This practice exam contains 50 questions that review the 50 Skills that can be found at ACT Math Prep. Take the test. Then check your answers and review the skills for any questions that were difficult.

  1. A sack of sweet yams could be equally divided among 4, 5, or 6 families. What is the smallest number of yams that could be in the sack?
    1. 24
    2. 30
    3. 56
    4. 116
    5. 120
  2. If 4(x + 1) = 8x, then x = ?
    1. 1
    2. 2
    3. 4
    1. 8
    2. –7
  3. If D = RT, then which of the following is an expression for T in terms of D and R ?
    1. DR
    2. DR
    3. RD
  4. Sora earned $5 on Monday. On each following day, through Friday, she earned 1 more dollar than the preceding day. What was her mean earnings for the five-day period, Monday through Friday?
    1. 4
    2. 5
    3. 6
    1. 7
    2. 8
  5. The measure of BFE in the figure below is 138°. The measures of 3 angles are given in terms of m, in degrees. What is the measure of BFD
      Posttest
    1. 110
    2. 115
    3. 120
    4. 125
    5. 130
  6. In the figure below, if a || b and the measure of x = 25, what is the measure of y?
      Posttest
    1. 25
    2. 50
    3. 90
    1. 108
    2. 133
  7. In ΔQRX, and the measure of Q is 34°. What is the measure of X?
    1. 112°
    2. 56°
    3. 54°
    4. 34°
    5. 17°
  8. For all x, (5x – 2)2 =?
    1. 5x2 + 4
    2. 25x2 – 2
    3. 25x2 – 10x + 2
    1. 25x2 – 20x + 4
    2. x2 – 25x – 2
  9. Which of the following lists gives the 3 largest prime numbers that are less than 50 ?
    1. 5, 7, and 11
    2. 7, 11, and 13
    3. 41, 43, and 47
    4. 39, 43, and 47
    5. 43, 47, and 49
  10. Which of the following lists the factors of 36 ?
    1. 1, 36
    2. 2, 3
    3. 36, 72, 144
    1. 1, 2, 3, 4, 6, 9, 12, 18, 36
    2. The empty set
  11. What is the lowest common denominator of ?
    1. 3
    2. 24
    3. 36
    4. 96
    5. 288
  12. In the xy coordinate plane below, which of the following points has coordinates (x, y) such that x = y – 2 ?
      Posttest
    1. A
    2. B
    3. C
    1. D
    2. E
  13. If the line through the points (5, –3) and (–2, p) is parallel to the line y = –2x – 3, what is the value of p ?
    1. 11
    2. 4
    3. 0
    4. –10
    5. –17
  14. In the standard (x, y) coordinate plane, what is the slope of the line parallel to the line with equation 6x + 8y = –12 ?
    1. – 6
    1. –2
    2. 4
  15. The graph below shows the number of Texas Hold 'em tournaments won by each of four players. According to the graph, what fraction of the tournaments did Reisner win?
    1. 0.30
    2. 0.25
    3. 0.20
    4. 0.15
    5. 0.10
  16. The number of yoga postures that Ben knows after a series of yoga classes can be modeled by the function P(c) = 3c + 8, where c represents the number of classes that he has attended. Using this model, how many postures would you expect him to know after 10 classes?
    1. 3
    2. 8
    3. 10
    1. 30
    2. 38
  17. What is the distance, in coordinate units, between the points (–4, 3) and (7, –2) in the standard (x, y) coordinate plane?
    1. 15
    2. 21
  18. In the equation , x represents a positive integer. As x gets larger and larger without bound, the value of y
    1. approaches 5
    2. approaches 2
    3. approaches 1
    1. approaches 0
    2. remains unchanged
  19. If a completely fenced in square lot requires 80 feet of fence, what is the area, in square feet, of the lot?
    1. 6400
    2. 1600
    3. 400
    4. 200
    5. 20
  20. In the figure below, what is the area of the shaded region?
      Posttest
    1. 17
    2. 32
    3. 39
    1. 49
    2. 54
  21. If the numerator of a fraction is 4 more than the denominator and the fraction equals , what is the numerator of the fraction?
    1. 3
    2. 6
    3. 9
    4. 10
    5. 14
    1. undefined
  22. In a standard (x, y) coordinate plane, three points Q, R, and S have coordinates (5, 2), (–1, 5), and (5, b). If the three points determine a right triangle, what is the value of b ?
    1. 0
    2. 2
    3. 4
    4. 5
    5. 6
  23. The lengths of the sides of a right triangle are consecutive integers, and the length of the shortest side is x. Which of the following could be used to solve for x ?
    1. (x)2 + (x + 1)2 = (x + 2)2
    2. x + x – 3 = x
    3. (x)( x + 1) = (x + 2)2
    1. (x + 2) – (x + 1) = x
    2. (x)2 + (x + 2)2 = (x + 4)2
  24. Which of the following sets of 3 numbers could be the side lengths, in meters, of a 45°–45°–90° triangle?
    1. 2, 2, 2
    2. 2, 2, 2
    3. 2, 2 , 2
    4. 2, 2 , 2
    5. 2, 2 , 4
  25. Edward casts a shadow 3 meters long. At the same time, Bella casts a shadow 2.5 meters long. If Edward is 2 meters tall, which of the following is closest to Bella's height, in meters?
    1. 0.6
    2. 1.2
    3. 1.7
    1. 2.6
    2. 3.4
  26. 71 is what percent of 1420 ?
    1. 2%
    2. 5%
    3. 20%
    4. 50%
    5. 700%
  27. What is the total cost of 4.3 pounds of flour at $1.10 per pound and 0.5 pounds of baking soda at $0.82 per pound?
    1. $5.14
    2. $4.27
    3. $3.34
    1. $2.46
    2. $1.78
  28. If x2y2 = 72 and xy = 8, what is the value of x + y ?
    1. 8
    2. 9
    3. 36
    4. 72
    5. 80
  29. What is the product of 3m4 and 2b4 ?
    1. 5mb8
    2. 6mb8
    3. 6(mb)4
    1. 5(mb)8
    2. 6(mb)8
  30. What is the sum of 4x2 and 5x2 ?
    1. 9x4
    2. 9x2
    3. 4x2 + 5x2
    4. 20x4
    5. 20x2
  31. If Brian spends 7 minutes warming up and 3 minutes per yoga posture, which of the following equations expresses the number of minutes Brian spends on warm-ups and postures when he does p postures?
    1. y = 7
    2. y = 7p + 7
    3. y = 10p
    1. y = 3p – 7
    2. y = 4p + 4
  32. On the first day of Professor Gil's Reading Shakespeare class, six students will select a seat from the six seats at a table in the classroom. How many different sitting arrangements of the six students are possible?
    1. 6
    2. 72
    3. 144
    4. 720
    5. 1440
  33. In the right triangle shown below, which of the following statements is true about ?
      Posttest
    1. cos L =
    2. sin L =
    3. tan L =
    1. cos L =
    2. sin L =
  34. Which of the following expressions could be used to find the length of the side marked with the question mark in the triangle shown below, with measurements as marked?
  35. (Note: The law of cosines states that for any triangle with vertices A, B, and C and the sides opposite those vertices with lengths a, b, and c, respectively, c2 = a2 + b2 – 2ab cos C.)

