If the sum of the consecutive integers from x to (x + 6), inclusive, is 56, what is x?
301
Answer: 5
There are 7 integers in the sum, so the average integer is 
In a consecutive set, the average is also the middle term. Thus, the 4th integer in this list is 8: {5, 6, 7, 8, 9, 10, 11}. x is 5.
Alternatively,
What is the average of 11.5, 12.25, 13, 13.75, 14.5, 15.25, and 16?
302
Answer: 13.75
Give the calculator a rest! The terms in this set are equally spaced 0.75 apart. In an evenly spaced set, the average of the set equals the middle term (if an odd number of terms) or the average of the two middle terms (if an even number of terms). This set has 7 terms, so the middle term is the 4th, which is 13.75.
| Quantity A | Quantity B |
| The average of –2x, –x, 0, x, 2x, and 3x. | 0 |
303
Answer: (D) The relationship cannot be determined from the information given.
In Quantity A, the 6 terms are evenly spaced x apart. The average of the set is the average of the two middle terms (since there are an even number of terms), which is 
Be careful! It is tempting to think this list is ordered from low to high, negative to positive, but that is only true if x is positive. If x is negative, the list is ordered high to low, positive to negative. Either way, the average is 
, but this could be greater or less than 0. In fact, x could even be 0, making all the terms and the average in Quantity A equal 0.
Raffle tickets with consecutive integers 1,014 to 2,345, inclusive, were sold. Each ticket has exactly one unique integer on it. How many tickets were sold?
304
Answer: 1,332
Subtract the low number from the high number and “add one before you are done.”
A, B, and C lie on a number line such that B is the midpoint between A and C.
| Quantity A | Quantity B |
| A | B |
305
Answer: (D) The relationship cannot be determined from the information given.
The number line could look like this:
Or like this:
Either A or B could be the greater number.
| Quantity A | Quantity B |
| A × B | C × D |
306
Answer: (D) The relationship cannot be determined from the information given.
From the number line, it is clear that A and B are both negative and C and D are both positive. Thus, both quantities are positive, but no scale is provided, so nothing is known about the relative values.
One possibility is that A = –3, B = –2, C = 4, D = 5. Since (–3)(–2) = +6 and (4)(5) = +20, Quantity B is bigger.
Another possibility is that A = –9, B = –7, C = 7, D = 8. Since (–9)(–7) = +63 and (7)(8) = +56, Quantity A is bigger.
On the number line above, B is twice as far from C as from A, and D is twice as far from B as from C. What is the ratio of length 
to length 
?
307
Answer: 
“B is twice as far from C as from A”: Label the distance between B and A as x. The distance between B and C is twice that, or 2x.
“D is twice as far from B as from C”: If the distance between C and D is 2x, then D is 2x + 2x = 4x from B, which is “twice as far…as from C.”
| Quantity A | Quantity B |
| |P| | Q |
308
Answer: (B) Quantity B is greater.
P is between –5 and 0, so |P| is between 0 and 5.
Q is between 5 and 10.
Thus, Q > 5 > |P|.
If the tick marks on the number line above are evenly spaced, what is the value of x?
309
Answer: 
is three tick marks to the right of 
In other words, 3 intervals equals
Each interval is 
, then.
x is three tick marks to the right of , so x is 
greater than 
Based on the number line above, which of the following must be true? Select all that apply.
[A] xy < 0
[B] zy < xy
[C] z – x > x – y
[D] z – y > z – x
310
Answer: [A], [B] and [D]
Note that only one number, 0, is labeled on the number line. From their relative positions, x and z are positive while y is negative. It is also true that z > x. Beyond that, however, the values are not known, and don't assume the figure is drawn to scale.
[A] TRUE: x is positive and y is negative, so xy < 0
[B] TRUE: zy < xy divided by y becomes z > x (flip the inequality sign because y is negative). It is true that z > x.
[C] UNCERTAIN: Both (z – x) and (x – y) are positive, but without known values, either could be greater.
