What is the reciprocal of 0.4?
201
Answer: 
or 2.5
The reciprocal of 0.4 is 
Alternatively, express 0.4 as a fraction:
Then flip 
to get .
What is the remainder when 1,827 is divided by 4?
202
Answer: The remainder is 3.
A number is divisible by 4 if the two-digit number at the end is divisible by 4. For example, 367 is not divisible by 4 because 67 is not divisible by 4. In contrast, 360 is divisible by 4 because 60 is divisible by 4.
So 1,827 is not divisible by 4, but 1,824 and 1,828 would be. Thus, when 1,827 is divided by 4, the remainder is 1,827 – 1,824 = 3.
In other words, we can drop everything but the last two digits:
1,827 
27 gives a remainder of 3 when divided by 4.
What is the remainder when 18,271 is divided by 3?
203
Answer: The remainder is 1.
A number is divisible by 3 if the sum of its digits is divisible by 3. For 18,271, the sum of the digits is 1 + 8 + 2 + 7 + 1 = 19. This is 1 more than a multiple of 3. If 18,271 were 1 less (i.e. 18,270), it would be a multiple of three. Thus, when 18,271 is divided by 3, the remainder is 1.
In addition to checking for whether a number is divisible by 3, you can also use the sum of the digits to determine the remainder when dividing by 3. Since the digit sum 19 divided by 3 gives a remainder of 1, so does the original number 18,271.
Given xy ≠ 0, simplify: 
204
Answer: 3xy2
Factor: x4 + 2x3 + x2
205
Answer: 
The price of a jacket was first reduced 20%, then was increased back to 90% of the original price. If the original price of the jacket was x, which of the following represents the amount of the most recent price increase?
(A) (0.1)(0.2)x
(B) (0.8)(0.9)x
(C) (0.8)(1.1)x
(D) (0.9 – 0.8)x
(E) (1-(0.8)(1.1))x
206
Answer: (D)
A reduction of 20% can be expressed as times (1 – 0.2), or simply times 0.8. So, after “the price of a jacket was first reduced 20%,” it cost 0.8x.
“90% of the original price” is 0.9x.
So the most recent price increase was from 0.8x to 0.9x, or an increase of 0.9x – 0.8x = (0.9 – 0.8)x, or 0.1x.
Simplify: 
207
Answer: 3-k+1 or 31-k
| Quantity A | Quantity B |
| The area of the shaded region. | 96 |
208
Answer: (C) The two quantities are equal.
The diagonal lines in the figure split each square exactly in half, creating 6 identical right triangles with legs of length 8. The shaded area is 3 triangles, and the unshaded area is 3 triangles. Thus, the shaded area is half the area of the 3 squares.
Quantity A: 
| Quantity A | Quantity B |
| The average of x, y, and z. | (The average of a, b, and c) ÷ 2 |
209
Answer: (C) The two quantities are equal.
Quantity A: 
An exterior angle of a triangle is equal in measure to the sum of the two opposite interior angles of the triangle, so the average of a, b, and c is
Quantity B: 120 ÷ 2 = 60
Jeremy cut a 64 foot rope into 5 pieces. If each piece is at least 10 feet long, what is the longest any of the 5 pieces could be?
210
Answer: 24 feet
To maximize the length of one piece, minimize the length of the other four. In other words, if four of the pieces were minimized to 10 feet long, all the “extra” length would be assigned to the one remaining piece.
64 – 4(10) = 64 – 40 = 24
If it takes 5 machines 1 hour to complete 3 job lots, how long will it take 2 machines to complete 2 job lots?
211
Answer: 
hours, or 100 minutes.
Rather than setting up equations with variables, it may be easier to reason through the given information step-by-step.
5 machines complete 3 lots in 1 hour.
1 machine completes 3/5 lot in 1 hour.
1 machine completes 1 lot in 5/3 hour.
2 machines complete 2 lots in 5/3 hour.
A certain substance reduces in mass by half every 20 years. If the mass 100 years ago was 3,200 milligrams, how many years from now will the mass be 25 milligrams?
