ANSWERS
Drill 1
Drill 2
Drill 3
Drill 4
1. B, C, D
2. 1
3. A, B
4. D
5. C, D, E
6. E
7. B
8. 2
9. A, C
10. B, D
11. D
12. 29,456
13. A, B, C
EXPLANATIONS
Drill 1
1. E When dividing by a fraction, flip the fraction and multiply: 
. Alternatively, you may estimate and realize that 3 divided by something slightly smaller than 1 must be slightly larger than 3.
2. C Use the Bowtie when adding or subtracting fractions: 
. Next, divide the fractions by flipping the numerator and denominator and multiplying: 
.
3. B In Quantity A, 
, and in Quantity B, 
. If you multiply the numerator and denominator of 
by 4, you obtain a common denominator: 
. Clearly 
, so Quantity B is greater.
4. B 5% of $800 is $40, thus, the maximum amount of money that could be in the account at the end of one year is $840; eliminate choices (C), (D), and (E). Similarly, the minimum amount that could be in the account at the end of one year is $800 plus 2% of $800, or $816; eliminate choice (A).
5. D Be careful: You’re not given the original price of either pair of shoes, and because you can’t assume they’re the same price, try Plugging In a variety of values. If the shoes in both quantities originally cost 10 dollars, then the change in price of the shoes in Quantity A is 5
dollars, and the change in price of the shoes in Quantity B is 3 dollars; Quantity A is greater, so eliminate choices (B) and (C). If the shoes in Quantity B originally cost 20 dollars, though, then the change in price is 6 dollars. Quantity B is now greater, so eliminate choice (A), and you’re left with choice (D), the correct answer.
6. E With his employee discount, Joey purchases an item for 90% of its regular price, so 90% of the regular price of this item is equivalent to $99 or 
. Solve for x to find that the regular price is $110.
7. B The percent change formula is 
. Remember that the “original” is the amount before the change. So, in Quantity A, the difference is 200 − 150 = 50, and the original is 200, which yields a 25%
change. In Quantity B, the difference is also 50, but the number changes from 150 to 200, so the “original” is 150, which yields roughly a 33.3% change. Thus, Quantity B is greater.
8. A Begin by Plugging In 7 for y, so 20 percent of x is 35. You could go on to solve for x, but a shortcut would be to say that 60 percent of x is three times 20 percent of x, so multiply 35 by 3 to get 105.
9. B You have far too many fractions to add quickly with the Bowtie. Instead, convert all of the fractions to the common denominator of 48: 
.
10. C If The Warm Muffin Bakery sells 10,000 muffins, it sells 6,000 cookies. If the Warm Muffin Bakery then sold 20,000 muffins, it would sell 12,000 cookies. The cookie sales would thus increase by 6,000.
11. B Convert the fractions to decimals. So 
, making Quantity A 0.625. In Quantity B, 
is about 0.333, making Quantity B about 0.658. Quantity B is greater.
12. B Convert the fractions to decimals to see which inequality is correct. You can divide them out (remember, numerator divided by denominator), but it might help to have some common fraction/decimal equivalents memorized. Starting with choice (A), 
, so this inequality is not true. Convert the fractions in choice (B): 0.1 < 0.11 < 0.125; this inequality is true.
13. C Translate the question into a percent formula. So, “245 widgets per week is 35 percent of Company B’s weekly widget output” means 245 = 
× B. Try Plugging In Quantity A into this formula. Does
× 700 = 245? Yes, so the quantities must be equal.
14. E Plug In for the variables. Let m = 3 and n = 5, and 
. Only choice (E) works. Alternatively, you could manipulate the fractions: 
Drill 2
1. A The question asks for a percent increase from the original price; be careful not to find 20% of $18,000, and reduce the higher total ($18,000) by that amount. Instead, you’ll need to find the amount that yields the higher total, when increased by 20%, though,
it’s much easier to just increase the price in Quantity B and compare it to the total in the problem: 10% of $14,400 is $1,440, so 20% must be $2,880; adding this to the base price of $14,400 yields a total of $17,280. That’s smaller than what you were looking for, so Quantity A is greater.
2. C Convert 0.0025 to a percent by sliding the decimal point two places to the right: 0.25%. Then convert 0.25 to a fraction to get 
%.
3. C There are two ways to go about this problem. One is to use the Bowtie method to compare fractions. 
versus 
yields a 16 versus 15.
Pretty close. 
versus yields 24 versus 20. Not as close, so eliminate it 
versus yields 56 versus 55, that’s really close on a percentage basis
because the number are bigger. Eliminate choice (A). 
versus 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. = 0.66. Keep it. When you start to divide 3 by four, 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 choice (A). The answer is 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.
