a > 3
| Quantity A | Quantity B |
 |
a |
151
Answer: (D) The relationship cannot be determined from the information given.
Algebra solution: 
If a > 3, a could be less than, equal to, or greater than 5.
Number testing solution:
When a = 4, 
Quantity A is greater.
When a = 6, 
Quantity B is greater.
| Quantity A | Quantity B |
 |
 |
152
Answer: (A) Quantity A is greater.
Quantity A: 
Quantity B: 
Or, notice that both simplified products include (6)(5). 7 times that product is greater than 4 times that product.
| Quantity A | Quantity B |
 |
 |
153
Answer: (B) Quantity B is greater.
Quantity A: 
Quantity B: 
| Quantity A | Quantity B |
| x2 + 4 | 4x |
154
Answer: (D) The relationship cannot be determined from the information given.
As a squared term, (x – 2)2 can never be less than 0. But (x – 2)2 can equal 0 when x = 2, or (x – 2)2 can be greater than 0 when x ≠ 2.
You could test numbers, but if you did not test x = 2, you might incorrectly conclude that Quantity A is always greater.
m < 1
| Quantity A | Quantity B |
| m + 1 | 3m – 2 |
155
Answer: (A) Quantity A is greater.
m must be less than 1, which is less than 3/2. Quantity A must be greater.
What is x?
7x + 10 = 4x – 5
156
Answer: x = –5
7x + 10 = 4x – 5
3x + 10 = –5
3x = –15
x = –5
If x ≠ 0, Which of the following is closest to –x on the number line?
(A) 0
(B) x
(C) –
x
(D) 
x
(E) –
x
157
Answer: (E) –
x
Pick a smart number for x. If x = –6, we are looking for the answer that is closest to +6 on the number line.
(A) 0. Distance = 6
(B) x = –6. Distance = 12
(C) –
x = 9. Distance = 3
(D) 
x = –2. Distance = 8
(E) –x = 4. Distance = 2
Any other smart number would give us the same result—(E) is smallest.
x = –1
| Quantity A | Quantity B |
| x2 | x – 1 |
158
Answer: (A) Quantity A is greater.
Quantity A: When x = –1, x2 is positive (specifically, +1).
Quantity B: When x = –1, x – 1 is negative (specifically, –2).
x = –1
| Quantity A | Quantity B |
| 2x + 1 | –2x + 1 |
159
Answer: (B) Quantity B is greater.
Calculate, avoiding silly errors:
Quantity A: 2x + 1 = –2 + 1 = –1
Quantity B: –2x + 1 = 2 + 1 = 3
x = –1
| Quantity A | Quantity B |
| 3x – 1 | 2x + 1 |
160
Answer: (B) Quantity B is greater.
Calculate, avoiding silly errors:
Quantity A: 3x – 1 = –3 – 1 = –4
Quantity B: 2x + 1 = –2 + 1 = –1
x = –1
| Quantity A | Quantity B |
| 2x – 1 | 3x + 1 |
161
Answer: (B) Quantity B is greater.
Calculate, avoiding silly errors:
Quantity A: 2x – 1 = –2 – 1 = –3
Quantity B: 3x + 1 = –3 + 1 = –2 (greater than –3)
Which two of the following numbers have a product that is greater than 10?
(A) –6
(B) –3
(C) 2
(D) 4
162
Answer: (A) and (B)
(A)(B) = (–6)(–3) = 18
For a positive product of two terms, we need to pair either the two positive choices, (2)(4), or the two negative choices, (–6)(–3). The product of the two negative choices is greater than 10.
Which two of the following numbers have a product that is less than –15?
(A) –6
(B) –3
(C) 2
(D) 4
163
Answer: (A) and (D)
(A)(D) = (–6)(4) = –24
For a negative product of two terms, we need to pair a positive choice with a negative choice. There ares four possibilities to check, but since the answer must be less than a certain value, it makes sense that it results from the smallest negative choice (–6) times the greatest positive choice (4).
Which three of the following numbers have a product that is greater than 40?
