What is the average of the set {1, 5, 9, 13, 17, 21, 25}?
251
Answer: 13
Notice that the terms in this set are evenly spaced: 5 is 4 greater than 1, 9 is 4 greater than 5, etc.
In an evenly spaced set, the middle term equals the average. There are 7 terms in this set, so the 4th term is the middle term: Average = 13.
Alternatively, add “easy pairs” (e.g., 13 + 17 = 30) and get a total of 91, then divide by 7.
Translate:
Shipping costs for a package are $5 for the first 5 pounds or less, plus an additional $0.40 for each additional pound.
252
Answer: If x ≤ 5, Cost = $5
If x > 5, Cost = $5 + $0.40(x – 5)
Let's call x the weight of the package, in pounds.
“Shipping costs for a package are $5 for the first 5 pounds or less,” so if x ≤ 5, Cost = $5.
“plus an additional $0.40 for each additional pound” over 5 pounds, which is an additional $0.40(x – 5). So if x > 5, Cost = $5 + $0.40(x – 5).
The 4 women competing in a race each finished in an average of 22 minutes and 35 seconds. The 3 men competing in the same race each finished in an average of 21 minutes and 45 seconds.
| Quantity A | Quantity B |
| The average race-completion time for all 7 racers. | 22 minutes and 8 seconds |
253
Answer: (A) Quantity A is greater.
If the number of men and women racers were equal, the average time for ALL racers would be the simple average of the men's and women's average times, or 22 minutes and 10 seconds (±25 seconds, exactly between each gender's average time).
But this group has slightly more women than men. The women's average time is greater than the simple average, so more women pull the average up. Therefore, the average time for all racers must be greater than 22 minutes and 10 seconds, which is already greater than Quantity B (22 minutes and 8 seconds).
How many different ways can the letters in the word “PLATE” be arranged?
254
Answer: 120
PLATE has 5 letters, all unique, so there are 5! = 120 different ways to arrange the letters.
How many different ways can the letters in the word “SPOON” be arranged?
255
Answer: 60
SPOON has 5 letters, but the letter O appears twice, so there are 
different ways to arrange the letters.
How many different ways can the letters in the word “FORK” be arranged?
256
Answer: 24
FORK has 4 letters, all unique, so there are 4! = 24 different ways to arrange the letters.
If (x – 3) = 4, what is the value of x?
257
Answer: x = 7
(x – 3) = 4, implies that x = 4 + 3 = 7. This is a linear equation, so there is one solution.
If 
, what is the value of x?
258
Answer: 7 or –1
implies that (x – 3)2 = 16, which can happen when (x – 3) = +4 or (x – 3) = –4. There are two solutions.
If x – 3 = 4, then x = 7.
If x – 3 = –4, then x = –1.
Check:
and 
, so x – 3 = 4 or –4.
{–5, –3, –2, 
, 0, 
, 
,1, 1, 2, 2, 4, 8, 9, 10}
If a 16th term is added to the 15-term set above, the new term would be in Quartile 3 if it were which of the following? Indicate all such values.
[A] 
[B] 
[C] 
[D] 3 [E] 
[F] 5
259
Answer: [B], [C], [D], and [E]
The set is already ordered from least to greatest. Split the 15-term set into quartiles, with 4 terms in Quartiles 1, 2 and 4, leaving space for the 16th term somewhere in Quartile 3:
The new term will fall in Quartile 3 if it is between 1 and 4. The answer choices in this range are 
, 
, 3, and 
How many different triangles are in the figure above?
260
Answer: 18
There are 8 small triangles: 
etc.
There are 8 medium triangles (comprised of 2 small triangles): 
etc.
There are 2 large triangles (comprised of 4 small triangles): 
How many different triangles are in the figure above?
261
Answer: 12
There are 6 small triangles: 
etc.
There are 2 medium triangles (comprised of 2 small triangles): 
There are 4 large triangles (comprised of 3 small triangles): 
etc.
| Quantity A | Quantity B |
 |
0.08 |
262
Answer: (A) Quantity A is greater.
Resist the urge to reduce 
to 
. The numbers are easier to compare if we make use of the 8 in both.
Quantity B: 
, which has the same numerator but larger denominator than the fraction in Quantity A. Larger denominator means smaller value.
What are the unique prime factors of 362?
