Solve for x.
351
Answer: x = 4 or –15
First, isolate the absolute value sign.
So, either (2x + 11) = 19 or (2x + 11) = –19. Solve both.
Each is a valid solution of the original equation, as we can see by plugging back in.
Solve for x.
352
Answer: x = 0
So, 
means that –4 ≤ x2 + 4 ≤ 4. Subtract 4 from all three sides of the inequality to isolate the x term: –8 ≤ x2 ≤ 0. But the fact that x2 is greater than a negative number is nothing new. A squared term can never be less than 0. Thus, the only valid solution to the inequality is x = 0.
Solve for x.
353
Answer: x = –4 or 52
, so either 
Solve both.
Each is a valid solution of the original equation, as we can see by plugging back in.
If 
, what is 
?
354
Answer: 4
Deal with the expression in the parentheses first. Remember that fraction bars are like parentheses, too, so first solve 
and 
separately.
If Sn = n2 + n – 1, what is S6 – S3?
355
Answer: 30
S6 = 62 + 6 – 1 = 36 + 6 – 1 = 41
S3 = 32 + 3 – 1 = 9 + 3 – 1 = 11
S6 – S3 = 41 – 11 = 30
An = 3An–1 + 4 for all n ≥ 2. If A3 = 34, what is A1?
356
Answer: 2
The subscript on A denotes a term's order in the sequence. If An is “this term,” then An–1 is “the previous term.” Each term in this sequence is based on the previous term, so given any term you can plug to get the next term, or backsolve to get the previous term.
A3 = 34 = 3A2 + 4, so 30 = 3A2, and A2 = 10.
A2 = 10 = 3A1 + 4, so 6 = 3A1, and A1 = 2.
for all n ≥ 3.
If S1 = 1 and S2 = 2, what is S4?
357
Answer: 9
The subscript on S denotes a term's order in the sequence. If Sn is “this term,” then Sn–1 is “the previous term” and Sn–2 is “the term before that.” Plug in to solve for each term in succession.
Each number in a sequence is 9 more than the previous term. If the 11th number in the sequence is 49, what is the 3rd number in the sequence?
358
Answer: –23
There are 8 “jumps” from the 3rd term to the 11th term, so think of the 3rd term as 8 jumps of 9 “back” or “down” from the 11th term.
3rd term = 11th term – (8 jumps) + (9 per jump) = 49 – 72 = –23
In a sequence, An = 4n – 1
| Quantity A | Quantity B |
| The units digit of A17 | The units digit of A93 |
359
Answer: (C) The two quantities are equal.
The units digits of the powers of 4 exhibit a repeating pattern. The powers are 4, 16, 64, 256, 1,024, etc., so the units digits pattern is {4, 6, 4, 6, 4, etc.} Because 1 is subtracted from each term, the units digits for An will have the pattern {3, 5, 3, 5, 3, etc.} When the subscript is odd, the units digit of An is 3. When the subscript is even, the units digit of An is 5.
In both quantities, the subscript is odd, so the units digits are equal.
If 
, what is f(20)?
360
Answer: 0.62 or 
If g(x) = 4x and f(x) = x3 + 5, what is g(f(5))?
361
Answer: 520
Start from the inside and work outward: First input 5 to the f function, then input the result to the g function.
f(5) = (5)3 + 5 = 125 + 5 = 130
g(f(5)) = g(130) = 4(130) = 520
Solve for k.
362
Answer: k = –11 or 19
First, isolate the absolute value sign.
So, either (4 – k) = 15 or (4 – k) = –15. Solve both.
Each is a valid solution of the original equation, as we can see by plugging back in.
If point C has integer coordinates such that 0 ≤ x ≤ 5 and 0 ≤ y ≤ 5, how many different triangles ABC can be constructed?
363
Answer: 33
Point C can be anywhere in the grid of integer coordinate points given by the inequalities. There are 6 possible x coordinates and 6 possible y coordinates (0, 1, 2, 3, 4, or 5). Thus, there are (6)(6) = 36 possible integer coordinate points in the region. However, to create a triangle ABC, Point C cannot coincide with either Point A or Point B, nor lie on the line joining them. Points (1,1) and (3,2) are invalid.
