1,296 ACT Practice Questions, 3rd Edition (2013)
Math Practice Section 1
60 Minutes—60 Questions
DIRECTIONS: Solve each problem, choose the correct answer, and then darken the corresponding oval on your answer document.
Do not linger over problems that take too much time. Solve as many as you can; then return to the others in the time you have left for this test.
You are permitted to use a calculator on this test. You may use your calculator for any problems you choose, but some of the problems may best be done without using a calculator.
Note: Unless otherwise stated, all of the following should be assumed:
1. Illustrative figures are NOT necessarily drawn to scale.
2. Geometric figures lie in a plane.
3. The word line indicates a straight line.
4. The word average indicates arithmetic mean.
1. A magician performing at children’s birthday parties charges $120.00 total for a one-hour performance with ten goodie bags for children at the party. She will provide additional goodie bags for $2.50 each. For an additional $25.00, she will also present a 15-minute laser light show. The magician is always paid on the day of the show, receives no tips or other additional payments, and never varies the length of the show. If the magician performs exactly four shows one weekend, presents the light show at three of those performances, and collects $635.00 total, how many additional goodie bags did she provide?
2. In an elite marathon runner’s training, total mileage consists of miles run at or faster than marathon pace and miles run slower than marathon pace. The table below shows miles run at or below marathon pace and total mileage for an elite marathon runner for each of 3 consecutive years.
In 2004, how many miles of the runner’s total mileage were miles run slower than marathon pace?
3. A 24-hour day is how many times as long as 60 seconds?
4. A student reads a pages per day for d days and then reads b pages per day for 2d days. In terms of a, b, and d, how many pages did the student read?
F. ad + 2b
G. ad + 2bd
H. 2ad + 2bd
5. A trapezoidal driveway has the dimensions, in yards, given in the figure below. What is the area, in square yards, of the driveway?
6. The graph below shows the number of people visiting a museum during the first 5 months of the year. How many people need to visit the museum during June for the mean of the first 6 months to equal the mean of the first 5 months?
7. A graduation cap is tossed upward. It is f feet above the ground s seconds after it has been thrown. The relationship between f and s is given by the equation f = 60s − 17s2 where 0 ≤ s ≤ 3.5. How many feet above the ground is the cap 3 seconds after it is thrown?
8. The highest and lowest test scores of five students in Mr. Canyon’s science class are listed below. Which student had the greatest range of scores?
9. Nita, Craig, and Chris catch a total of 300 fish on their trip. If Chris catches 45% of the fish and Craig catches 25 fish, what fraction of the 300 fish does Nita catch?
10. Given that f(x) = 4x2 and g(x) = 3 −, what is the value of f(g(4)) ?
11. In the grid shown below, each small square has a side length of 1 unit. In the shaded region, each vertex lies on a vertex of a small square. What is the area, in square units, of the shaded region?
12. A ramp rises 6 inches for each 24 inches of horizontal run. This ramp rises how many inches for 62 inches of horizontal run?
13. What is the value of yx + (2x − 2y) when x = 2 and y = −3?
14. In the figure below, the circle with center O is tangent to , , , and . The measure of angle BDE is 75° and the measure of DEA is 105°.
The lines in which of the following pairs of lines are necessarily parallel?
F. I only
G. II only
H. III only
J. I and II only
K. I, II, and III
15. The day a clothing store puts out a batch of brand-name T-shirts it sells 95 shirts at $4.10 per shirt. However, each day the shirts are on the rack, the store reduces the price of the shirts by $0.02 and consequently sells 1 additional shirt with each price reduction. If x represents the number of $0.02 price reductions, which of the following expressions represents the amount of money, in dollars, that the store will take in daily in sales of these brand name T-shirts?
A. (4.10 + 2x)(95 + x)
B. (4.10 − 2x)(95 + x)
C. (4.10 + 0.02x)(95 + x)
D. (4.10 − 0.02x)(95 + x)
E. (4.10−− 0.02x)(95 + 0.02x)
16. The expression x2 − 7x + 12 is equivalent to:
F. (x − 12)(x + 1)
G. (x − 4)(x − 3)
H. (x − 4)(x + 3)
J. (x − 6)(x − 2)
K. (x − 6)(x + 2)
17. When x = 5 and y = 2, the expression
18. The minutes and seconds on a 60-minute digital timer are represented by 3 or 4 digits. What is the largest product that can be obtained by multiplying the digits in one of these representations?