      Posttest
    1. 412 + 322 – 2(41)(32) cos 67
    2. 352 + 782 – 2(41)(32) cos 67
    3. 352 + 782 – 67(41)(32) cos 2
    4. 412 + 322 – 67(35)(67) cos 41
    5. 352 + 782 – 2(41)(32) cos 32
  36. There are 16 buttons in a bag: 8 are red, 4 are blue, and 4 are white. What is the probability that a button selected at random from the bag is NOT white?
    1. 1
  37. What is the product of the 2 solutions of the equation x2 + 2x – 24 = 0 ?
    1. –24
    2. –10
    3. –2
    4. 8
    5. 14
  38. What are the coordinates for the minimum point in the graph of y – 5 = (x – 4)2 ?
    1. (4, 5)
    2. (–4, 5)
    3. (4, –5)
    1. (–4, –5)
    2. (0, 0)
  39. A circle in the standard (x, y) coordinate plane has center (4, 9) and radius of 9 coordinate units. Which of the following is an equation of the circle?
    1. (x – 4)2 – (y – 3)2 = 9
    2. (x + 4)2 + (y + 9)2 = 9
    3. (x – 4)2 – (y – 9)2 = 81
    4. (x – 4)2 + (y – 9)2 = 81
    5. (x + 4)2 – (y – 9)2 = 81
  40. The ratio of the radii of two circles is 2 : 3. What is the ratio of their areas?
    1. 2 :3
    2. 4:9
    3. 8:27
    1. 2:π
    2. π:3
  41. Which of the following are solutions to |q – 5| = 10 ?
    1. 15 only
    2. –5 only
    3. –15 only
    4. –5 and –15 only
    5. 15 and –5 only
  42. What is the next term after 12 in the geometric sequence – , 3, –6, 12, …?
    1. 24
    2. 0
    1. –24
    2. –48
  43. 137°F is the point at which ethanol will boil. Since Fahrenheit and Celsius temperatures are related by the formula F = C + 32, to the nearest degree, which of the following is the boiling point of ethanol in degrees Celsius? (Note: F = C + 32)
    1. 58°C
    2. 76°C
    3. 105°C
    4. 247°C
    5. 279°C
  44. (5x + 2) – (8x – 5) is equivalent to
    1. (–3x – 3)
    2. (–3x + 7)
    3. (3x – 3)
    1. (3x – 7)
    2. (x – 3)
  45. When x = –3, what is the value of the equation y = 2x2 + 4 ?
    1. –2
    2. –14
    3. 12
    4. 22
    5. 40
  46. If (x – 2)(x + 3) = 0, then x could equal
    1. –3
    2. 0
    3. 2
    1. II only
    2. I and II
    3. I and III
    1. II and III
    2. I, II, and III
  47. Which of the following is a value that satisfies log6 216 = x ?
    1. 0
    2. 1
    3. 2
    4. 3
    5. 4
  48. If the product of 3 and (2i2 + x) equals –9, then x = ?
    1. 0
    2. –1
    3. –2
    1. i
    2. 2i
  49. What is the slope of a line parallel to the line through (–6, 3) and (5, 8) in the standard (x, y) coordinate plane?
    1. –2
    2. 5
    3. 16
  50. If m = –2 and n > 0, which of the following is greatest?
    1. mn
    2. m2n
    3. m3n
    1. m4n
    2. m5n