263
Answer: 2 and 3
362 has the same prime factors as 36. From the factor tree, the prime factors are 2 and 3.
362 = (2232)2 = 2434. There are no new prime factors.
How many factors does 36 have?
264
Answer: 9
From the factor tree, 36 = 2232. These prime factors can be combined to create all the other factors of 36. These factors can include no, one, or two 2’s (3 possibilities) and no, one, or two 3’s (3 possibilities). Thus, there are 3 × 3 = 9 possible ways to combine the prime factors. Alternatively, list the factors:
A factory can produce 190 widgets per metric ton of zinc, and a metric ton of zinc costs $2,060. What is the cost for the zinc used to produce 450 widgets, rounded to the nearest hundred dollars?
265
Answer: $4,900
Zinc required: 450 widgets/190 widgets per ton = 45/19 metric tons
Cost per metric ton of zinc = $2,060
Total cost = (45/19)($2,060) = $4,878.95. Round up to $4,900.
This is a really tough one for mental math, but we could get close:
45/19 = 2 + 7/19 ≈ 2 + 1/3
2 × $2,060 = $4,120
1/3 of $2,060 ≈ 1/3 of $2,100 = $700
The sum is $4,820—off by only a little.
A triangle has a perimeter of 24.
What is the maximum area of the triangle?
266
Answer: 
To maximize area for a given perimeter, make a regular polygon (one with all sides equal and all angles equal). For a triangle with perimeter 24, this is an equilateral triangle with sides 8. One of these sides will be the base of the triangle. The height of an equilateral triangle is always 
times the side length. Here, the height is 
A triangle has one side that is 6 inches and another that is 15 inches. What is the maximum area of the triangle?
267
Answer: 45 inches2
If two sides of a triangle are given, the area is maximized then those two sides are placed perpendicular to each other. Doing so makes the given sides the base and height:
Any other angle between the 6 and the 15 makes the height less than 6.
The sum of five integers is 17. The integer 10 is then included in the set. What is the average of the complete set of six integers?
268
Answer: 4.5
In order to answer, you don't need to know what the original five integers are individually. Only the sum is required.
The sum of all six integers is just the sum of the first five integers (17), plus the newly included integer 10.
Set S: {11, 18, 45}
If 4 were added to each of the terms in the set above, the standard deviation of the set would ________.
(A) decrease
(B) remain the same
(C) increase
269
Answer: (B) remain the same
After adding 4 to each of the terms in the set, the average of the terms in new Set S′ would be 4 greater than the average of the terms in original Set S. So, if each term increases by 4 and the average increases by 4, the difference between each term and the average would remain the same. Standard deviation is simply a measure of these differences for the whole set, so the standard deviation also would remain the same.
Set S: {11, 15, 31}
If each of the terms in the set above were multiplied by –3, the standard deviation of the set would ________.
(A) decrease
(B) remain the same
(C) increase
270
Answer: (C) increase
When each term in the set is multiplied by –3, the average of the new set would become –3 times the original average. The absolute value of the difference between each term and the average of the set would also be |–3| = 3 times the original difference.
Standard deviation is simply a measure of these differences for the whole set, so the fact that these differences increase by a factor of 3 means that the standard deviation would increase.
Don't be fooled by the negative sign. The numbers all get farther apart when multiplied by –3.
Set Q: {–5, 20, 145}
If each of the terms in the set above were multiplied by 
, the standard deviation of the set would ________.
(A) decrease
(B) remain the same
(C) increase
271
Answer: (A) decrease
When each term in the set is multiplied by 
, the average of the new set would become times the original average. The positive difference between each term and the average of the set would also be of the original difference.
Standard deviation is simply a measure of these differences for the whole set, so the fact that these differences decrease by a factor of 5 means that the standard deviation would decrease.
Set R: {1, 2, 3, 4, 5}
If each of the terms in the set above were multiplied by 2 then had 2 subtracted from the it, the standard deviation of the set would ________.
(A) decrease
(B) remain the same
(C) increase
272
Answer: (C) increase
The average of the terms in Set R is 3 (the middle term in the evenly spaced set). The positive differences between each term and the average are 2, 1, 0, 1, and 2, respectively.
The new Set R′ would be {0, 2, 4, 6, 8}. The average would be 4, and the positive differences between each term and the average would be 4, 2, 0, 2, and 4, respectively.