[D] TRUE: Subtract z from both sides to get – y > – x. Divide by –1 (don't forget to flip the inequality sign!) to get y < x. This must be true because y is negative and x is positive. Alternatively, (z – y) is the distance between y and z on the line, which is clearly greater than the distance between x and z, (z – x).
Simplify: 
311
Answer: 
Simplify: 
312
Answer: 
Simplify: 
313
Answer: 75
What is the length of the diagonal of a square with side length 
?
314
Answer: 16
The diagonal of a square is always 
times the side length. For this square, the diagonal is 
The diagonal of a square is 10. What is the area of the square?
315
Answer: 50
The diagonal of a square is always times the side length. Conversely, the side length is always the diagonal divided by 
For this square, the side length is 
Area of a square is the side length squared: area 
The area of a rectangle is 60 and the diagonal length is 13. What is the perimeter of the rectangle?
316
Answer: 34
When the diagonal of a rectangle is 13, you should at least test the possibility that the diagonal creates two 5–12–13 right triangles. If so, the rectangle area would be (5)(12) = 60, which is exactly what this question specifies. Thus, perimeter = 2(L + w) = 2(12 + 5) = 34.
If two sides of a triangle are each 8 inches and the third side is x inches, what are possible values for x?
317
Answer: 0 < x < 16
The sum of any two side lengths of a triangle will always be greater than the third side. The sum of 8 and 8 is 16, which must be greater than third side x. The sum of x and either side 8 must be greater than the other side 8: x + 8 > 8, or x > 0.
| Quantity A | Quantity B |
The area of a triangle with side lengths  , 3, and  |
3 |
318
Answer: (B) Quantity B is greater.
The triangle side lengths in Quantity A are special: , 3, and can be written as 
, and 
In other words, the side lengths are in the ratio 
, which you should recognize as the ratio of the side lengths in a 30–60–90 triangle. Multiplying by doesn't change any angles;s it just increases the size of the triangle. Thus, the base and height of this right triangle are and 3, so the area 
What is the area of the triangle above?
319
Answer: 
The height and hypotenuse of the right triangle are known. Use Pythagorean Theorem to determine the base.
The area of the triangle is 
| Quantity A | Quantity B |
| a | b |
320
Answer: (B) Quantity B is greater.
The longest side of a triangle is opposite the largest angle, and the smallest side is opposite the smallest angle.
6 is the shortest side of the triangle, so a° is the smallest angle of this triangle. 9 is the longest side of the triangle, so b° is the largest angle of this triangle. Thus, b > a.
In the figure above, what is x?
321
Answer: 40
The sum of the angles in a triangle is 180:
180 = x + 87 + 53
180 = x + 140
40 = x
In a triangle with angles 45°, 45°, and 90°, a side length measures 
What could the other side lengths be?
322
Answer: 5, 
, and/or 10
In a 45–45–90 triangle, the ratio of the side lengths is x : x : 
The known side could be the hypotenuse, in which case the sides are 5 : 5 : 
Or, the known side could be one of the perpendicular legs, in which case the hypotenuse is 
, and the side ratio is 
| Quantity A | Quantity B |
 |
2 |
323
Answer: (C) The two quantities are equal.
This is a right triangle, so by Pythagorean Theorem, the hypotenuse is 
The sides of this triangle are in the ratio 1 :
: 2, which you should recognize as a property of a 30–60–90 triangle. The shortest side, 1, is across from the smallest angle: y = 30. The middle side, , is across from the middle angle: x = 60.
Therefore, 
x = 1,234,567.89
| Quantity A | Quantity B |
| The digit in the hundredths place of x | The digit in the hundredths place of x |
324
Answer: (A) Quantity A is greater.
Quantity A: The hundredths place of 1,234,567.89 is 9.
Quantity B: The ten thousands place of 1,234,567.89 is 3.
9 > 3
Which of the following equals 48,290?
Select all that are equal.
[A] 482.9 × 102
[B] 48.29 × 10–3
[C] 0.4829 × 104
[D] 482,900 × 10–1
[E] 0.04829 × 106
325
Answer: [A], [D], and [E]
What is equal to 48,290?