4. B Twenty cigars bought individually would cost $40, so apply the percent change formula—
—to determine Quantity A. In this case, the difference is $10, and the original, because it’s a percent decrease, is
$40: 
×100 = 25, so Quantity A is 25%. Quantity B is greater.
5. D Try Plugging In a possible value for 
. If = , then 
, which is closest to coordinate D.
6. E This question is really asking about place value. Start with the greatest place: the thousands. So, 6 × 1,000 means a 6 in the thousands place. Eliminate choices (A), (B), and (C). Next, 4 × 100 means the next digit should be 4. Eliminate choice (D), and select choice (E).
7. E To calculate this expression, break it into pieces: 
= 
= 
.
8. B Solve each equation by translating into algebra. The first is 
. Reduce and multiply to find a = 18. The second is 
. Multiply both sides by
100 then divide by 90 to find b = 20. Quantity B is greater.
9. E Plug in a value for x: If x = 20, then the expression 
becomes 
. Now Plug in 20 for x in the answer
choices; only choice (E) hits your target answer of 4. Alternatively, you could factor the expression: 
.
10. C The first thing you need to do is to clean up these expressions. You have 15th, 50th, and decimals, so it is very difficult to compare values. 
can be reduced to 
. 0.3n is the same as 
. Change your first expression from 
to 
. 
is pretty close to 
or
, the first expression, but a bit smaller. Because 
is clearly the smallest expression and you need only concern yourself with the smallest, the second smallest, and the biggest, you can ignore 
. Convert 
, and convert your other expressions to 20ths as well. You now have 
, and 
. The difference between the smallest and largest is 
. Three times the difference between the two smallest is also
3. The answer is choice (C).
11. D Plug in some real numbers to compare quantities. For example, plug in 10 for the number of games in the season. For the first 5 games of the season, try an average of 10 points for David (for a total of 50 points), which makes an average of 12 points for Antonio (for a total of 60 points). Next, try a total of 0 points
for each player for the second half of the season; now Antonio’s average for the season is 6 points, and David’s is 5. Because 90% of 6 is 5.4, Quantity A is greater, so eliminate choices (B) and (C). Finally, try a total of 100 points for each player for the second half of the season; now Antonio has scored a total of 160 points in 10 games, for an average of 16 points, and David has scored a total of 150 points in 10 games, for an average of 15 points. Because 90%
of 16 is 14.4, Quantity B is now greater, so eliminate choice (A), and you’re left with choice (D).
12. D The percent change formula is 
, so plugging the numbers from the problem into the formula yields 
= 
= 
× 100 = 20. If you selected choice (E), you may have used the wrong value as the original: Remember, in a percent increase, the original number is the smaller value.
13. D Solve this problem in chunks. To find the numerical value for 0.004 percent, divide by 100: 0.004 ÷ 100 = 0.00004. The reciprocal of that is 
. So of the result is ×25,000 = 5,000.
14. D Plug in values for m. When m = 1, Quantity B is larger. When m = 12, Quantity A is larger.
Drill 3
1. B To find exact values for p and q, apply percent translation: 
, so 
, 27p = 10,000, and p = 370.37; p
is q percent of 100, so q = 370.37 as well. Quantity B is greater. Alternatively, you could avoid the calculation altogether and Ballpark this one all the way through: 100 is more than 25% (or ) of p, so p must be less than 400—and so must q.
2. B This is a good problem for Plugging In. If y = 2, then the expression becomes equal to −. Choice (B) is the only choice that gives you − when you replace y with 2.
3. 
The total mixture contains three cups, so the second bowl must contain two cups. This 2-cup bowl of nuts divided into even thirds consists of
cups peanuts, cups cashews, and cups almonds. Combining this with the 1-cup mixture of 
cup peanuts and cup cashews results in 
cups peanuts in a 3-cup mixture. So, 
= 
of the new nut mixture is peanuts.
4. C To find 16 percent of 83, multiply 83 by 0.16. To find 83 percent of 16, multiply 16 by 0.83. Both expressions yield 13.28, so choice (C) is correct.
5. B Set up a proportion so that the original ratio equals the final ratio: 
. Cross-multiply and then divide both sides by 1024 to find x = 600.
6. 
One easy way to solve this problem is to multiply both sides of the equation by 9, which gives you (w + 1) = 
, or (w + 1) = 
. Now subtract 1 from both sides to get 
.
7. A, B, and E
Anything that’s a multiple of 30 will work here because of the two denominators of the two fractions in the question. Choices (C), (D), and (E) are all multiples of 30, but the remaining answers are not. You can also treat this as a PITA problem
and try each answer. Take 
of each of the answer choices, then take 
of what’s left. Do you wind up with an integer? If so, eliminate it and keep working until you’ve tried every answer.