212
Answer: 40 years from now
Set up a Population Chart:
| Year | milligrams |
| –100 | 3,200 |
| –80 | 1,600 |
| –60 | 800 |
| –40 | 400 |
| –20 | 200 |
| Now | 100 |
| +20 | 50 |
| +40 | 25 |
At a certain school, there are 24 students per teacher and 2 administrative staff members for every 3 teachers. If there are 6 administrative staff members, how many students attend the school?
213
Answer: 216
Admin : Teachers : Students = 2 : 3 : (3)(24) = 2 : 3 : 72
If there are 6 administrative staff members, the ratio above must be multiplied by 3. Remember to multiply each term in the ratio, so their relationships to one another don't change.
2(3) : 3(3) : 72(3) = 6 : 9 : 216
Simplify: 44 + 44 + 44 + 44
214
Answer: 45 or 210 or 1,024
44 + 44 + 44 + 44 = 4(44) = 41(44) = 41+4 = 45
Since 4 is not a prime number, an equivalent alternative is:
45 = (22)5 = 22×5 = 210
If you know your powers of 2 (or use the calculator), you know that 210 works out to be 1,024, or approximately 1,000. A “kilobyte” is actually 1,024 bytes of data.
Simplify: 44 + 44
215
Answer: 29 or 512
If you know your powers of 2 (or use the calculator), you know that 29 works out to be 512.
If z ≠ 0, simplify 
216
Answer: 1
What is the median of 25, 26, 29, 99, and 501?
What is the average (arithmetic mean) of 25, 26, 29, 99, and 501?
217
Answer: Median = 29, Mean = 136
The median is the middle number in the set when arranged from least to greatest. In this set of 5 terms, the median is the 3rd largest (or, equivalently, the 3rd smallest).
ab < 0 and 
| Quantity A | Quantity B |
| 4b | –2a |
218
Answer: (B) Quantity B is greater.
Multiply both sides by ab to eliminate the denominators. Since ab is negative, this requires the inequality direction to be flipped.
y ≠ 0 and x > 0 and 
| Quantity A | Quantity B |
| 2x | 7y2 |
219
Answer: (A) Quantity A is greater.
Since y ≠ 0, y2 is positive. The problem also states that x is also positive. So multiply both sides by y2x to eliminate the denominators, and there is no need to flip the sign of the inequality.
Even or Odd?
Even × Odd + Odd × Odd = ____
220
Answer: Odd
Order of operations is PEMDAS: Parentheses, Exponents, Multiplications/Division, and Addition/Subtraction. So do the multiplication in this problem before the addition. Parentheses can serve as a visual reminder of what to do first:
(Even × Odd) + (Odd × Odd) = Even + Odd = Odd
Simplify: 
221
Answer: 3
Simplify: 
222
Answer: 4
1,024 is a number you should recognize, as it is a power of 2.
1,024 = 210
12 is what percent of 1,000?
223
Answer: 1.2%
Translate and solve:
Alternatively, 1% of 1,000 is 10. So, because 12 is 1.2 times 10, the answer is 1.2 times 1%, or 1.2%.
224
Answer: 108,999
Use the calculator or compute on paper. For the latter, add vertically in each digit place, starting at the right side (the units digit). Don't forget to “carry the 1” when a digit sum is greater than 9.
Also, you should “sanity check” answers you get even when using the calculator, as numbers can easily be mis-entered. Just over 89,000 plus almost 20,000 should be about 109,000.
| Quantity A | Quantity B |
| 2635 | 2753 |
225
Answer: (B) Quantity B is greater.
26 is a factor of both quantities, so cancel/ignore it when comparing.
Quantity A: 35 = 243
Quantity B: 21 53 = 2 × 125 = 250
| Quantity A | Quantity B |
| 2442 | 2552 |
226
Answer: (B) Quantity B is greater.
The first term in each product has a base of 2. The exponent in Quantity B is greater, so 25 > 24.