(Note: When the timer displays 16:15, the product of the digits is (1)(6)(1)(5) = 30.)
19. The difference of 2 integers is 6. The sum of the same 2 integers is 42. What is the lesser of the 2 integers?
20. The area of the square in the figure below is 324 square centimeters, and the two small isosceles right triangles are congruent. What is the combined area, in square centimeters, of the 2 small triangles?
21. Jasper wants to measure the altitude of his kite. He ties the kite string to a spike driven into the ground and measures the angle between the string and the ground. Then, he creates 2 similar triangles by adjusting the distance between an 8 foot pole and the spike until the angle created by a piece of string tied to the top of the pole and to the spike with the ground is the same as the angle he measured previously. The length of the string to the kite is 85 feet and the length of the string to the pole is 17 feet. Which of the following is closest to the height, in feet, that the kite is above the ground?
22. For what value of x, if any, is the equation (x − 1)2 = (x − 7)2 true?
K. There is no value of x for which the equation is true.
23. ABC, shown below in the standard (x,y) coordinate plane, is equilateral with vertex A at (0,w) and vertex B on the x-axis as shown. What are the coordinates of vertex C?
E. (2w, w
24. The diagonal of a square quilt is 4 feet long. What is the area of the quilt in square feet?
25. A painter needs to reach the top of a tall sign in the middle of a flat and level field. He uses a ladder of length x to reach a point on the sign 15 feet above the ground. The angle formed where the ladder meets the ground is noted in the figure below as θ. Which of the following relationships must be true?
26. The equation is true for what real value of a?
27. In rectangle ABCD below, is 16 inches long and is 12 inches long. Points E, F, and G are the midpoints of , and respectively. What is the perimeter, in inches, of pentagon CDEFG?
Use the following information to answer questions 28−30.
The table below details a recent census report about the commuting habits of U.S. workers age 16 or over for the years 2004, 2005, and 2006.
28. To the nearest percent, what percent of all U.S. workers age 16 or over in 2004 was female?
29. The circle graph (pie chart) below represents the 2006 means of transportation for U.S. workers age 16 or over for the 4 transportation types listed. To the nearest degree, what is the measure of the central angle for the “Public” sector?
30. Expressed in millions of people, what was the average growth per year for female U.S. workers age 16 or over from 2004 to 2006, rounded to the nearest 0.1 million?
31. Two hoses attached to separate water sources are available to fill a cylindrical swimming pool. If both hoses are used, the time it will take to fill the pool can be represented by the following equation: , where T1 and T2 represent the time needed for hoses 1 and 2, respectively, to fill the pool on their own, and Tc represents the time needed for hoses 1 and 2 to fill the pool working together. If hose 1 alone can fill the pool in exactly 20 minutes and hose 2 alone can fill the pool in exactly 60 minutes, how many minutes will it take to fill the pool if both hoses work simultaneously?
32. A 5-sided die, which has sides 2, 3, 4, 5, and 6, is thrown. What is the probability that the die will NOT land on a prime-numbered face?
33. For f(x,y) = 7x + 9y, what is the value of f(x,y) when and x = 3?
34. What is the length, in coordinate units, of a diagonal of a square in the standard (x,y) coordinate plane with vertices at points (0,0), (4,0) and (4,4)?
35. What is the value of a if log4a = 3?
36. A certain 18-quart stockpot is filled completely with water and exposed to a heat source so that the water boils away at a constant rate. The water remaining in the stockpot can be approximated by the following equation: y = 18 − 0.2x, where x is the number of minutes that the pot has been heated for 0 ≤ x ≤ 90, and y is the number of quarts remaining in the pot. According to this equation, which of the following statements is true about this stockpot?
F. After 0.2 minutes, 1 quart of water has boiled away.
G. After 1 minute, 0.2 quarts of water have boiled away.
H. After 18 minutes, 0.2 quarts of water have boiled away.
J. After 18 minutes, 1 quart of water has boiled away.
K. After 36 minutes, 18 quarts of water have boiled away.
37. The volume of a right circular cone with the bottom removed to create a flat base can be calculated from the following equation: V = πh(R2 + r2 + Rr), where h represents the height of the shape and R and r represent its radii as shown in the figure below:
This formula can be used to determine the capacity of a large coffee mug. Approximately how many cubic inches of liquid can the cup shown below hold if it is filled to the brim and its handle holds no liquid?
