A water reservoir empties at a constant rate of 520,000 gallons per day, but is refilled by periodic rain storms. On average, if it rains every 5 days, how many gallons per rainstorm are required to replace the used water?
451
Answer: 2.6 million
On average, if it rains every 5 days, each storm must replace the water used over a period of 5 days.
520,000 gallons per day × 5 days = 2,600,000 gallons = 2.6 million gallons
At an ice cream shop, customers can order a single scoop of chocolate, vanilla, or strawberry ice cream in either a cone or a cup. If candy sprinkles are optional, how many different orders are possible?
452
Answer: 12
Choice of flavors: 3 options (chocolate, vanilla, or strawberry)
Choice of vessel: 2 options (cone or cup)
Choice of sprinkles: 2 options (yes or no)
Customers can order flavor AND vessel AND sprinkle option. “And” means multiply: 3 × 2 × 2 = 12 order combinations.
What is x in terms of y and z?
453
Answer: 
By Pythagorean Theorem, x2 + z2 = y2, so x2 = y2 – z2, and therefore .
What is d in terms of a?
454
Answer: 
By Pythagorean Theorem, a2 + (2a)2 = d2. Remember to square both the 2 and the a in the parentheses! So, a2 + 4a2 = d2. Simplifying, 5a2 = d2 and 
Therefore, .
| Quantity A | Quantity B |
 |
1 |
455
Answer: (A) Quantity A is greater.
Both 
and 
are greater than 1, so their product will also be greater than 1.
Simplify:
456
Answer: π
What is 45.57 as a fraction?
457
Answer: 
The decimal goes to two places: the first is the tenths 
place, the second is the hundredths 
place. Put the entire number (with no decimal) over 100.
In the decimal form of 
, how many terms are in the repeating cycle of digits to the right of the decimal point?
458
Answer: Two
The decimal form is 
. There are two terms (3 and 7) in the repeating cycle of digits after the decimal point.
Any two-digit number over 99 repeats in this simple pattern. Three-digit numbers overs 999 repeat the same way, and so on.
ab ≠ 0
|a + b| = |a| + |b|
| Quantity A | Quantity B |
| ab | 0 |
459
Answer: (A) Quantity A is greater.
Since ab ≠ 0, neither variable can be zero. Each is either positive or negative.
If |a + b| = |a| + |b|, then a and b must have the same sign. If they are both positive, the absolute value bars don't do anything, and can be removed. It is true that a + b = a + b. If a and b are both negative, the absolute value bars mean “multiply by –1 to turn the negative into a positive” and the equation becomes -(a + b) = -a + (-b), which is also true. However, if only one of the variables is negative, |a + b| < |a| + |b|. For example, |5 + (–3)| < |5| + |–3|, since 2 < 5 + 3.
Quantity A: ab = (pos)(pos) or (neg)(neg) = positive in either case
102 is what fraction of 17?
460
Answer: 6, or 
Write the question as an equation: 102 = what fraction? × 17, so what fraction? 
. It may seem weird that the “fraction” is an integer, but all numbers can be written as fractions. This fraction just happens to reduce to an integer.
102 is 6 times 17, so it is of 17.
4 is 5% of what number?
461
Answer: 80
Write the question as an equation: 
what number?
So, 
what number? The answer is 80. Alternatively, 5% or 
of some number is 4. Thus 4(20) = 80 = the number.
Annual number of visitors to a museum
| Year | Percent change from previous year |
| 2010 | –5% |
| 2011 | +10% |
| 2012 | –2% |
What was the cumulative percent change in annual visitors to the museum from 2009 to 2011?
462
Answer: 4.5%
Use a Smart Number of 1,000.
If 1,000 people visited the museum in 2009, then 5% fewer, or 950 people, visited in 2010. In 2011, 10% more than 950 people visited the museum: 950 + (0.10)(950) = 950 + 95 = 1,045 people visited in 2011.