[A] EQUAL. 482.9 × 102 = 48, 290
[B] NOT. 48.29 × 10–3 = 0.04829
[C] NOT. 0.4829 × 104 = 4,829
[D] EQUAL. 482,900 × 10–1 = 48, 290
[E] EQUAL. 0.04829 × 106 = 48, 290
What is 
, rounded to the nearest integer?
326
Answer: 9
or
0.00021 is what percent of 0.007?
327
Answer: 3%
Normally, for an “X is what percent of Y?” question, you would do 
in your calculator. While you could just plug these numbers into the calculator and divide, the risk of misplacing the decimal is high, so a secondary “human calculator” check is a good idea.
Shift the decimal in both top and bottom by 5 places before dividing, eliminating decimals altogether.
What is 0.000002 × 50,000?
328
Answer: 0.1
While you could just plug into the calculator and multiply, the risk of misplacing the decimal means that a secondary “human calculator” check is a good idea.
“Trade” decimal places. Move the decimal 5 to the right in the first number and 5 to the left in the second number, getting:
0.2 × 0.5
Half of 0.2 is 0.1.
| Quantity A | Quantity B |
The ten thousandths digit of  |
The hundred thousandths digit of  |
329
Answer: (A) Quantity A is greater.
is a “terminating” decimal, as there are only prime factors of 2 and 5 in the denominator: 20 = (2)(2)(5). Where does it terminate? 
= 0.55, so every digit past the hundredths place equals 0. Thus, Quantity B is 0.
is a repeating decimal, as there are prime factors other than 2 or 5 in the denominator: 99 = (3)(3)(11). Determine the repeating pattern: 
The hundred thousandths digit is the 5th digit after the decimal, so Quantity A is 3. (Either digit after the decimal would have been greater than 0, actually.)
Change the improper fraction 
to a mixed number.
330
Answer: 
is 
short of 
Thus, 
as a mixed number is 
Alternatively, divide 3 into 89. It goes in evenly 29 times. (29)(3) = 87, so 89 – 87 = 2 thirds are left over.
Simplify: 
331
Answer: 
Simplify: 
332
Answer: 
or 
| Quantity A | Quantity B |
 |
 |
333
Answer: (B) Quantity B is greater.
Both quantities are close to a benchmark value of 
Quantity A: 
is slightly less than 
, as the numerator is smaller.
Quantity B:
is slightly greater than 
, as the denominator is smaller.
Quantity B > 
> Quantity A.
Simplify: 
334
Answer: 
Don't split the denominator! You must add the numbers in the denominator before dividing. Parentheses can serve as a visual reminder, as Parentheses come first in the PEMDAS order of operations.
Given x ≠ 2, simplify: 
335
Answer: 
of the fish in a certain tank are clownfish. 
of the other fish were first placed in the tank yesterday. Placing those fish in the tank increased the total number of fish in the tank by what fraction?
336
Answer: 
(8)(5) = 40, which is a Smart Number for the total number of fish in the tank, as it is divisible by every denominator in the constraints. If there are 40 fish in the tank, 
of the fish are clownfish. There are 40 – 15 = 25 fish of other types. If 
of these 25 others were placed in the tank yesterday, 
fish were placed in the tank yesterday. There are 40 fish now, so before the 10 were added there were 40 – 10 = 30. The number of fish increased by 
of the original number of fish.
If x is even and y is odd, what is 4x + yx + y2?
(A) Definitely odd
(B) Definitely even
(C) Could be either odd or even
337
Answer: (A) Definitely odd
If all variables are integers, and the product abcd is odd, what is abc + bcd + acd?
(A) Definitely odd
(B) Definitely even
(C) Could be either odd or even
338
Answer: (A) Definitely odd
Since the product abcd is odd, none of the variables can be even. A single even integer would introduce a factor of 2 that would make the resulting product even. So, each variable is odd.
If m, n, and p are consecutive integers such that m < n < p, what is m2 + n2 + np?
(A) Definitely odd
(B) Definitely even
(C) Could be either odd or even
339
Answer: (A) Definitely odd
Since consecutive integers alternate O, E, O, E…etc., there are two cases:
Both wind up odd.
x is a prime number and y is a positive multiple of x. What is x – y?