38. Which of the following is the set of real solutions for the equation 9x + 12 = 3(3x + 4)?
F. The set of all real numbers
K. The empty set
39. The expression equals:
40. A thin slice is cut from a bagel, creating the cross-section represented below. The diameter of the bagel is 144 mm and the width from the inner edge of the bagel to the outer edge is uniformly 56 mm. Which of the following is closest to the area, in square millimeters, of the shaded empty space inside the cross-section of the bagel?
41. For all nonzero real numbers a, b, and c, what is the value of a0 + b0 + c0?
B. a + b + c
42. In the figure below, ABCD is a rectangle, AB = AE, and E, F, G, and H lie on AD. Of the angles BEA, BFA, BGA, BHA, and BDA, which one has the greatest tangent?
Use the following information to answer questions 43−45.
The graph of y = f(x) is shown in the standard (x,y) coordinate plane below with points V, W, X, Y, and Z labeled.
43. The y-intercept of the graph of y = f(x) is located at which of the following points?
44. The function y = f(x) can be classified as one of which of the following types of functions?
H. Absolute value
45. If y = f(x) is to be reflected across the line y = x, which of the following graphs represents the result?
46. If a is a factor of 32 and b is a factor of 45, the product of a and b could NOT be which of the following?
47. For each positive integer k, let ko be the sum of all positive odd integers less than k. For example, 6o = 5 + 3 + 1 = 9 and 7o = 5 + 3 + 1 = 9. What is the value of 17o × 4o?
48. If (a,−3) is on the graph of the equation x − 4y = 14 in the standard (x,y) coordinate plane, then a =?
49. For all t > 0, f(t) = . Which of the following is true about f(t) ?
A. It increases in proportion to t.
B. It increases in proportion to t2.
C. It decreases in proportion to t.
D. It decreases in proportion to t2.
E. It remains constant.
50. In the figure below, X is on . If the angle measures are as shown, what is the degree measure of YXZ?
51. Points (2, − 2) and (3,10) lie on the same line in the standard (x,y) coordinate plane. What is the slope of this line?
52. What is the degree measure of an angle that measures radians?
53. Which of the following gives the equation for the circle in the standard (x,y) coordinate plane with a center at (4,−8) and a circumference of 10π square coordinate units?
A. (x − 4)2 + (y + 8)2 = 25
B. (x − 4)2 + (y + 8)2 = 100
C. (x + 8)2 + (y − 4)2 = 25
D. (x + 8)2 − (y − 4)2 = 100
E. (x + 8)2 + (y − 4)2 = 100
54. For some x and y that satisfy the equation xy = −x2, which of the following is FALSE?
H. x2 + y2 = −2xy
J. x2 = y2
K. x3 − y3 = 0
55. Rectangle ABCD lies in the standard (x,y) coordinate plane with corners at A(4,2), B(6,−1), C(1,−4), and D(−1,−1), and is represented by the 2 × 4 matrix . ABCD is then translated, with the corners of the translated rectangle represented by the matrix . What is the value of n?
56. Whenever a > 0, which of the following real number line graphs represents the solutions for x to the inequality |x − a|≤?
57. Three different functions are defined in the table below.
The diagram below uses three functions. The only values for p, q, r, s, and t are 1 and 0. Which of the following inputs (p, q, r, s, t) will produce the output 0?
58. Whenever x and y are both integers, what is (6.0 × 10x)(5.0y × 10y)expressed in scientific notation?
F. 30.0 × 100xy
G. 30.0 × 100xyy
H. 30.0 × 10xy
J. 3.0 × 10x + y + 1
K. 3.0 × 10xy
59. The points P, Q, R, and S lie in that order on a straight line. The midpoint of is R and the midpoint of is Q. The length of is x feet and the length of is 4x − 16 feet. What is the length, in feet, of ?
60. The circle below has an area of 64π cm2. A central angle with measure 24° intercepts minor arc . What is the length of minor arc , in centimeters?
Answers and Explanations
MATH PRACTICE 1 ANSWERS
MATH PRACTICE 1 EXPLANATIONS
1. B The magician receives 4($120) + 3($25) = $555 in payment for performances and light shows, leaving $635 − $555 = $80 in payment for additional goodie bags. Since each costs $2.50, she provides = 32 bags. Choice (A) calculates all 4.75 hours worked at the $120 rate. Choice (C) divides the $120 and $2.50 from the problem without answering the question, and choice (D) miscalculates based on three performances instead of four. Choice (E) assumes the entire $635 is for additional goodie bag payment.