The second term in each product has an exponent of 2. The base in Quantity B is greater, so 52 > 42.
Quantity B is the product of two larger terms than in Quantity A.
Alternatively, ignore the common factor of 24:
Quantity A: 42 = 16
Quantity B: 2 × 52 = 50
| Quantity A | Quantity B |
| 38 | 94 |
227
Answer: (C) The two quantities are equal.
Put Quantity B in terms of a prime base.
Quantity B: 94 = (32)4 = 32×4 = 38
General rule: Squared base and halved exponent means equal. (9 = 32 and 
)
What percent of 4 is 6?
228
Answer: 150%
Translate and solve:
Put the number answer back into the question to check: 150% of 4 is 6.
Alternatively, think: 6 is 4 plus half of 4, so 6 is 100% + 50% of 4, or 150% of 4.
| Quantity A | Quantity B |
 |
 |
229
Answer: (B) Quantity B is greater.
Both quantities start with 
. In Quantity A, this is multiplied by a positive fraction less than 1, reducing the value, and then again. In Quantity B, this is increased by adding a positive fraction, and then again. The exact values don't matter: Quantity B > > Quantity A.
| Quantity A | Quantity B |
| 11% of 37 | 37% of 11 |
230
Answer: (C) The two quantities are equal.
Quantity A: 
Quantity B: 
| Quantity A | Quantity B |
| 26% of 32 | 8 |
231
Answer: (A) Quantity A is greater.
26% is a bit over 25%, which equals 
. So Quantity A is a bit over of 32, or a bit over 8.
At 29 miles per gallon, it takes ____ gallons to make a 319 mile trip.
232
Answer: 11
, so 
. Notice that the units “check out”: gallons cancel on the left, so both sides of the equation are in terms of miles.
What is the integer equivalent of 211?
(The answer should not be in exponent form.)
233
Answer: 2,048
While the calculator could generate the answer, it is faster if you have memorized the powers of 2 up to 1,024 = 210.
211 = 21 210 = 2 × 1,024 = 2,048
Notice that 210 ≈ 1,000, so 211 ≈ 2,000.
What is the integer equivalent of 220?
(The answer should not be in exponent form.)
234
Answer: 1,048,576
While the calculator is helpful, you can reduce the number of buttons you need to push by memorizing the powers of 2 up to 1,024 = 210.
220 = (210)2 = (1,024)2 = 1,048,576
Notice that 210 ≈ 1,000, so 220 = (210)2 ≈ 1,000,000 (= 1,0002).
At a certain university, all of the students major in either engineering or liberal arts (no students major in both). If the ratio of engineering students to liberal arts students is 3/4, what fraction of the students are liberal arts students?
235
Answer: 
, where x is an integer.
The total number of students is 3x + 4x = 7x.
At a certain university, half of the students major in either engineering or liberal arts, while half of the students major in something else. No students major in more than one field. If the ratio of engineering students to liberal arts students is 3/4, what fraction of the students are engineering students?
236
Answer: 
, where x is an integer.
The number of students majoring in engineering or liberal arts is 3x + 4x = 7x. These students account for half of the students at the university, so the total number of students is 2(7x) = 14x.
| Quantity A | Quantity B |
| The units digit of (8)(3)(7)(5)(9) | 0 |
237
Answer: (C) The two quantities are equal.
(8)(3)(7)(5)(9) has factors of 5 and 2 (in the 8), so it is a multiple of 10. The units digit is therefore 0.
| Quantity A | Quantity B |
| 59% of $104 | $60 |
238
Answer: (A) Quantity A is greater.
Quantity A: 59% of $104 = 0.59 × $104 = $61.36
It can easily be seen that 60% of $100 would equal $60, but a smaller percent of a larger dollar amount is worth calculating to be certain.
If you want to try mental math, note that 60% of $104 is 60% of $100 plus 60% of $4, or $60 + $2.40 = $62.40. One percent of $104 is $1.04. Subtracting $1.04 from $62.40 still leaves you above $60.