[A] –6
[B] –3
[C] 2
[D] 4
164
Answer: [A], [B], and [D]
[A][B][D] = (–6)(–3)(4) = 72
For a positive product of three terms, we need either (pos)(pos)(pos), which is not possible with the choices offered, or (neg)(neg)(pos). Thus, we must include [A] and [B], and we maximize the product by selecting the greatest positive remaining choice, [D].
Which three of the following numbers have a product that is less than –30?
[A] –6
[B] –3
[C] 2
[D] 4
165
Answer: [A], [C], and [D]
[A][C][D] = (–6)(2)(4) = –48
For a negative product of three terms, we need either (neg)(neg)(neg), which is not possible with the choices offered, or (neg)(pos)(pos). Thus, we must include [C] and [D], and we minimize the product by selecting the most negative remaining choice, [A].
What is the units digit of 7248?
166
Answer: 6
In a product (which is fundamentally what an exponent represents), to find the units digit, only pay attention to the units digits of the numbers you are working with. Drop any other digits. Thus, only consider the 2 in the units digit of 72. The powers of 2 are: 2, 4, 8, 16, 32, 64, 128, 256, 512, etc. Notice that the units digits repeat in a 4-term cycle: [2, 4, 8, 6]. The 48th power is at the end of the twelfth 4-term repeat cycle, so 7248 has a units digit of 6.
The cost of item A is $0.17, and the cost of item B is $0.25. What is the total cost of 50 item A's and 21 item B's, in dollars?
167
Answer: $13.75
50($0.17) + 21($0.25) = $8.50 + $5.25 = $13.75
Alternatively, some “mental math”:
Half of 100($0.17) + 21 quarters = Half of $17 + $5.25 = $8.50 + $5.25
If a group of 10 students includes exactly six seniors and exactly five women, what are the minimum and maximum possible numbers of women seniors?
168
Answer: Maximum = 5, Minimum = 1
Maximum: Occurs if all of the women are seniors. Note that not all of the seniors be women, as there are fewer women than seniors in the group. Group = 5 senior women, 1 senior man, and 4 non-senior men.
Minimum: Occurs when the number of women non-seniors is maximized. There are 4 non-seniors in the group, all of which could be women, leaving 1 woman who must be a senior. Group = 5 senior men, 1 senior woman, and 4 non-senior women.
How many lines in the plane are equidistant from the parallel lines above?
169
Answer: 1 line, halfway between the two parallel lines
How many points in the plane are equidistant from the parallel lines above?
170
Answer: An infinite number of points
Any point on the dotted line below is equidistant from the parallel lines. Some example points are shown below.
In the figure above, lines m and n are parallel.
In order to determine all of the angles above, what information would suffice?
171
Answer: Any one of the angles
Knowing any one of the angles would allow us to determine all the angles. In such a figure (parallel lines cut by a transversal), the relationship among all eight angles is known:
For example, if x = 55, you could label all of the angles as either 55° or 125°, according to the figure above.
Simplify:
am × an – m
172
Answer: an
When multiplying terms that have a common base, add the exponents:
am × an – m = am + n – m = an
Simplify:
33 + 33 + 33 + 33 + 33 + 33 + 33 + 33 + 33
173
Answer: 35
33 + 33 + 33 + 33 + 33 + 33 + 33 + 33 + 33 = 9(33)
= 32(33)
= 32 + 3
= 35
x and y are positive integers such that xy = 16
| Quantity A | Quantity B |
| The absolute difference between x and y | 0 |
174
Answer: (D) The relationship cannot be determined from the information given.
If x and y are both positive such that xy = 16, there are the following possibilities:
| x | y | Absolute difference = |x – y| |
| 1 | 16 | 15 |
| 2 | 8 | 6 |
| 4 | 4 | 0 |
| 8 | 2 | 6 |
| 16 | 1 | 15 |
The absolute difference between x and y can be 15, 6, or 0. Do not assume, just because the variables are different letters, that x cannot equal y! x might equal y in this case, or it might not.
| Quantity A | Quantity B |
| Two times the average of x, y, and z | The average of 2x, 2y, and 2z |
175
Answer: (C) The two quantities are equal.
Quantity A: 
Quantity B: 
If the positive integer x is divisible by 6, is x divisible by 3?