Since the slope of the line joining the points is 
, the only other integer coordinate point on the line (and on the grid) is (5,3), as we can see by drawing the line. So the possible locations for Point C are 36 – 3 = 33 in number.
An airline has sold 20% more tickets for a flight than there are seats available. If 10% of the ticket holders do not arrive at the airport for the flight, what fraction of the ticket holders who do arrive at the airport will definitely not get a seat on the flight?
364
Answer: 2/27
Because no actual numbers are given, just percents, a Smart Number of 100 is useful. Suppose there are 100 seats on the flight, and the airline has sold 20% more than 100, or 120 tickets. Of those 120 ticket holders, 10% or 12 ticket holders do not show up for the flight. Thus, 120 – 12 = 108 people arrive at the airport, but 108 – 100 = 8 of them will definitely not get a seat on the flight.
(It is possible that even more ticket holders will not get a seat—maybe some of them don't make it through security—but we know that 8 ticket holders will definitely not get seats.)
| Quantity A | Quantity B |
| The area of a rectangle with length 3 and width 5. | The area of a parallelogram that has a pair of parallel sides 5 in length with a distance of 3 between them. |
365
Answer: (C) The two quantities are equal.
Quantity A: Area of a rectangle = wL = (5)(3) = 15.
Quantity B: Area of a parallelogram = bh, where b is the length of a pair of parallel “base” sides and h is the perpendicular distance between these bases. Here, the area of the parallelogram = bh = (5)(3) = 15.
| Quantity A | Quantity B |
| The area of rectangle A above. | The area of trapezoid B above. |
366
Answer: (B) Quantity B is greater.
Quantity A: Area of rectangle A = wL = (4)(14) = 56
Quantity B: Area of trapezoid 
What is the perimeter of the polygon below?
367
Answer: 24
The perimeter of a polygon is the sum of all side lengths. Starting at the top and going clockwise, Perimeter = 3 + 5 + 4 + 3 + 4 + 5 = 24.
What is the surface area of the wedge-shaped solid below? (The base and back rectangles are at a right angle to one another.)
368
Answer: 150
First, note that the front and back are 5–12–3 right triangles, so the sloped top edges of the wedge are 13 long.
Bottom Rectangle: Area = wL = (3)(12) = 36
Front Triangle: 
Back Triangle: Area = 
Vertical Rectangle: Area = wL = (5)(3) = 15
Sloped Top Rectangle: Area = wL = (13)(3) = 39
Total Surface Area = 36 + 30 + 30 + 15 + 39 = 150
The figure above shows a symmetrical solid, similar to an extruded “+” sign. All surfaces of the solid meet at right angles, and each edge on the front “+” face is 1 long. What is the surface area of the solid?
369
Answer: 106
The surface area consists of two “+” shaped faces (front and back), and twelve 1 by 8 rectangles (sides).
The front and back faces can be seen as five 1 by 1 squares:
Front and Back: (5)(1)(1) each, so 5 + 5 = 10 total.
Sides: (12 side rectangles)(1)(8) = 96
The total surface area is 10 + 96 = 106.
| Quantity A | Quantity B |
| The sum of the interior angles of a rhombus. | The sum of the interior angles of a square. |
370
Answer: (C) The two quantities are equal.
The side lengths are irrelevant, as are the different individual angles. The sum of the interior angles of any polygon only depends on number of sides.
Sum of the interior angles = 180°(n – 2), where n = the number of sides in the polygon.
Both a rhombus and a square have 4 sides, so they have the same interior angle sum: 180°(4 – 2) = 360°.
| Quantity A | Quantity B |
| The area of the rhombus | 102 |
371
Answer: (B) Quantity B is greater.
A rhombus has two sets of parallel sides, and four sides of equal length. Each side of this rhombus is 40/4 = 10.
The area of a rhombus in general is bh, where b = 10 and h is < 10 if the rhombus “leans” at all, i.e. bottom left and top right angles less than 90°. In these cases, bh = (10)(less than 10) = less than 100.
The area of the rhombus is maximized if all the angles are equal to 90° (i.e. if the shape is a square, which is a subset of rhombus). In that case, the area = s2 = (10)(10) = 100.