Fill in the blank:
32n = 3n × ____
239
Answer: 3n
Fill in the blank:
3n+2 = 3n × ____
240
Answer: 32
Last year, the tax revenue for Town A was $120,120. If tax revenue increased by 15% this year, what was tax revenue this year?
241
Answer: $138,138
An increase of 15% means that the revenue this year was 1.15 times the revenue last year: $120,120 × 1.15 = $138,138.
If you want to try mental math, treat 15% as 10% plus 5%. 10% of $120,120 is $12,012. 5% is half of that, or $6,006. Add these two number to get $18,018, and finally add that to $120,120 to get $138,138.
The arithmetic mean of Set P is 14 and the standard deviation of Set P is 2.3.
If x is a number in Set P that is three standard deviations below the mean, what is the value of x?
242
Answer: 7.1
Three standard deviations is 3 × 2.3 = 6.9. If x is three standard deviations below the mean of 14, then x = 14 – 6.9 = 7.1.
To increase a number by 125%, multiply by ____.
243
Answer: 1.25, or 5/4
The fractional equivalent of 1.25 is 5/4.
To increase a number by 10% three times sequentially, multiply by ____.
244
Answer: (1.1)3 or 1.331
To increase a number by 10%, multiply by 1.1. To do so again, multiply by 1.1 again. To increase by 10% three times sequentially, multiply by (1.1)(1.1)(1.1).
To take a 25% discount, then a 10% discount, multiply by _____.
245
Answer: 0.675, or 27/40
To decrease a number by 25%, multiply by (1 – 0.25) = 0.75.
To decrease a number by 10%, multiply by (1 – 0.10) = 0.9.
To make these decreases sequentially, multiply by (0.75)(0.9) = 0.675.
Notice that the order in which the discounts are taken doesn't matter!
Alternatively, use fractional equivalents:
0.75 = 3/4 0.9 = 9/10
The product is 27/40.
Given x ≠ 0, simplify: 
246
Answer: x
Translate into an equation:
A percent of B is C
247
Answer: 
“A percent” translates to “ 
”
“of B” translates to “ × B”
“is C” translates to “ = C ”
Check: If A percent is 5% and B is 40, then 5% of 40 is 2, which is equal to C. Plug these numbers into the equation to make sure it is true:
True.
Translate into an equation:
Julie has 10 more course credits than Mike and Larry combined.
248
Answer: J = 10 + M + L
“Julie has” translates to “J =”
“10 more course credits than” translates to “10 +” [Add to whatever noun(s) follow the “more…than.”]
“Mike and Larry combined” translates to “M + L” [Since Mike and Larry follow the “more…than,” the 10 is added to them.]
Check: If Mike has 5 credits and Larry has 8, combined they have 13, and Julie has 10 more than that, or 23. Plug these numbers into the equation:
23 = 10 + 5 + 8. True, so the equation is correct.
Translate into an equation:
The average rainfall on Monday, Tuesday, and Wednesday was 3 times the average rainfall on Friday and Saturday.
249
Answer: 
“The average rainfall on Monday, Tuesday, and Wednesday” is 
“was” translates to “=”
“3 times” translates to “3 × ” [Multiply by what follows.]
“ the average rainfall on Friday and Saturday” is 
At a certain theater, mezzanine tickets cost $110 and balcony tickets cost $20. The 20 tickets purchased by a group had a total cost between $1,850 and $2,010. How many balcony tickets did the group buy?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
250
Answer: (D) 3
Try pairs of numbers:
| Mezzanine | Balcony | Total Cost |
| 15 | 5 | 15($110) + 5($20) = $1,750 |
| 16 | 4 | 16($110) + 4($20) = $1,840 |
| 17 | 3 | 17($110) + 3($20) = $1,930 |
| 18 | 2 | 18($110) + 2($20) = $2,020 |
You can also start by assuming all 20 tickets were mezzanine, for a cost of $2,200, then subtract $90 for each cheaper balcony ticket you swap in ($110 – $20 = $90) until you are in the right range.