2. J Subtract the number of miles run at or faster than marathon pace during 2004 from the number of total miles run in 2004. 4,982 − 1,150 = 3,832.
3. E 60 seconds is 1 minute. There are 60 minutes in 1 hour, so there are 60 × 24 = 1,440 minutes in 24 hours.
4. G The total pages read by the student equals the number of pages read per day times the number of days at each rate. (a × d) + (b × 2d) = ad + 2bd. Choice (F) omits the d in the second term and choice (H) applies the 2 to both terms. Choice (J) drops the addition sign between the terms. Choice (K) multiplies the two terms instead of adding them.
5. C To find the area of the trapezoid, split it into a rectangle with dimensions of 12 and 6 and a triangle with a height of 12 and a base of 5, then add the areas of the two smaller shapes: (6)(12) + (5)(12) = 72 + 30 = 102. Choice (B) gives only the area of the rectangular portion and choice (A) gives the perimeter of the whole figure. Choice (E) doubles the area. Choice (D) finds the hypotenuse of the triangle and multiplies it by the height of the triangle.
6. H Estimating should help you eliminate choices (F), (J), and (K). Find the mean (or average) of the first five months by finding the total number of visitors and dividing by 5: (200 + 300 + 200 + 350 + 200) ÷ 5 = 1,250 ÷ 5 = 250. Now try out the answer choices and see which one gives a six-month average equal to 250. Choice (F) doesn’t work because you need to divide the total by 6 this time. Choice (G) is the mode. Choice (H) gives you 1,500 ÷ 6 = 250. Choice (J) is double what it should be. Choice (K) is the total of the first five months.
7. A Substitute 3 seconds into the equation f = 60(3) − 17 (3)2 and solve. f = 180 − 153 = 27 feet.
8. K To find the test-score range for each student, subtract the highest test score from the lowest. Emily’s score range is 89 − 70 = 19 points. The range for Cleo’s and David’s scores are 18 points each, and Alicia’s score range is 17 points. Brandon’s score range is 12 points.
9. D Subtract Craig’s and Chris’s fish to determine how many fish Nita catches. For Chris, 45% of 300 = .45 × 300 = 135 fish. 300 − 135 − 25 = 140 fish that Nita catches make a: .
10. G Find the value of g(4) and substitute it into the function given for f(x). g(4) = 3 − = 3 − 2 = 1 and f(1) = 4(1)2 = 4, choice (G). Choice (F) stops at the value of g(4) and choice (K) finds f(4). The other choices make small math errors within each function.
11. D Estimate by calculating the area of a 6 × 6 square surrounding the shaded figure, then counting and subtracting the unshaded squares within that 36 unit area: roughly 18 squares. Subtract 36 − 18 = 18. Answer choice (D) is closest.
12. F Because the problem asks you about the rise after two different horizontal runs, sketch one run of 24 with a ramp of 6. Sketch a second run of 62 and a ramp of x. Make each a triangle by drawing a third line from the top of the ramp to the end of the run. The two triangles you’ve sketched have three common angles, so they’re similar. To solve, set up a proportion of rise/run. You know the rise after 24 inches is 6, so 6/24 = x/62. Solve and you’ll find that x = 15.5, choice (A).
13. E Substitute the given values of x and y into the expression and simplify, using the rules of order of operations (PEMDAS). (−3)2 + [(2 × 2) − (2 × −3)] = 9 + [4 − (−6)] = 9 + (4 + 6) = 19. Choice (C) drops the second negative sign in the final term, subtracting 6 from 4 instead of adding. Choice (B) subtracts 9 from 10 in the last step instead of adding. Choice (A) negates the value of 2x − 2y. Choice (D) finds the value of xy in the first term of the expression instead of yx.
14. G Because DEA measures 75° and BDE measures 105°, and the sum of these angles is 180°, you know that and are parallel. Eliminate choices that do not include II, choices (F) and (H). Since there is no way to determine the measures of ABD or BAE, it cannot be concluded that and are parallel. Therefore, eliminate choices (J) and (K).