The area of the rhombus cannot equal or exceed 102.
| Quantity A | Quantity B |
| The radius of a circle with area 8π | The circumference of a circle with radius  |
372
Answer: (B) Quantity B is greater.
Quantity A: Area = πr2 = 8π, so r2 = 8. 8 is less than 9, so the radius is less than 
which is less than 3.
Quantity B: Circumference 
If the 118° angle is a central angle of the circle above, what is y?
373
Answer: 59
If the 118° angle is a central angle of the circle above, the y° angle is an inscribed angle bounded by the same arc on the circle. For a given arc, an inscribed angle is always 
the central angle. 
A circular pizza is cut into 12 identical slices, as shown. What is x?
374
Answer: 30
The point of each slice is a central angle of the circle. The ratio of the central angle to 360° is equal to the proportion of the circle enclosed.
Each slice 
of the circle 
Thus, 
x is 
of the circle, so:
A circular pizza is cut into 12 identical slices, as shown. If each slice is 3π square inches in area, what is the diameter of the whole pizza?
375
Answer: 12 inches
Each of the 12 pieces is 3π square inches in area, so the area of the whole pizza is (12)(3π) = 36π square inches.
Area of the circle = πr2 = 36π. Thus r = 6 inches.
The diameter is twice the radius, or (2)(6) = 12 inches.
What is the area of the shaded sector of the circle above?
376
Answer: 14π
Because all three lines are the same length, meet at the same point, and extend to the circle edge, all must be radii of the circle (and the point at which they meet must be the center of the circle). If the radius is 6, the area of the whole circle is πr2 = 36π.
The shaded sector has a central angle of 360° – 70° – 150° = 140°.
Thus, Sector 
.
In the figure above, both shapes are rectangles, and all lines are either horizontal or vertical. What is the area of the shaded region?
377
Answer: 144
The area of the larger rectangle is wL = (16)(14) = 224
The area of the smaller rectangle is wL = (16 – 2 – 4)(14 – 3 – 3) = (10)(8) = 80.
The area of the shaded region is the difference between the larger and smaller rectangle areas: 224 – 80 = 144.
| Quantity A | Quantity B |
| x + y | z |
378
Answer: (C) The two quantities are equal.
Label the bottom right angle of the triangle b°. From the sum of the angles in the triangle, it is known that x + y + b = 180, or b = 180 – x – y. From the sum of the angles that form a line, it is known that b + z = 180, or b = 180 – z. Setting these two expressions for b equal to one another:
180 – x – y = 180 – z
–x – y = –z
x + y = z
Or, note that z is an exterior angle of the triangle, which is always equal to the sum of the two opposite interior angles of the triangle, x + y.
In the figure above, lines m and n are parallel. What is x?
379
Answer: 84
Because lines m and n are parallel, the transversal crosses both at the same angle. Thus, the angle directly below x° is also 96°.
Angle x and the newly labeled angle form a straight line, so their sum is 180: x + 96 = 180, so x = 180 – 96 = 84.
| Quantity A | Quantity B |
 |
d |
380
Answer: (D) The relationship cannot be determined from the information given.
Angles a, b, c, and d form a complete circle, or a total of 360°. Since b and c form a line, b + c = 180, so the b > c constraint indicates that b > 90 while c < 90. Similarly, d > 90 while a < 90.
Quantity A: 
The two quantities would be equal if 
, or 
, or 
.
The two quantities are equal if d = 120.
If 90 < d < 120, Quantity A is greater.
If 120 < d < 180, Quantity B is greater.
What other information can be inferred from the figure above?
381
Answer: The two almost-horizontal lines are parallel; all of the other angle measures can be determined.
The vertical angles (opposite the labeled 63° angles where two lines intersect), are also 63°. The other angles form a line with one of these 63° angles (they are supplementary), and can be labeled 180 – 63 = 117°.
Finally, because the transversal crosses both of the two almost-horizontal lines at the same angle, these two lines are parallel.
What is a + c + e?
382
Answer: 180
Notice that b° = e°, because these two angles are vertical angles, i.e. opposite each other where two lines intersect.
Thus, a + c + e = a + c + b. This is useful because the angles a, b, and c form a line, or are supplementary. The sum of such angles is 180.