Standard deviation is simply a measure of these differences for the whole set, so the fact that these differences would double means that the standard deviation would increase. We can ignore the subtraction and just pay attention to the multiplication by 2.
Set R: {–50, –49, –48}
If each of the terms in the set above had 50 subtracted from it, then was doubled, the standard deviation of the set would ________.
(A) decrease
(B) remain the same
(C) increase
273
Answer: (C) increase
The average of the terms in Set Z is –49 (the middle term in the evenly spaced set). The positive differences between each term and the average are 1, 0, and 1, respectively.
The new Set Z′ would be {–200, –198, –196}. The average would be –198, and the positive differences between each term and the average would be 2, 0, and 2, respectively.
Standard deviation is simply a measure of these differences for the whole set, so the fact that these differences would double means that the standard deviation would increase. We can ignore the subtraction and just pay attention to the doubling.
In a set of 1,000 consecutive multiples of 3, what is P62 – P12?
(Remember, P62 is the 62nd percentile of the set, where each percentile is comprised of 
terms.)
274
Answer: 1,500
If each percentile is composed of 10 terms, and P62 and P12 are 50 percentiles apart (62 – 12 = 50), then P62 and P12 are (10 terms/percentile)(50 percentiles) = 500 terms apart.
The terms are consecutive multiples of 3. Terms that are 1 apart (i.e. adjacent terms) differ by 3. Terms that are 2 terms apart differ by (2)(3) = 6. Terms that are 500 terms apart differ by (500)(3) = 1,500.
If two fair 6-sided dice are rolled together, what is the probability of rolling numbers that have an even product?
275
Answer: 
An even product results from (E)(O) or (E)(E) or (O)(E). The only way to get an odd product is (O)(O). Thus, out of these 4 basic cases, each of which is equally probable, 3 case result in an even product.
Alternatively, think of all the possible number pairs. There are 6 number possibilities for the first die, and 6 for the other, for a total of (6)(6) = 36 possible pairings. How many of these are (O)(O), the “failing” cases? There are 3 odd possibilities for the first die, and 3 for the other, for a total of (3)(3) = 9 possible (O)(O) pairings. Thus, 36 – 9 = 27 pairings have an even product, and 
If two fair 6-sided dice are rolled together, what is the probability of rolling numbers that have an even sum?
276
Answer: 
An even sum results from (E + E) or (O + O). An odd sum results from (E + O) or (O + E). There are no other possibilities. Thus, out of these 4 basic cases, 2 result in an even sum. 
If two fair 6-sided dice are rolled together, what is the probability of rolling numbers that have a product of 12?
277
Answer: 
Using the numbers 1 through 6, what pairs have a product of 12? (2 × 6) or (3 × 4) or (4 × 3) or (6 × 2). Note that the order of the numbers doesn't matter for the product (i.e. 12 = 3 × 4 = 4 × 3}, but the fact that the 3 could occur on either die, while the 4 is on the other, doubles the chance of pairing 3 with 4. The same is true for 2 and 6.
In all, there are 6 number possibilities for the first die, and 6 for the other, for a total of (6)(6) = 36 possible pairings. As shown above, 4 of these pairings have a product of 12, so the probability is 
| Quantity A | Quantity B |
The area of a regular hexagon with each side equal to  |
The area of a square with each side equal to  |
278
Answer: (A) Quantity A is greater
A hexagon has 6 sides, so the perimeter of the hexagon is x. A square has 4 sides, so the perimeter of the square is also x.
For a given perimeter, area is maximized by making a polygon as regular as possible (all sides the same, all angles the same) and as multi-sided as possible. In the extreme, consider a regular polygon with 200 sides—it is almost a circle, which is the shape with maximum area for a given perimeter. Thus, for perimeter x, a regular hexagon has a greater area than a regular square.
A fair coin is flipped three times. What is the probability of flipping tails exactly once?
279
Answer: 
On each flip, either a head (H) or tails (T) will result. There are 2 possible outcomes of 1 flip. Thus, there are 2 × 2 × 2 = 8 ways the series of three flips could go.
There are 3 ways that tails could be flipped “exactly once”: THH or HTH or HHT. Therefore, the probability is .