8. D Sure, you could use algebra here, but most likely you won’t wind up with an answer that looks anything like choice (D). Instead, try plugging in a value for x. Make x something easy such as 1000, which makes y = 500. Plug x = 1000 into each
answer choice, and it turns out that only choice (D) gives you the correct value of 500.
9. 5 First, translate the English into math. Thirty-five hundredths is 0.35 and four thousandths is 0.004. Now, do the division on your on-screen calculator; the first place after the decimal point is the tenths digit, so the answer is 5.
10. C and E
To solve this question, plug in the answers as the number of seniors to see if the rest of the class adds up to 150. Start with choice (C). If S = 24, then 
and J = 40; since 40
= 50% of F, then F = 80; since the sum of the students is 24 + 40 + 80 = 144, which is less than 150, try larger numbers and eliminate choices (A), (B), and (C). For choice (D), if S = 25, then 25 =
J, thus there are 41.67 juniors, which is incorrect since it is impossible to have a fraction of a student. For choice (E), if S = 27, then
J, thus J = 45; since 45 = 50% of F, then F = 90; since 27 + 45 + 90 = 162, which is at least 160, choice (E) is the only correct answer.
11. E Since the problem doesn’t give you prices, Plug In; since the question involves percents, use 100. If the starting price for Melange was $100, then, after the 20% decrease, the price was $80; increase that by 50%, or , and the
ending price was $120. Since the ending prices for Melange and Blue were the same, the ending price for Blue was also $120. Now, work backwards. Declining by 50% is the same as being cut in half; if Blue was $120 after being cut in half, it must have been $240 before the decrease. Since that $240 was the result of a 20% increase, translate the question “240 is 120% of what” to get the equation 240 = 
× x; solve for x to get the starting price for Blue, $200. Finally, now that you have both starting prices, translate the question into 20 = 
× 100, and solve for x to get your answer, 200.
12. 15 We know that the difference between 
and of the container is 3 liters, so set up the following equation to solve for the volume of the container:
Since the container is still full, it has 45 liters in it. Therefore, you’ll need to pour 15 liters back in.
13. E Plug in the answers. Subtract 12 percent from each answer choice to find the one that gives you 220,000. Start with choice (C):
$196,429
$221,200 − .12(221,200) = 194,656
$246,400 − .12(246,400) = 216,832
$250,000 − .12(250,000) = 220,000
Drill 4
1. B, C, and D
It’s useful to use PITA in this problem. Note that increasing something by 10 percent is the same as multiplying by 1.10, and decreasing something by 5 percent is the same as multiplying by 0.95. Definitely use your on-screen calculator on a problem
such as this. Because you are dealing with a range, it will save you time to work from the top down until you reach a correct answer, then work your way from the bottom up until you reach a correct answer. Each answer in between the smallest correct answer and the largest correct answer must also be correct.
2. 1 Sometimes dealing with fractions is easier than dealing with decimals, and sometimes vice versa. Here are both methods. When multiplying decimals, count how many total digits or decimal places there are to the right of the decimal point, then put the decimal point that
many places to the left once you’ve done the multiplication. So (0.05)(0.5) = 0.025 (2 digits for 0.05 and 1 digit for 0.5, so 3 total digits and therefore 3 places to the left).
(5)(0.005) = 0.025
They equal the same thing, so the answer is 1. When dealing with fractions, just remember that for each decimal place, you add another zero on the bottom. So 0.05 = 
, because you put two zeroes on the bottom, one for each of the two digits to the right of the decimal point. Or, you could convert the decimals to fractions and then multiply:
Again, you get the same fractions on the top and bottom.
3. A and B You can use your on-screen calculator, but it will probably be faster to ballpark on at least some of these—for example, think of 
as slightly more than 9, or 
as slightly less than 6. In order to figure out which fractions are greater than one, just figure out if the top part of the fraction is bigger than the bottom part:
: top bigger than 12, bottom smaller, so choice (A) works;
: top bigger than 6, bottom smaller, so choice (B) works;
: top smaller than 10,000, bottom bigger, so eliminate choice (C);
: top smaller than 24—you might think of it as less than 12 times less than 2—bottom bigger, so eliminate choice (D).
4. 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 – 
)
F = (
)100 = 90 times, and the Q will arrive 30% fewer times than 90: Q = (1 – 
) B = (
)90 = 63 times. Translating the question “the Q train’s frequency is what percentage of the F train’s”
gives Q = (
)F and thus 63 = ()100, which means that the Q’s frequency is 63% of the F’s. The correct answer is choice (D).