The number of visitors increased from 1,000 in 2009 to 1,045. This is an increase of 45 per 1000, or 4.5 per 100, which is 4.5%.
Read carefully! We can ignore the change in 2012.
At what point on the coordinate plane will the lines 0 = x – 2 and y = 4 intersect?
463
Answer: (2, 4)
0 = x – 2 is equivalent to x = 2. This is a vertical line through +2 on the x-axis. Likewise, y = 4 is a horizontal line through +4 on the y-axis. These lines intersect at (2, 4).
Alternatively, if x = 2 and y = 4, then the point (x, y) must be (2, 4).
Given x > 0, simplify:
464
Answer: xa-b
Alternatively,
As long as x > 0, xa-b is positive, so it equals the square root of its square. (That's not true for negative numbers.)
Solve for x and y:
3x + 5y = 9
2x + 5y = 11
465
Answer: x = –2 and y = 3
Because the coefficient on the y term is the same in both equations, elimination (of the y terms) is easier than substitution. Subtract the whole 2nd equation from the 1st:
Plug x = –2 into either equation:
3(–2) + 5y = 9
–6 + 5y = 9
5y = 15
y = 3
What is the domain of the function
466
Answer: All real numbers except –2.
The domain of a function is the set of all permissible inputs (here, the x values). The fraction 
is defined for all real numbers except x = –2. Fractions with zero as the denominator are undefined.
If $1,000 is invested at a simple annual interest rate of 1.5%, what is the value of the investment after 4 years?
467
Answer: $1,060
The formula for simple interest is 
, where P is the principal amount invested, r is the annual interest rate, and t is the number of years.
For this question: 
In which year was the number of visitors to the museum least?
| Annual number of visitors to a museum | |
| Year | Percent change from previous year |
| 2010 | –5% |
| 2011 | +10% |
| 2012 | –2% |
(A) 2009
(B) 2010
(C) 2011
(D) 2012
468
Answer: (B) 2010
There were 5% fewer visitors to the museum in 2010 than 2009, so 2009 cannot be the year with the least visitors. Likewise, there were 10% more visitors to the museum in 2011 than 2010, so 2011 cannot be the year with the least visitors. Eliminate (A) and (C).
To compare 2010 and 2012, use a Smart Number of 1,000 for the number of visitors in 2010.
| Year | Percent change from previous year | Number of visitors (Smart Number) |
| 2010 | –5% | 1,000 |
| 2011 | +10% ( = +100) | 1,100 |
| 2012 | –2% ( = –22) | 1,078 |
There were more visitors in 2012, so 2010 was the year with the fewest visitors.
What is the quadratic formula, and what is its purpose ?
469
Answer: 
, where a, b, and c are real numbers and a ≠ 0 in a quadratic equation of the form ax2 + bx + c = 0. Plugging a, b, and c into the equation yields the solution(s) of the quadratic equation.
Often, we can find the solutions in other ways (e.g., by factoring), but occasionally the quadratic formula is the only option.
For what values of x is y a real number?
470
Answer: x ≥ –3
y is a real number when 
is a real number. The +x – 4 part of the expression is a real number for all real x values. But 
is only a real number when x + 3 ≥ 0, or x ≥ –3.
If $200 is invested at an annual interest rate of 4%, compounded quarterly, what will the value of the investment be at the end of 6 months?
471
Answer: $204.02
For compound interest, 
, where V is the value of the investment at the end of t years if P is the amount invested at an annual interest rate of r percent and compounds n times per year. 
Alternatively, think “4% annually but compounded quarterly is like increasing 1% every 3 months.” In 6 months, two such 1% increases occur:
200 + 2.00 = 202
202 + 2.02 = 204.02
If $1,000 is invested at a simple annual interest rate of 10%, what will the value of the investment be at the end of 2.5 years?
472
Answer: $1,250
For simple interest, 
, where V is the value of the investment at the end of t years if P is the amount invested at an annual interest rate of r percent. Here, 
6 people are traveling on an airplane, each with 2 bags. If the bags could be placed either in the cabin or the cargo area, with no restrictions, how many possibilities are there for the number of bags in the cabin?