15. D Multiply the price by the number of shirts sold. The price of the shirts will be the original price: $4.10 − the $0.02 discount × the number of discounts (x). The number of shirts sold is the original 95 shirts + 1 shirt for each day the price is reduced: (x). Multiply (4.10 − 0.02x)(95 + x) to get the number of shirts sold on any given day. If you’re stuck on a problem like this or if you can’t figure out exactly what the answer should be, use process of elimination: you know that the price of the shirts goes down but the number sold goes up. Therefore, you’ll want a (–) in your first set of parentheses and a (+) in your second set of them—this allows you to eliminate choices (A) and (C). Then, notice that the $0.02 will be a change in the price, not the number of items sold, so you can eliminate (E). Compare choices (B) and (D) and consider which contains numbers from the problem—in no part of the problem is there a 2, only $0.02, so choice (D) is the best answer choice.
16. G To factor x2 − 7x + 12, find two numbers that multiply to +12 and add to −7. Those numbers are −4 and −3. The factored expression is (x − 4)(x − 3).
17. A Substitute the values given for x and y in the equation make a: . Find a common denominator for all of the fractions by looking at the smallest multiple of 70, 35, and 7. 70 is the smallest denominator. The equation becomes . If you’re not sure how to simplify all of these fractions, plug the values in to your calculator to see that your answer will be roughly 0.543. Check the answer choices against this value to find that choice (A) is the only one that works.
18. G To find the greatest possible product, first determine the largest number of minutes and seconds possible, which is 59:59. After that, the timer will roll over to its maximum value of 60:00. (5)(9)(5)(9) = 2,025. Choice (F) results from (5 × 9) + (5 × 9) and choice (H) takes the product of (59)(59) instead of separating the values into digits first. Choice (J) finds the number of seconds in an hour and choice (K) assumes the largest display to be 99:99 which it cannot be because, as the problem states, this is only a 60-minute timer.
19. A Since you are looking for 2 numbers, one of which is bigger by 6, that add up to 42, set up an equation: 2x + 6 = 42, and solve for x, which is the lesser of the 2 numbers, 18. If you chose (E), be careful! That’s the greater of the two numbers and you’re looking for the lesser of the two numbers.
20. J The area of the square is 324, so each side is 18. Since the two right triangles are isosceles, their respective bases and heights are congruent. For each of the triangles, A = bh = (18)(18)=162. There are two triangles, so the total area is 162 × 2 = 324. Choice (G) gives the area of only one triangle, and choice (K) gives the area of the entire figure. Choice (F) assumes 324 to be the area of the entire figure and divides by three, assuming that the three smaller shapes are equal in area. Choice (H) doubles the value of choice (F).
21. B Since the triangles are similar, set up a proportion:. So, x = 40. Choice (A) is 17 + 8 = 25. Choice (C) is 85 + 17 = 102. Choice (D) is 85 + 17 + 8 = 110. Choice (E) flips one side of the proportion:.
22. J Rewrite the equation as (x − 1)(x − 1) = (x − 7)(x − 7) and multiply everything out: x2 − 2x + 1 = x2 − 14x + 49. Subtract x2 from each side: −2x + 1 = −14x + 49. Add 14x to each side: 12x + 1 = 49. Subtract 1 from each side: 12x = 48. Divide both sides by 12: x = 4. If you try the answer choices in the equation, x = 4 gives you (4 − 1)2 = (4 − 7)2. This simplifies to (3)2 = (−3)2 or 9 = 9, making the equation true.
23. D Draw a horizontal line at point A to split BAC in half, creating two 30°-60°-90° triangles. The y value at point A, which is w, is equal to the y value halfway between points B and C, so the y value at point C is twice as big: 2w. Eliminate choices (A), (C), and (E), where the y value is not 2w. The ratio of the sides of a 30°-60°-90° triangle is 1: :2 from smallest to largest, so the length of the leg adjacent to the 30° angle is times the length of the shorter leg, or w. Choice (B) neglects to multiply the x value by .
24. G The diagonal of a square makes two triangles with angle measures 45°, 45°, and 90°, and sides in the proportion x:x:x . Since we know that the hypotenuse of one of the triangles is 4, we know that the legs of that triangle, which are also the sides of the square, measure 4 feet. The area of the square is side2 = 16feet2.
25. A SOHCAHTOA tells you that sinθ=. Eliminate choices (B) and (C), which confuse the sides of a right triangle. =tanθ, so choice (E) can also be eliminated. Choice (D) refers to the angle measure of θ itself, rather than the trigonometric relationship between the sides of the triangle.
26. J Since 15 is a whole number, will most likely be a whole number, and a a perfect square. Try some perfect squares to see which yields 15 in the equation. .