176
Answer: Yes
A number is always divisible by any of its factors. Since x is divisible by 6, x has the factors of 6, including 2 and 3. Thus, x is divisible by both 2 and 3.
For example: x is divisible by 6, so x could be 6, 12, 18, 24, 30, 36, etc.
All of the possible values of x are divisible by 3.
If the positive integer x is divisible by 6, is x divisible by 12?
177
Answer: Maybe
In order for x to be divisible by 12, it would need to have the factors of 12: (2)(2)(3). Since x is divisible by 6, it has the necessary factor of 3 and one of the necessary factors of 2. However, it is uncertain whether x has the second necessary factor of 2.
For example: x is divisible by 6, so x could be 6 or 24.
24 is divisible by 12, but 6 is not.
If y = 2x + 1 and y = -x + 4, solve for x and y.
178
Answer: x = 1 and y = 3
A point being on a line is the same as the equation “working” when you plug in the (x, y) coordinates for that point. For example, the points (1, 3) and (3, 7) are both solutions for y = 2x + 1, as both points are on that line. So for a solution to “work” for two equations, it would have to be a point lying on both lines. Here, the only such point is the intersection of the two lines, at (1, 3).
Algebra: Since both 2x + 1 and -x + 4 equal y, they equal each other.
2x + 1 = -x + 4
3x = 3
x = 1
Plug x = 1 into either equation to get y = 3.
| Quantity A | Quantity B |
| The shaded area of rhombus ABCD. | The non-shaded area of rhombus ABCD. |
179
Answer: (A) Quantity A is greater.
Notice that the two shaded triangles are both 3–4–5 right triangles.
If sides AB and CD have length S, then so must BC and AD (because ABCD is a rhombus). Therefore the non-shaded portions of BC and AD have length 2.
The non shaded area is a rectangle: lw = 4 × (5 – 3) = 8
Which of the following is greatest?
(A) 1 + 2 – 3 + 4 – 5 + 6 – 7 + 8
(B) 1 – 2 + 3 – 4 + 5 – 6 + 7 – 8
(C) 1 – 2 – 3 – 4 – 5 + 6 + 7 + 8
(D) 1 + 2 + 3 + 4 – 5 – 6 – 7 – 8
180
Answer: (C)
Of course, you could just compute.
(A) 1 + 2 – 3 + 4 – 5 + 6 – 7 + 8 = 6
(B) 1 – 2 + 3 – 4 + 5 – 6 + 7 – 8 = –4
(C) 1 – 2 – 3 – 4 – 5 + 6 + 7 + 8 = 8
(D) 1 + 2 + 3 + 4 – 5 – 6 – 7 – 8 = –16
Tedious calculations can lead to silly errors, so strategic elimination is a good double-check.
(B) Pair terms: (1 – 2) + (3 – 4) + (5 – 6) + (7 – 8) = sum of four negatives < 0
(D) Pair terms: (1 – 5) + (2 – 6) + (3 – 7) + (4 – 8) = sum of four negatives < 0
In (A) and (C), the negative terms are generally smaller in absolute value than the positive terms, so expect positive results and be most careful in calculating (A) and (C).
If 35 percent of the rooms at a hotel have an ocean view, what is the ratio of the number of rooms with an ocean view to the number of rooms without?
181
Answer: 
What is 135% of 140?
182
Answer: 189
100% of 140 = 140
10% of 140 = 14
30% of 140 = 3(14) = 42
5% of 140 = 14/2 = 7
So, 135% of 140 is 140 + 42 + 7 = 189.
Alternatively, calculate 1.35 × 140 = 189.
What is the sum of all the integers from 11 to 30, inclusive?
183
Answer: 410
Sum of consecutive integers 
The number of integers is First – Last + 1 = 30 – 11 + 1 = 20.
Alternatively, with a calculator: 11 + 12 + 13 + 14 +…+ 27 + 28 + 29 + 30, but even with a calculator, this method is more time consuming and error-prone.
| Quantity A | Quantity B |
 |
 |
184
Answer: (B) Quantity B is greater.
Note that the last term in each sum is +2. Ignore 10/5 in Quantity A and ignore 2 in Quantity B.