Factor: ab + 2b – 3a – 6
280
Answer: (a + 2)(b – 3)
Group terms with similarities, and then factor out the shared element:
ab + 2b – 3a – 6
(ab – 3a) + (2b – 6)
a(b – 3) + 2(b – 3)
(a + 2)(b – 3)
To check, FOIL the factored expression back to its original form.
Factor: 
281
Answer: 
Check: When FOILing, the Outer and Inner terms will each be 
, and there are two of them, so they will sum to ax, the middle term in the original expression. The First and Last terms are just the squares of the respective terms in the factored form.
The “length” of a positive integer is the number of non-unique prime factors it has. For example, 60 = (2)(2)(3)(5) has a length of 4. What is the maximum length of the numbers between 1,500 and 2,000, inclusive?
(A) 6
(B) 7
(C) 8
(D) 9
(E) 10
282
Answer: (E) 10
To maximize the “length” of a positive integer while also limiting its value, use as many factors of 2 (the smallest prime) as possible, and avoid using large primes, such as 47 or 89. The powers of 2 near our number range are: 29 = 512, 210 = 1,024, and 211 = 2,048.
Since 2,048 > 2,000, length of 11 is not possible. What about length of 10? If all 10 primes are 2, the resulting number is too small (1,024), but try using a 3, the next smallest prime. After a little trial and error: 31(29) = 3(512) = 1,536. This is in the range, and has a length of 1 + 9 = 10.
Solve for x: x2 + 6x – 216 = 0
283
Answer: x = 12 or –18
The factors of 216 are:
Since 216 is negative in the quadratic, the factor numbers will have opposite signs. Look for the factor pair that differs by 6, in order to create the +6x middle term of the quadratic: (x – 12)(x + 18) = 0
What is the greatest integer less than 65 that has at least 3 unique prime factors?
284
Answer: 60
Consider the numbers less than 65, starting with the greatest such number. Factor and count unique prime factors.
| 64 = 26 | 1 unique prime factor: 2 |
| 63 = (3)(3)(7) | 2 unique prime factors: 3, 7 |
| 62 = (2)(31) | 2 unique prime factors: 2, 31 |
| 61 is prime! | 1 unique prime factor: 61 |
| 60 = (2)(2)(3)(5) | 3 unique prime factors: 2, 3, and 5 |
What are the first 15 prime numbers?
285
Answer: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
| Quantity A | Quantity B |
 |
 |
286
Answer: (B) Quantity B is greater.
Since Quantity A mixes decimals and fractions, you could solve using either form. Because Quantity A is compared to 1/2, the question might be rephrased as “How does the denominator compare to 2(1.75) = 3.50?”
Quantity A: 
4.06 > 3.50, so Quantity A is less than 1/2.
What could be the value of x if…
…x2 = 36?
287
Answer: 6 or –6 (top question), but only +6 (bottom question)
x2 = 36 has two solutions: x = 6 and x = –6. Plug back in to see why.
62 = 36 and similarly (–6)2 = 36.
has only one solution: x = 6. Notice that 
In the figure above, 
, and the length of AE is 7.5.
What is the length of BD?
288
Answer: 5
Line segment BD drawn within triangle ACE parallel to side AE creates a Similar Triangle BCD. Similar triangles have the same three angle measures, and the ratio of corresponding sides of the two triangles is constant.
In equation form, this means that the given side ratio is equivalent to the ratio of BD to AE: 
So,
Based on the figure above, what is x – b?
289
Answer: 40
Angles that form a line sum to 180c, so x + a = 180
The sum of the angles in a triangle is 180°, so a + b + 40 = 180.
The two expressions that equal 180 can be set equal to each other and solved.
x + a = a + b + 40
x = b + 40
x – b = 40
Alternatively, an exterior angle of a triangle is equal to the sum of the two opposite interior angles of the triangle. This means that x = b + 40, or x – b = 40.
Two right circular cylindrical glasses hold the same volume of tea when filled to the brim. The glass with diameter 5 is how many times the height of the glass with radius 5?
290
Answer: 4
In order to hold the same volume of tea, one glass is tall and narrow (diameter 5, so radius 5/2), the other glass is short and wide (diameter 10, since radius 5).
Tall cylinder volume 
Short cylinder volume 
The volume of tea is equal:
Intuitively, the glass with radius 5 has twice the radius, so the area of its base is four (= 22) times as big. So the other glass must be four times as tall, to compensate.