27. A Use the Pythagorean theorem to find the length of . The midpoints cut each side of the rectangle in half, so AF = 6 and AE = 8. Right triangle AFE, then, is a 6:8:10 triangle, and EF = 10. FG is also 10, and the perimeter of the pentagon is 10 + 10 + 8 + 12 + 8 = 48 inches. Choice (B) is the perimeter of the rectangle. Choice (C) is the area of a triangle with the same base and height as the rectangle. Choices (D) and (E) are the areas of the pentagon and rectangle, respectively.
28. H In 2004, there were 60 million female workers age 16 or over and there were 130.9 million total such workers. The percent of the workers who are female is found by ×100 ≈ 46%.
29. E Since the chart shows that 8.4% of the workers took public transportation, the degree measure of the central angle for the “Public” sector must also be 8.4% of the 360° in a circle: × 360 ≈ 30°. Choice (A) is the percent of workers who took public transportation in 2006. Choice (B) is the measure of the central angle for the workers who walked in 2006. Choice (D) is the central angle that would result from using the public transportation percentage from 2004.
30. J In 2006, there were 63.6 million female workers. In 2004, there were 60.0 million female workers. The total growth was 3.6 million. The growth happened over a 2-year period so the average growth per year is 3.6 ÷ 2 = 1.8 million female workers per year. Choice (F) is the difference between the number of female workers in 2004 and 2005 divided by 2. Choice (G) is the result of dividing 3.6 by 4 years rather than 2 years. Choice (H) is the average growth in the number of male workers between 2005 and 2006. Choice (K) is the difference between the number of female workers in 2004 and 2006.
31. C To solve, enter the data into the given equation and solve using a common denominator. , so Tc, the combined time, is 15 minutes. Choice (A) divides the two numbers given while choice (E) finds their average. Choices (B) and (D) make calculation errors. Choice (E) is also unreasonable because the combined time must be less than the time of either hose working alone.
32. H The numbers 2, 3, and 5 are prime, while 6 and 4 are not. The probability that the die will NOT land on a prime number is 2 non-prime numbers out of 5 total numbers, or . If you picked choice (G), you left out the NOT.
33. D Find y by substituting x = 3: y = . Now find the value of 7x + 9y: 7 × 3 + 9 × = 21 + 25 = 46. If you chose (A) or (B), you may have forgotten to square the fraction or to multiply it by 9. Choice (C) comes from not squaring the . Choice (E) comes from squaring only the 5 in the numerator.
34. H Sketch a graph of points (0,0), (4,0) and (4,4). You’ll notice that the fourth point of the square is at (0,4). Since you need to find a diagonal, draw a line from (0,0) to (4,4), and notice you now have two 45°-45°-90° triangles, with the square’s sides as their legs and the diagonal as their hypotenuse. The hypotenuse will be the length the question asks for, the sides of a 45-45-90 triangle are in the proportion x:x:x, so c = 4. If you chose (G), you selected the length of a side of the square, not one of its diagonals. If you chose either (J) or (K), you may have been using the ratio of sides in a 30-60-90 triangle rather than that of a 45-45-90 triangle.
35. B The question is asking, 4 raised to the 3rd power = ? 4 × 4 × 4 = 64.
36. G An easy way to do this problem is to test each answer choice to find the pair of numbers that satisfies the given equation. For choice (G), y = 18 − 0.2(1) = 17.8 or 0.2 quarts of water have boiled away. None of the remaining answers makes the provided equation true.
37. D The second figure shows that the small and large diameters of the coffee cup are 4 and 6 respectively, so the radii are 2 and 3. Plugging the numbers into the equation given, V=π5. 5(32 + 22 + 3ç2)π5.5(19)≈109. Choice (A) is the portion of the equation inside the parentheses and choice (C) neglects to multiply by π. Choice (B) uses a height of instead of 5, or 5.5. Choice (E) plugs the diameters into the equation instead of the radii.
38. F Simplify the right side of the equation: 3(3x+4) = 9x + 12. What can x be in the equation 9x + 12 = 9x + 12? Any real number.
39. E First, figure out : the big fraction bar means division, so this is . To subtract and , get a common denominator of 12:. Now you have , but you divide fractions by flipping the fraction on the right and multiplying, so this becomes. This is or 9. Now combine the three remaining fractions:. You now have 9÷, which becomes . If you chose one of the other answers, be careful—watch your order of operations, and be careful putting these fractions into your calculator.