If the equation of a line is y = –3x + 6, what is the x-intercept of the line?
383
Answer: x-intercept = 2
Watch out! This question is not about the y-intercept of the line, which is the constant +6 from the given line equation, given in standard slope-intercept form.
This question is about the x-intercept, which is where the line crosses the x-axis. At that point, the y coordinate is 0, so this question is really “What is x when y = 0?”
y = –3x + 6
0 = –3x + 6
–6 = –3x
x = 2
What is the distance between (1, 1) and (7, –2) on the coordinate plane?
384
Answer: 
Draw the points on the coordinate plane, along with the right triangle for which the point to point distance is the hypotenuse.
The horizontal leg of the triangle is |7 –1| = 6.
The vertical leg of the triangle is |–2 – 1| = 3.
By Pythagorean Theorem, the distance (hypotenuse) is 
Fiona bicycled 20 miles on a trail at 10 miles per hour, then walked on the same trail back to her starting place at 4 miles per hour. What was her average speed for the round trip, in miles per hour?
385
Answer: 
Average speed is total distance over total time.
Time spent bicycling: 
Time spent walking: 
Her total trip was 20 + 20 = 40 miles, and it took 2 + 5 = 7 hours.
Average rate for the round trip:
miles per hour 
miles per hour
In a fruit salad, there are 7 grapes for every 2 oranges, and 3 oranges for every pineapple. If 4 pineapples are in the salad, how many grapes are in the salad?
386
Answer: 42
You can solve in steps:
There are 3 oranges for every pineapple, and 4 pineapples in the salad. There are (3)(4) = 12 oranges in the salad.
There are 7 grapes for every 2 oranges, so there are 7/2 grapes per orange. There are (7/2)(12) = 42 grapes in the salad.
Or, you can set up ratios and create a common term:
Since there are 4 pineapples, double this ratio to 42 : 12 : 4, and see that there are 42 grapes.
A soup recipe requires 2 cups of chopped tomatoes, 1 cup of white beans, 4 cups of chopped spinach, and 4 cups of broth. If there are 6 cups of white beans in a large batch of soup, how many cups of ingredients total are in the batch?
387
Answer: 66
The ratio of (tomatoes : beans : spinach: broth) is given as (2x : 1x : 4x : 4x). The total number of cups of ingredients is 2x + 1x + 4x + 4x = 11x.
In the large batch of soup, there are 6 cups of beans, so x = 6. Thus, there are 11x = 11(6) total cups of ingredients in the batch.
At a photography exhibit, the ratio of color photographs to black and white photographs displayed is 1 to 8. If there are 72 photographs of these two types in the exhibit, how many color photographs are on display?
388
Answer: 8 (not 9!)
The ratio of (color : b&w) is given as (1 : 8), or using the unknown multiplier, (1x : 8x). There are 1x + 8x = 9x = 72 photographs on display, so x = 8.
Thus, there are 1x = 1(8) color photographs on display.
Be careful! The 1 : 8 ratio does not mean that 1/8 of the photographs are in color. It means that 1/9 of the total are color photographs and 8/9 of the total are black and white.
In a certain school building, 
of the classrooms are carpeted, and the remainder have tile floors. If 12 classrooms have tile floors, how many classrooms are in the school building?
389
Answer: 21
If 
of the classrooms are carpeted, then 
of the classrooms have tile floors. If x is the number of classrooms in the building:
The ratio of boys to girls in a science class is 7 to 5. If there are 21 boys in the class, how many girls are in the class?
390
Answer: 15
Set up a labeled proportion: 
Cross multiply: 7x = 105
x = 15
The ratio of adults to children at a picnic is 4 to 3. If there are 49 people at the picnic, how many of the people are adults?
391
Answer: 28
Use the unknown multiplier, where x is an integer: 
49 = Adults + Children = Total
49 = 4x + 3x
49 = 7x
x = 7
There are 4x = 4(7) = 28 adults at the picnic.
Harvey checked out books from the library, and the ratio of hardcover books to paperback books was 5 to 4. If the library permits patrons to check out no more than 30 books at a time, what is the maximum possible number of paperback books Harvey checked out?
