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

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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