40. H To find the area of the shaded empty space, you must first find its radius. Since the width of the bagel from the center to the edge is 56 mm, the width, or diameter, of the space, is the total diameter of the cross-section minus the width of the bagel itself on both sides. 144 − 2(56) = 32. The radius of a circle is half its diameter, so r = 16 mm. Area = πr2 = π (16)2 ≈ 800. Choice (J) calculates the area using the diameter of 32 instead of the radius. Choice (K) finds the area of the entire bagel including the shaded space. Choice (F) finds the circumference of the shaded space, and choice (G) finds the outer circumference of the bagel.
41. E Any non-zero number raised to the 0 power = 1. So a0 = 1, b0 = 1, and c0 = 1. 1 + 1 + 1 = 3.
42. F Use SOHCAHTOA: tan = . For each of the angles, the opposite side is AB. Since AB is constant for each angle, the length of the adjacent side is all that matters when determining which angle has the greatest tangent value. The angle with the shortest adjacent side, BEA, has the greatest tangent.
43. E The y-intercept of a graph is the point where the graph meets up with (intercepts) the vertical (y) axis. This graph meets up with the y-axis at point (0,1). Choices (B) and (D) are the two x-intercepts of the graph. Choices (A) and (C) do not lie on either axis.
44. G This graph is symmetrical with a vertex at point X. The squared term of a quadratic equation gives it the smooth curve about the vertex. If you have a graphing calculator, you can enter a simple quadratic equation like y = x2to confirm your choice. Choice (F) would require a wave-like graph and choice (H) would form a graph with a sharp V shape. Cubic functions, as in choice (J), do not create symmetrical graphs. Choice (K) refers to a function that is a straight line, which this is clearly not.
45. D To reflect a graph over the line y = x, switch the x and y coordinates of each point on the graph. For this graph, Point V (−2,1) becomes (1,−2) and point Z (0,1) becomes (1,0). Eliminate any graph that does not include these three points. Note that point X (−1,−1) remains the same. You can also visualize the reflection by drawing the line y = x on the graph and imagine folding the paper along that line. Where the graph meets the other half of the page is where the reflection will be. Choice (A) reflects the graph across the x-axis and choice (C) reflects it across the y-axis. Choice (B) shifts the graph up two places. Choice (E) moves the vertex to (1,1).
46. J Factor 32 and 45. 32 has factors 32, 16, 8, 4, 2, and 1. 45 has factors 45, 15, 9, 5, 3, and 1. Answer choice (F) is 32 × 45, (G) is 32 × 9, (H) is 16 × 5, (K) is 1 × 1, and (J) cannot be made by multiplying 1 number from the set of factors of 32 by 1 number in the set of factors of 45.
47. C 17o = 15 + 13 + 11 + 9 + 7 + 5 + 3 + 1 = 64 and 4o = 3 + 1 = 4. So 17o × 4o = 64 × 4 = 256. Choice (A) is 17o ÷ 4o. Choice (B) is 72 × 2; this would be correct if you were adding all even integers less than k, instead of the odd ones. Choice (D) is 81 × 4; this would be correct if the problem said “less than or equal to k.” Choice (E) is 136 × 6; this would be correct if you were adding all integers less than k, not just the odd ones.
48. H There’s no need to graph the equation or to put it into y = mx + b form. Substitute a into the equation for x and −3 into the equation for y. So, a − 4(−3) = 14 or a + 12 = 14. So, a = 2. Choice (F) is the result of substituting afor y and −3 for x. Choice (K) is the result of a sign error when substituting to get a− 12 = 14.
49. E Substitute values to see what happens as t grows larger. When t = . When . Even when . Choices (A), (B), (C), and (D) mirror terms from the question but do not accurately describe what happens as t changes. If you work this problem algebraically, it will give you the same result: .
50. K Since there are 180° in a straight line, the sum of the two angle measures must be equal to 180. (3x + 2) + (x + 28) = 180. Simplify the equation: 4x + 30 = 180, 4x = 150, x = 37. To find the measure of YXZ, substitute 37for x in the expression 3x + 2, and solve. Choice (G) represents the value of x. Choice (H) is the value of YXW.
