good number, such as *x *= 4. Now use *x* = 4 to read the problem again and solve for the target. The problem states that “Mara has six more than twice as many apples as Robert.” If Robert has 4 apples, then Mara must have 14. Next, the problem states that Mara has “half as many apples as Sheila.” That means that Sheila must have 28 applies. The question asks for the number of apples that Robert, Sheila, and Mara have combined, so add 4 + 14 + 28 = 46 apples. This is the target number, so circle it. Now, Plug In *x* = 4 for all of the variables in the answer choices, and use the scratch paper to solve them, eliminating any answer choice that does not equal 46. Choice E yields 46. 7(4) + 18 = 46. This is the correct answer.

Start solving this problem by assessing all the information that is given to you. A 20-gallon water jug is 20% full, so there are 4 gallons in the water jug. The question is asking how many days it will be before the jug is 85% full. 85% of 20 gallons is 17 gallons, so that is the number we are looking for. After the first three days, 50% of the total water in the jug is added. There are 4 gallons in the jug, so after three days, 2 more gallons are added, making a total of 6 gallons. After another three days, 50% of 6 gallons is added, so 3 gallons are added, which increases the total amount of water in the jug to 9 gallons. After three more days, 50% of 9 gallons is added, so 4.5 gallons are added, increasing the total to 13.5 gallons. After another three days, the total is increased by 50% of 13.5, which is 6.75 gallons, which will increase the total to more than 17 gallons. So there are 4 increases of three days apiece, for a total of 12 days. Choice (C) is correct.

There are 3 terms in the sequence, and they repeat. The question asks about the product of the 81st, 82nd, 83rd, 84th, and 85th terms. Use the fact that the values of the terms in the sequence repeat after every third term to determine the value of the 81st term. Divide 81 by 3 to find that there are 27 complete iterations of the sequence. The 81st term is at the end of one of these repetitions, so its value is –5. Therefore, the 82nd term is –2, the 83rd is 3, the 84th is –5, and the 85th is –2. The product is (–5) x (–2) x 3 x (–5) x (–2) = 300.

While the relationship among the can prices is provided, no actual numbers are supplied, so try plugging in some numbers for can prices. A good number to choose for the cost of the large cans is the value of the small can multiplied by the value of the medium can, or $5 x $7 = $35. This means the medium can costs $35/$5 = $7, and the small can costs $35/$7 = $5. The amount of money needed to buy 200 medium cans is 200 x $7 = $1,400. Now PITA (Plug In The Answers). Start with (C). If the customer purchases 72 small cans, that will cost 72 x 5 = $360. If the customer purchases 72 small cans, she also purchases 72 large cans, so 72 x $35 = $2,520, which is more than the $1,400 spent on medium cans. This number is too great, so eliminate (C), (D), and (E). Choice (B) also works out to be too great, which leaves (A). 35 small cans x $5 a can = $175. Then, 35 large cans x $35 = $1,225. Add those values: $1,225 + $175 = $1,400, the same price as the medium cans. Choice (A) is correct.

The first thing you need to do is determine whether the order matters. In this case it does (because we're arranging paintings on the wall). This is a permutation question. We have three slots to fill because we're arranging three paintings. There are 6 paintings that could fill the first slot, 5 paintings that could fill the second slot, and 4 paintings that could fill the third slot. So, we have 6 x 5 x 4, which equals 120.

This is a simultaneous-equation question. Both quantities ask for the value of *x *+ *y*, so try to combine the equations to find that value. If you multiple 3*x *+ 4*y *= 12 by 3, the result is 9*x *+ 12*y *= 36. This can be subtracted from the other equation to find that 2*x *+ 2*y *= –6. Divide both sides of the equation by 2 to find that *x *+ *y *= –3. Quantity A, then, is equal to –3. Quantity B is now (–3)^{–2}, which can be rewritten as 1/(–3)^{2}= 1/9. Therefore, Quantity B is greater than Quantity A, and the correct answer is B.

Plug In The Answers, starting in the middle with (C). If each

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