If Point P is the center of the circle in the figure above, what is x?
291
Answer: 60
There are two ways to determine the answer.
(1) Since AP and PB are both radii of the circle, triangle ABP is an isosceles triangle, so ∠ABP is also 30. Thus, the remaining angle in the triangle, ∠APB, is 180 – 30 – 30 = 120. Since x and ∠APB form a straight line, x = 180 – ∠APB = 180 – 120 = 60.
(2) Recognize that for minor arc BC, 30° labels the inscribed angle, and x° labels the central angle. For a given arc, the inscribed angle is half of the central angle, or the central angle is twice the inscribed angle. 2(30) = 60.
What is x if point P does NOT lie on any side of the triangle and the area of the triangle is maximized?
292
Answer: 60
Because the center of the circle does NOT lie on any side of the triangle, x cannot be 90. However, x could be any other value greater than 0 and less than 180.
But given the circular limits for this triangle, area is maximized by making the triangle as regular as possible (all sides the same, all angles the same). In other words, an equilateral triangle maximizes the triangle area. Thus, x = 60, as do the other two angles in the triangle.
The formal proof that the equilateral triangle maximizes the area in this case is not trivial, but you're not responsible for that proof. Just remember that regular polygons (such as the equilateral triangle) maximize the area under constraints (being inside a circle, or having a constant perimeter, etc.).
What is the maximum area of the triangle if the radius of the circle is 1?
293
Answer: 1
Because the center of the circle lies on one side of the triangle, the opposite angle of the triangle is a right angle: x = 90. The hypotenuse of this right triangle is 2(radius) = 2. To maximize the area of the triangle, make the two perpendicular sides of the right triangle the same length, i.e. make a 45–45 –90 triangle. This makes the height as large as possible. For a hypotenuse of 2, the other side lengths would be 
Which quadrants will 
go through?
294
Answer: I, II, and III
The y-intercept of this line is 
, which is positive. The line crosses the y-axis between quadrants I and II.
The slope of the line is +4, which means the line slopes up into quadrant I and down into quadrant II, passing also down into quadrant III.
Which quadrants will y = –3x + 7 go through?
295
Answer: I, II, and IV
The y-intercept of this line is 7, which is positive. The line crosses the y-axis between quadrants I and II.
The slope of the line is –3, which means the line slopes up into quadrant II and down into quadrant I, passing also down into quadrant IV.
Which quadrants will 
go through?
296
Answer: I, III, and IV
The y-intercept of this line is –13, which is negative. The line crosses the y-axis between quadrants III and IV.
The slope of the line is +1/2, which means the line slopes up from left to right: from quadrant III into quadrant IV, and finally up into quadrant I.
| Quantity A | Quantity B |
| The maximum possible circumference of a spherical ball that would fit inside a 10 by 11 by 14 rectangular box. | 14π |
297
Answer: (B) Quantity B is greater.
The ball cannot have a diameter greater than 10, the smallest dimension of the box. The maximum circumference of the ball is like the “equator” of the ball, where d = 10. Circumference = πd = 10π.
If a right circular cylindrical can has radius 3.75 and height 11, what is the surface area of the curved surface of the can (that is, excluding the flat bases)?
298
Answer: 82.5π
If the curved surface of the can were unrolled, it would be a rectangle with height = height of the original can, and width = circumference of the flat circular ends of the can.
Rectangle width = can circumference = 2πr = 2π(3.75) = (7.5)π.
Rectangle area = wh = (7.5π)(11) = 82.5π
x, y, and z are integers. If 
is a negative integer, xz > 0, and y is even, which of the following statements could be true?
Select all such statements.
[A] x is even
[B] x is odd
[C] x is positive
[D] z is negative
299
Answer: [A], [C], and [D].
is a negative integer, so x and y have opposite signs.
xz > 0, so x and z have the same sign.
| x | y | z |
| + | - | + |
| - | + | - |
Both choice [C] and [D] could be true.
Since = an integer, then x = y(integer) = even(integer) = even. Choice [A] must be true. Because x cannot be odd, choice [B] is eliminated.
What is the sum of the 30 smallest even positive integers?
300
Answer: 930
The first even integer is 2, the second is 2(2) = 4, and the third is 3(2) = 6, so the thirtieth even integer is 30(2) = 60. What is the sum of 2, 4, 6,…, 58, and 60?