51. A Use the formula for the slope of a line, m=. You can start with either point, so long as you start with the same point in both the numerator and denominator. In this case, . Watch your signs and remember to put the y values on top. Slope measures how steep something is; how quickly the height rises. Choice (C) reverses the x and y values. Choice (E) confuses signs when dividing. Choices (B) and (D) confuse signs in calculating the numerator.
52. J One radian = , or 180° = π radians. Multiply radians = = 84°.
53. A The equation of a circle with center (h,k) and radius r is defined as (x − h)2 + (y − k)2 = r2. Here, h = 4 and k = −8. Eliminate choices (C), (D), and (E), which all flip the values of h and k. To find the radius, use the circumference 10π given in the problem. C = πd = 10π, so the diameter is 10. The radius is half of that, or 5, so to complete the equation, r2 = 25. Choice (B) squares the diameter instead of the radius.
54. K Choose values for x and y that make xy = −x2 true. For example, x = 2 and y = −2. Choices (F), (G), (H), and (J) are all true with these numbers. However, choice (K) is false because (2)3− (−2)3 = 16.
55. A Compare the points of the original rectangle with the first matrix to see that the x values of A, B, C, and D run along the top row and their y values run along the bottom row. For the translated rectangle ABCD, plot the points you know: B(3,−3), C(−2,−6), and D(−4,−3). When you’ve got those points, note the relationship between them. In your figure, it should be clear that the distance from point C to point D will be the same as the distance from point B to point A. From C to D, the point shifts to the left 2 and up 3. Now do the same thing to point B to get A(1, 0) and n = 0, choice (A). Choices (B), (C), (D), and (E) are in the range of numbers of the problem, but do not translate properly.
56. K To solve this equation, you’ll need to remove the absolute value signs. Remember, since absolute value means only the “positive distance from 0,” the value inside the absolute value signs can be either positive or negative. To solve |x − a|≤3, therefore, remember that when you remove the absolute value sign, this becomes two separate inequalities: x − a≤3 and x − a≥−3. Don’t forget to flip the inequality sign in the second equation when you introduce the negative! These expressions simplify to x ≤ a + 3 and x ≥ a − 3. Combine these two equations to get a − 3 ≤ x ≤ a + 3 which is represented in choice (K).
57. B Start from the output 0. Since a CHANGE function needs to have input 1 to get output 0, the output of the IF function needs to be a 1. A 1 is the output of an IF function either if the first input is 1 and the second input is 1, or if the first input is 0 and the third input is 1. The first input could be 1 if both p and q are 1; however, no answer choices contain p and q values which are both 1. Therefore, p and q could be either 0 and 1, or 1 and 0, or 0 and 0, to yield an output of 0. This doesn’t narrow any answer choices. Since you know that the first input of the IF function is 0, the second input could be either 1 or 0, so the input of the next CHANGE function doesn’t matter. s and t, however, must both be 1, since the output of the BOTH function must be a 1 in order to make the third term in the IF function a 1. Choice (B) is the only answer choice that has s and t both as 1.
58. J First, scientific notation requires that the first term be a number less than 10. (F), (G), and (H) are out. Multiply the terms in the given expression: 30 × 10x + y = 3.0 × 10x + y + 1. Remember that when you multiply exponents with the same bases the base remains the same and the exponents are added together. The exponent 1 is added because 10 is being multiplied by itself one more time, to make up for taking a factor of 10 out of 30.
59. A The midpoint of a line segment divides that line segment in half; therefore, if the length of is x, the length of is 2x. The question also states that the midpoint of is Q, so = . Substitute for the lengths to get 4x −16 = 2x. (4x −16 is given in the problem as the length of .) Solving for x, you get 2x = 16, or x = 8. The length of is x + x + 4x −16 = 8 + 8 + 32 − 16 = 32. To understand this problem, draw a straight line, mark points P, Q, R, and S along it, and label the lengths as you find them. Choice (D) gives the value of x, not the length of Choice (C) finds the length of 4x −16. Choice (B) miscalculates x to be 4, and choice (E) is the value of the miscalculated x.
60. H Find the radius of the circle using the area formula: A = πr2. So, 64π = πr2 and r = 8. To find the length of a minor arc, first find the circumference of the circle: C = 2πr = 2π(8) = 16π. The minor arc is of the circumference of the circle. So, the length of the minor arc is . Choice (F) attempts to use the radius of the circle, 8, to find the fractional part of the circle. Choice (G) is similar to (F) but uses half of the radius. Choice (J) is the area divided by the central angle. Choice (K) is the radius multiplied by the central angle.