Quantity A: 
Quantity B: 
Alternatively, use decimals for Quantity B: 
3 rabbits each weigh an average of 8 pounds. What would a 4th rabbit need to weigh in order to reduce the average weight of the 4 rabbits to 7.5 pounds?
185
Answer: 6 pounds
Sum = Average × # of terms, so the total weight of the first 3 rabbits is 8 × 3 = 24 pounds.
If x is the weight of the 4th rabbit, and the average weight of all 4 is to be 7.5 pounds,
Alternatively, consider that the 4th rabbit only has 1/4 “influence” on the average weight of all 4 rabbits, yet is reducing the average weight by 0.5 pounds. To do so, it would have to be 4(0.5 pounds) = 2 pounds lighter than the average weight of the other rabbits.
| Quantity A | Quantity B |
| (–15)2(12)3 | (15)3(–12)3 |
186
Answer: (A) Quantity A is greater.
Quantity A: (–15)2(12)3 = positive × positive = positive
Quantity B: (15)3(–12)3 = positive × negative = negative
Simplify: 
187
Answer: 
| Quantity A | Quantity B |
| 3(72 + 112) + 25 | 147 + (121)(3) + 42 |
188
Answer: (A) Quantity A is greater.
Only do enough math to recognize similarities between the two quantities.
Quantity A: 3(72 + 112) + 25 = 3(49 + 121) + 25
Quantity B: 147 + (121)(3) + 42 = 3(49 + 121) + (22)2
Only the last additive term is different, and 25 > 24.
What is 160% of 45?
189
Answer: 72
100% of 45 = 45
10% of 45 = 4.5
60% of 45 = 6(4.5) = 27
So, 160% of 45 is 45 + 27 = 72.
Alternatively, calculate 1.6 × 45 = 72.
Which of the following is the graph on the number line for all values of x such that x3 < x2?
190
Answer: (B)
You could try values in the different ranges shown:
x = –3 works: –27 < 9
x = –1 works: –1 < 1
x = –1/2 works: –1/8 < 1/4
x = 0 does NOT work: 0 = 0
x = 1/2 works: 1/8 < 1/4
x = 1 does NOT work: 1 = 1
x = 3 does NOT work: 27 > 9
Algebra solution: x3 < x2 is true when x3 – x2 < 0, or x2(x – 1) is negative.
x2 is positive for non-zero x values. x – 1 will be negative when x < 1. So generally, x < 1 is the range where x3 < x2. However, x2(x – 1) equals 0 when x = 0 or 1, so these values are not allowed.
Simplify: 
191
Answer: 53 or 125
Alternatively, remember why this rule works:
This process works perfectly with numbers. With variables, you have to put in absolute value signs to force the outcome to be positive: 
Simplify: 
192
Answer: 
Any non-zero number to the 0th power equals 1.
Simplify: 
193
Answer: 
Simplify: (3x3)2
194
Answer: 32x6 or 9x6
(3x3)2 = (31x3)2 = (31 × 2)(x3 × 2) = 32x6 or 9x6
Simplify: 
195
Answer: 
Factor: x2 + 5x – 6
196
Answer: (x + 6)(x – 1)
6 can be factored as either (3)(2) or (6)(1). To achieve a middle term of 5x, note that 3 and 2 sum to 5, whereas 6 and 1 differ by 5. Since the last term is –6, the factor numbers must have different signs, so pick the factor pair that differs by 5: 6 and –1 are the factor numbers.
x2 + 5x – 6 = (x + 6)(x – 1)
Given x ≠ –4, simplify: 
197
Answer: x – 4
Recognize the “difference of two squares” common quadratic:
Alternatively, express 2.25 as a fraction:
Then flip 
to get 
What is 
in decimal form, rounded to the nearest 0.01?
198
Answer: 1.42
Because the thousandths digit is 4 or less, round down to the nearest 0.01.
What is the reciprocal of 2.25?
199
Answer: 
The reciprocal of 2.25 is 
What is the reciprocal of 3.5?
200
Answer: 
The reciprocal of 3.5 is 
Alternatively, express 3.5 as a factor:
Then flip 
to get .