473
Answer: 13
Watch out! This is not a combinations question about how many different arrangements are possible for the bags (for which it would matter whose bags are in the cabin and whose are in cargo, for example). This is simply: How many answers are possible for the question, “What is the number of bags in the cabin?”
Each person could carry 0, 1, or 2 bags into the cabin. The maximum number of bags in the cabin is (2 bags/person)(6 people) = 12 bags. The minimum number of bags in the cabin is (0 bags/person)(6 people) = 0 bags. The possibilities are 0, 1, 2,…, 11, 12 bags. There are 13 possibilities for the number of bags in the cabin.
If 
, what is g(–10)?
474
Answer: 2.3
If the area of triangle ABD is 54, what is the area of triangle CDE?
475
Answer: 24
Line segment CE, drawn inside triangle ABD and parallel to AB, creates a similar triangle CDE. The height of triangle CDE is 2/3 the height of triangle ABD. Because the triangles are similar, the base of triangle CDE is also 2/3 the base of triangle ABC.
Area of triangle ABC = bh = 54
Area of triangle CDE = 
Which of the following lines are perpendicular to y = –3x + 4? Indicate all such lines.
476
Answer: [C] and [D]
Lines are perpendicular when their slopes are negative reciprocals of one another. That is, the product of the two slopes equals –1.
The slope of y = –3x + 4 is –3. The negative reciprocal of –3 is 
.
[C] 
is already in slope-intercept form. Slope = .
[D] 3y = x + 4 is equivalent to 
, which has slope = .
| List X consists of 100 unique numbers. | |
| Quantity A | Quantity B |
| The range of list X. | The interquartile range of list X. |
477
Answer: (A) Quantity A is greater.
The range of any list is the difference between the greatest term and the least term. The interquartile range of any list is the difference between the third quartile Q3 and the first quartile Q1.
Because all of the numbers in list X are unique, i.e. no repeats in the list, Q3 < greatest term and Q1 > least term. Thus, Q3 – Q1 < greatest term – least term.
Which one of the following lists is summarized by the box-and-whisker plot above?
(A) 1, 4, 5, 6, 6, 7, 7, 8
(B) 4, 4, 4, 5, 5, 5, 6, 9
(C) 1, 2, 3, 5, 6, 7, 7, 9
478
Answer: (C)
A box-and whisker plot shows the full range of the data (i.e. the least and greatest term in the set) with the outer-most lines, or whiskers. Here, minimum = 1 and maximum = 9. That is enough to select (C).
Also, the boxes in the box-and-whisker plot indicate the 2nd and 3rd quartiles, so the line between the boxes is the median. In a set with 8 terms, each quartile contains 2 terms, and the median is the average of the 4th and 5th largest terms. For (C), that median is 5.5.
Incidentally, the left edge of the boxes is Q1, the average of the greatest term in the 1st quartile and the smallest term in the 2nd quartile, or 2.5 for (C). The right edge of the boxes is Q3, the average of the greatest term in the 3rd quartile and the smallest term in the 4th quartile, or 7 for (C).
| Quantity A | Quantity B |
 |
 |
479
Answer: (B) Quantity B is greater.
There are several ways to answer. Use the calculator:
…and so 
…is greater.
Or note that 
and 
, as a larger denominator means a smaller fraction.
Finally, we can cross-multiply upwards:
12 < 14, so is greater.
| Quantity A | Quantity B |
 |
 |
480
Answer: (A) Quantity A is greater.
There are several ways to answer. Use the calculator:
…and 
…, so 
is greater.
Or, note that each numerator is 2 less than its respective denominator. 
, since a larger denominator means a smaller fraction. So Quantity A is 1 – a smaller amount and Quantity B is 1 – a larger amount. Quantity A is greater.
If 
for all non-zero x, what is the value of k(20) – k(–9)?