5. C, D, and E
To solve this question, remember that for every space the decimal in the base number moves to right, the power of ten should decrease by 1, and vice versa. For choice (A), 100013 = 1015, so the decimal should
not have moved. In choice (B), moving the decimal 1 space to the right means that 1015 should decrease by 1 to become 1014. Eliminate choices (A) and (B). For choice (C), moving the decimal 1 space to the left means that the power should increase by 1; this is correct. For choice (D), 10014 is equivalent to 1015, so this is correct. For choice (E), moving the decimal 3 spaces to the
left means that the power should increase by 3; this is correct. The correct answers are (C), (D), and (E).
6. E To solve this question, convert the fractions into decimals and carefully add them. 
= 0.3 + 0.43 + 0.017 = 0.747; this is your target answer. Choice (A) equals 0.7 + 0.4 + 0.07 = 1.17. Choice (B) equals 0.6 + 0.12 + 0.037 =
0.757. Choice (C) equals 0.07 + 0.4 + 0.007 = 0.477. Choice (D) equals 0.07 + 0.004 + 0.7 = 0.774. Choice (E) equals 0.32 + 0.4 + 0.027 = 0.747. The correct answer is choice (E).
7. B To solve this question, remember that the formula for finding the result of periodic increases at a certain rate is (Original Amount)(1 + rate)number of periods. In this case, the final amount would be 1000(1.15)5. The correct answer is choice (B).
If you forgot the formula, you could calculate the final amount after 5 years, and then calculate all the answers for a match.
8. 2 To solve this question, remember the rules of exponents. When an exponent is outside a parentheses, it gets multiplied to any exponents inside the parentheses; thus 
= x3. Next, since when two numbers of the
same base are divided, the exponents are subtracted, x3 divided by x5 = x3 − 6 = x−3. Since a negative exponent is equal to its reciprocal, then x-3 = 
= 8 –1 = 
and thus x3 = 8. Thus, x = 2, the correct answer.
9. A and C To solve this question, plug in the answers. In choice (D), if Ben, B, sold 6500 copies and B = R, then 6500 = R, so Regina, R, sold 7800 copies, thus giving a difference of 7800 − 6500 = 1300. Since this is too small, eliminate choices (D), (E), and (F) and try larger numbers. In choice (C), if B = 9000, then 9000 = R, so R = 10,800, thus giving 10,800 − 9000 = 1800; since this is bigger than
1500, keep choice (C). In choice (B), if B = 11,244, then 11,244 = R, so R = 13,492.8 copies; eliminate choice (B), as it is impossible to sell a fraction of an album. In choice (A), if B = 12,000, then 12,000 = R, so R = 14,400 and 14,400 − 12,000 = 2400.
Since this is bigger than 1500, keep choice (A). The correct answers are choices (A) and (C).
10. B and D
To calculate the interest earned, multiply the principal by the annual interest rate: in one year, Carmen will earn $7000 × 0.06 = $420 in interest, so choice (B) works. To calculate her interest for any part of the year, divide $420 by the appropriate fraction of
a year. At the end of April, of the year has passed, so Carmen will have earned × $420 = $140; eliminate choice A. At the end of six months, Carmen will have earned × $420 = $210; eliminate choice (C). At the end of three
months, of the year has passed, so Carmen has earned × $420 = $105; there will be a total of $7000 + $105 = $7105 in the account, so choice (D) works.
11. D Since you know there are 200 lbs remaining after the grocery stores and farmer’s market get their shares, you cannot plug in your own number. So plug in the answers, and start with middle choice (C):
12. 29,456
First set up the first part of the question to calculate Mabel’s car cost without interest. Let Mabel = m and Rose = r. So, 1.12r = m and r + m = 53,000. Substitute: Since r + 1.12r = 53,000, then 2.12r = 53,000.
Solve: r = 25,000 and m = 28,000. Now, you can multiply m by the interest rate. (28,000)(.052) = 1,456 of interest. Add it to the price of the car to get 28,000 + 1,456 = 29,456.
13. A, B, and C
The problem has percents in the answers and no real values in the question, so it’s a Hidden Plug In; the hidden variable in this case is Evangeline’s weekly salary. To find an easy number, try multiplying the denominators in the problem: 8 × 6
× 9 = 432, so Evangeline makes $432 per week. Now work the problems in bite-sized pieces. She spends 
of $432, or $162, on rent, leaving her with $432 − $162 = $270. Of that, she spends 
, or $45, on food, leaving her with $270 − $45 = $225. If she wants to put 
of her weekly salary, or $192, into a savings account, then she can only spend $225 − $192 = $33 on entertainment. Finally, use your on-screen calculator to determine that 33 is just over 7.6% of 432, so choices (A), (B), and (C) will all work.