392
Answer: 12
Use the unknown multiplier, where x is an integer: 
30 ≥ Hardcover + Paperback = Total
30 ≥ 5x + 4x
30 ≥ 9x
Because x must be an integer, the greatest possible value for x is 3. Harvey could have checked out no more than 4x = 4(3) = 12 paperback books.
The members of a running club ran a total of 1,226 miles. If no member of the club ran less than 26 miles, what is the maximum number of running club members?
393
Answer: 47
To maximize the number of runners, we need to minimize the number of miles each member ran.
Number of Runners = 
= at most 47.154. Since there obviously cannot be a fractional number of running club members, the maximum is 47.
A presentation will be given on Thursday then repeated on Friday and Saturday, in a room that can accommodate 40 audience members. Of the 110 people to be scheduled to attend the presentation, 12 prefer to attend on Thursday, 18 prefer to attend on Friday, and 80 prefer to attend the presentation on Saturday. What is the minimum number of people who will not get to attend the presentation on their preferred day?
394
Answer: 40
Only 12 and 18 people, respectively, prefer to attend the presentation on Thursday and Friday. The room can accommodate these people, so they will get to attend on their preferred day.
The limiting factor is Saturday: 80 people – 40 seats = 40 people who will have to attend on either Thursday or Friday instead of their preferred day.
At a resort, 1/2 of the rooms have a lake view and 2/3 of the rooms have a hot tub. If 1/3 of the rooms with a lake view have a hot tub, then rooms that have neither a lake view nor a hot tub are what fraction of the total number of rooms at the resort?
395
Answer: 0
Use a Smart Number and a Venn Diagram. 18 is a Smart Number to use for the total number of rooms, as it is a multiple of all the fraction denominators.
“1/2 of the rooms have a lake view”: (1/2)(18) = 9
“2/3 of the rooms have a hot tub”: (2/3)(18) = 12
“1/3 of the rooms with a lake view have a hot tub”: (1/3)(9) = 3
Place the 3 in the middle first, then work out the outer numbers.
There are 9 + 3 + 6 = 18 rooms that have a hot tub, a view, or both. Thus, there are 18 – 18 = 0 rooms that have neither.
If it does not rain today, there is a 75% chance that the high temperature will be greater than 90°F. If it does rain today, there is a 50% chance that the high temperature will be greater than 90°F. If there is a 40% chance of rain, what is the probability that the high temperature will be greater than 90°F?
396
Answer: 65%
Chance of rain and high temperature above 90°F: (0.40)(0.50) = 0.2
Chance of NO rain and high temperature above 90°F: (1 – 0.40)(0.75) = (0.6)(0.75) = 0.45
Chance of high temperature above 90°F: 0.2 + 0.45 = 0.65 = 65%
The probability of flipping at least one head in a series of flips of a fair coin is calculated to be 
How many times will the coin be flipped?
397
Answer: 4
“Flipping at least one head” will NOT happen if all the coin flips produce tails. So if the probability of at least one head is 15/16, the probability of all tails is 1 – 15/16 = 1/16. The probability of flipping tails…
…once is 
…twice in a row is 
…three times in a row is 
…four times in a row is 
The coin would need to be flipped 4 times for the probability of flipping at least one head to be 15/16.
534,360 people travel through a train station per year. What is the average number of travelers through the train station per hour? (Assume the station operates continuously 365 days a year; be ready to use a calculator.)
398
Answer: 61
Use the calculator.
This is a tough one for mental math, but in a pinch you could estimate.
| Quantity A | Quantity B |
| f + b + d | a + e |
399
Answer: (A) Quantity A is greater.
Notice that b° = e°, because these two angles are vertical angles, i.e. opposite each other where two lines intersect.
Thus, f + b + d = f + e + d. This is useful because the angles f, e, and d form a line, or are supplementary. The sum of such angles is 180.
The same could be said for angles a, f, and e. Since a + f + e = 180 and the figure indicates that f > 0, then a + e < 180.
Quantity A: f + b + d = 180
Quantity B: a + e < 180
| Quantity A | Quantity B |
 |
 |
400
Answer: (C) The two quantities are equal.
Quantity A: 
Quantity B: 