(A) Definitely odd
(B) Definitely even
(C) Could be either odd or even
340
Answer: (C) Could be either odd or even
If x is a prime number, it could be even (2 only) or odd (all the other prime possibilities). If y is a positive multiple of x, it could be either an even number or an odd number times x. Thus, there are several cases:
If j and k are integers such that j2 is even and k3 is odd, what is 2k + 5jk + 3j?
(A) Definitely odd
(B) Definitely even
(C) Could be either odd or even
341
Answer: (B) Definitely even
Since j is an integer and j2 is even, j itself must be even.
Since k is an integer and k3 is odd, k itself must be odd.
If f is an integer, what is f2 + 15f + 50?
(A) Definitely odd
(B) Definitely even
(C) Could be either odd or even
342
Answer: (B) Definitely even
If f is even, then f2 + 15f + 50 = E2 + 15E + 50 = E + E + E = E
If f is odd, then f2 + 15f + 50 = Odd2 + 15(Odd) + 50 = O + O + E = E
Either outcome is even.
Alternatively, factor:
f2 + 15f + 50 = (f + 5)(f + 10) = (f + Odd)(f + Even)
Since one term adds Odd and the other adds Even, there will always be one even and one odd term in the product. Both (E)(O) and (O)(E) are even.
If x and y are integers, xy = z and x + y = 11, what is z2 – 2z?
(A) Definitely odd
(B) Definitely even
(C) Could be either odd or even
343
Answer: (B) Definitely even
Since x + y = 11 = odd, x and y must be odd and even, respectively, or vice versa. x and y cannot both be odd or both be even, as x + y would be even in those cases.
The product xy will always be even, as both (E)(O) and (O)(E) are even. Thus, z is even.
z2 – 2z = E2 – 2E = E – E = E
Is zero odd, even, or neither? Why?
344
Answer: Zero is even.
Why?
(1) An even number is a number that, when divided by 2, yields an integer result. Note that 
an integer.
(2) On the number line, odd and even integers alternate. That is, every even integer is between two odd integers. Every odd integer is between two even integers. Keeping this pattern, the even integer zero is between the odd integers –1 and +1. The odd integer 1 is between the even integers 0 and 2.
If x and y are positive integers and 3xy2 = 1,200, what is xy?
345
Answer: 20
From the factor tree for 1,200 and the given constraint, 3xy2 = 312452.
y cannot have a factor of 3, as y2 would then have two factors of 3, but there is only one factor of 3 in 1,200. Thus, 3x = 31, and x = 1. Consider the remaining terms and factors:
y2 = 2452
y2 = (22)252
y2 = (22 × 5)2
y = 22 × 5 = 20
So x = 1, y = 20, and xy = 20.
Round the following numbers to the nearest 0.01.
(A) 1.378
(B) 0.2547
(C) 9.355
346
Answer: (A) 1.38 (B) 0.25 (C) 9.36
When rounding to the nearest 0.01 (hundredth),
• Round UP when the thousandths digit is 5, 6, 7, 8, or 9.
• KEEP the hundredths digit as-is when the thousandths digit is 0, 1, 2, 3, or 4. (This is just like truncating every digit after the 0.01 place.)
(A) The thousandths digit is 8, so round the 7 in the hundredths place UP to 8. Result: 1.38
(B) The thousandths digit is 4, so truncate every digit after the hundredths place. Result: 0.25
(C) The thousandths digit is 5, so round the 5 in the hundredths place UP to 6. Result: 9.36
What is 0.472 as a fully reduced fraction?
347
Answer: 
Use the place value of the last digit in the decimal as the denominator: Here, 0.472 ends with the thousandths place, so put 1000 in the denominator. Put the decimal's digits in the numerator. Then simplify. 472 is divisible by 8, as is 1,000.
Simplify: 
348
Answer: 1
Simplify, and express the answer as a mixed fraction.
349
Answer: 
It is simplest to sum the whole number and the fractional parts of these terms separately:
Solve for x.
350
Answer: x = –2 or 8
, so either (3 – x) = 5 or (3 – x) = –5. Solve both.