481
Answer: 2
Simplify:
3(y – x) – 5(2x – y)
482
Answer: 8y – 13x
If y < x and y > -x, which of the points on the coordinate plane below are solutions?
Indicate all such points.
483
Answer: C and D
The solutions lie under the line y = x (because y < x) and above the line y = -x (because y > -x), in the shaded area below. Only points C and D are in this region.
| Quantity A | Quantity B |
 |
0 |
484
Answer: (B) Quantity B is greater.
Since 1.44 = 144 ÷ 100, and both 144 and 100 are perfect squares, 1.44 is a perfect square. The 
symbol just means take the square root twice.
Alternatively, notice that (1.1)2 = 1.21, so 
and Quantity A will be negative.
Events A and B are mutually exclusive. If P(A or B) = 0.67 and P(B) = 0.11, what is P(A)?
485
Answer: 0.56
Events A and B are mutually exclusive, which means either A can happen or B can happen, but if one event happens the other cannot happen. In such cases:
P(A or B) = P(A) + P(B)
0.67 = P(A) + 0.11
P(A) = 0.67 – 0.11
P(A) = 0.56
If 
, what is 
?
486
Answer: –1
What is the mode for the data above?
487
Answer: 6–10 years of teaching completed
The mode of any set of data is the measurement that appears most frequently in the set. In this set of data, “Years of Teaching Completed” is the measurement, and “Number of Teachers” indicates the number of times each measurement appears in the set (i.e. how many teachers have that level of teaching experience). The highest bar indicates the mode: 30 teachers have completed 6–10 years of teaching.
Which of the following could be the median of the data set above, in years of teaching completed?
(A) 3
(B) 9
(C) 11
(D) 17
(E) 24
488
Answer: (C) 11
There are 10 + 25 + 10 + 20 + 15 = 80 teachers represented in the chart. The median “years of teaching completed” is the average for the teachers with the 40th highest and 40th lowest experience levels. Both of these teachers are in the 11–15 years of teaching completed category, so the average could be 11 to 15, inclusive. Only (C) is in this range.
What is the average (arithmetic mean) number of diners per table? Round your answer to the nearest 0.1.
489
Answer: 2.3
Total number of tables = 2 + 4 + 7 + 3 + 6 = 22
Total number of diners = 2(0) + 4(1) + 7(2) + 3(3) + 6(4) = 0 + 4 + 14 + 9 + 24 = 51
Average number of diners per table = 
diners per table. Rounded to the nearest 0.1, the answer is 2.3.
Events A and B are independent. If P(A or B) = 0.6 and P(A) = 0.2, what is P(B)?
490
Answer: 0.5
For any two events A and B, P(A or B) = P(A) + P(B) – P(A and B). The last term is subtracted because the probability that both events happen is included in both P(A) and P(B).
If A and B are independent events, then P(A and B) = P(A)P(B). Substitute and solve:
P(A or B) = P(A) + P(A) – P(A)P(B)
0.6 = 0.2 + P(B) – 0.2 × P(B)
0.6 – 0.2 = P(B) – 0.2P(B)
0.4 = 0.8P(B)s
P(B) = 0.4/0.8 = 4/8 = 0.5
In the figure above, what is the area of the shaded region minus the area of the unshaded triangle?
491
Answer: 8
If not for the 2 by 4 rectangle created by drawing a dotted line as shown, the shaded and unshaded areas would be equal.
The shortened rectangle and the triangle have the same base and height, but the area of the rectangle is base × height, whereas the area of the triangle is 
base × height. Therefore the triangle is half the area of the rectangle and the shaded area must also equal half the area. Thus, the actual shaded area is bigger only by that 2 by 4 rectangular area: (2)(4) = 8.
Which library has the greatest percent of hardcover books?
492
Answer: Library B
This can be solved visually. Hardcover books are the light gray segment of the bar graph. In Library B, there are clearly more hardcover than paperback books. Thus, the percent of hardcover books in Library B is greater than 50%. The opposite is true in Libraries A and C, which have fewer than 50% hardcover books.
At which library is the percent of books that are paperback closest to 60% of the books?
493
Answer: Library C
The number of books is not known, as the axis is not labeled. However, the grid lines are evenly spaced, so count them.
At Library C, the number of paperbacks is just over 6 gridline units. The total number of books is around 10 gridline units. 6 of 10 is 60%.
At Library B, paperbacks account for less than half of the books, and at Library A, paperbacks account for more than 80% of the books.
Which library has the most paperback books?
494
Answer: Library A
This can be solved visually. Paperback books are the black segment of the bar graph. This segment is longest for Library A.
Which of the following functions describes the parabola in the figure shown?
(A) (x – 1)2 + 2
(B) (x + 1)2 – 2
(C) (x – 2)2 + 1
(D) (x + 2)2 – 1
(E) (x + 2)2 + 1
495
Answer: (C) (x – 2)2 + 1
In general, if the “base” of the U-shaped parabola is at x = k, there is a (x – k)2 term in the function. If the U-shape were upside down, the squared term would be negative: -(x – k)2. Also, if the base of the U-shaped parabola is at y = t, there is a + t term in the function. Here, the U is right side up, k = 2, and t = 1, so (x – k)2 + t is the function.
Alternatively, note some easy points the function passes through: (2, 1) and (0, 5). Plug into the choices, eliminating any that don't work for these points. Only (C) and (E) work for (0, 5). Of these, only (C) works for (2, 1).
| Quantity A | Quantity B |
| k | m |
496
Answer: (A) Quantity A is greater.
In general, for a parabola centered on the y-axis, there is a cx2 term in the function, where c is a constant. When the parabola is like a right-side-up U, c is positive. When the U is upside down, c is negative. The more narrow the U, the greater the absolute value of c.
Here, both parabolas are right-side-up U shapes, so k and m are both positive. The top/inner parabola is more narrow than the other, so k > m.
The function above is defined for all x ≥ –4. Which of the following equations describes the function shown?
(A) y = (x + 4)2
(B) y = x2 + 4
(C) 
(D) 
(E) 
497
Answer: (E)
It helps to know that equations with a variation on the y = x2 form are parabolas centered on a vertical axis (which look like a U or upside-down U), whereas equations with a variation on the 
form are parabolas (or half-parabolas, as shown) centered on a horizontal axis. This would rule out (A) and (B). Likewise, the functions in (A) and (B) are defined for all x, not just x ≥ –4. Equation (C) would be invalid for x < 0, and equation (D) would be invalid for x < 4, as both would attempt to take the square root of a negative number.
Alternatively, note some easy points the function passes through: (–4, 0) and (0, 2). Plug into the choices, eliminating any that don't work for these points. Only (A) and (E) work for (–4, 0). Only (E) works for (0, 2).
| Quantity A | Quantity B |
| ab | r2 |
498
Answer: (B) Quantity B is greater.
The graph of the equation (x – a)2 + (y – b)2 = r2 is a circle with radius r centered at the point (a, b). Even if you didn't know this, you might make an inference from the bigger circle with the given equation. It is centered at (2, 3) and has radius equal 5.
The center of the smaller circle is in quadrant IV, so a is positive and b is negative. Thus, ab is negative. Since r2 must be positive, Quantity B is greater.
What is the value of |–1| + |2| + |–3| – |4| – |–5|?
499
Answer: –3
Be careful about sign! First, break each number out of absolute value bars, remembering that each negative number will become positive.
If ab < 0 and bc > 0, what is the sign of ac3?
500
Answer: Negative
If ab < 0, then a and b have opposite signs. If bc > 0, then b and c have the same sign. This implies that a and c have opposite signs, so ac is negative.
ac3 = (ac)(c2). Notice that c2 must be positive, regardless of the sign of c.
Thus, ac3 = (neg)(pos) = negative.