Answers

  1. E   Rather than randomly trying numbers, just "Use the Answers"! Divide each choice by 4, 5, and 6. Only choice E divides evenly (no decimal or remander).
  2. G   This "hard" algebra question seems impossible to many students. "Use the Answers" makes it easy! Just try the choices for x and see which one works. Choice G is correct, since 42 + 1 = 82 = 64.
  3. D   "T in terms of D and R" means "solve for T, use algebra to get T alone." So, just divide both sides by R to get T alone, which gives = T.
  4. J   Write out the earnings for the five days: 5, 6, 7, 8, 9. Then to find the "mean," which is just a fancy word for average, add them up and divide by 5.
  5. 5 + 6 + 7 + 8 + 9 = 35 ÷ 5 = 7

  6. B   The question tells us that BFE= 138° and the diagram shows us that BFE also equals 6m, so we know that 138 = 6m, and therefore m = 23. According to the diagram, BFD = 5m and must thusly (I always wanted to say thusly) be 5(23) = 115.
  7. K   Use the strategies. Mark x = 25. Therefore, in the triangle, the missing angle is 180 – 108 – 25 = 47. And in the pair of parallel lines, there are two kinds of angles, big and little, and a big + a little = 180. Here 47 is a littler angle, and y is a bigger angle, and therefore equals 180 – 47 = 133.
  8. A   When a picture is described, draw it. When you see a triangle with two sides equal, mark the angles opposite the equal sides as equal. So R = 34 also. Then since a triangle has 180 degrees, the third angle of the triangle must be 180 – 34 – 34 = 112.
  9. J   (5x – 2)2 – (5x – 2)(5x – 2) = 25x2 + 10x – 10x + 4 = 25x2 – 20x + 4. Careless Error Buster: When you FOIL, remember the middle term!
  10. C   You can make a list of the prime numbers less than 50, counting down from 50, or you can "Use the Answers." Just make sure to avoid the careless error of choosing 49 as a prime number (it's divisible by 7, so it's not prime). Either way, the three largest prime numbers less than 50 are 41, 43, and 47.
  11. J   The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. You can use our pairs strategy to factor 36, or you can just "Use the Answers" and see which one works. Remember to underline vocab words to avoid a careless error. Choice G lists the prime factors and choice H lists some multiples, but we want factors.
  12. B   Many kids flee this question, but with our Skills it's easy. Reread Skill 11 to see that the lowest common denominator is the least common multiple of 3, 8, and 12. So just "Use the Answers" and try each choice. Choices B, D, and E are multiples of 3, 8, and 12, but choice B is the lowest and is therefore correct.
  13. K   "Use the Answers"! Simply test each point in the equation to see which one works. Point E has coordinates (2, 4), so 2 = 4 – 2 which works.
  14. A   Parallel lines have equal slopes. The slope of line y = –2x – 3 is –2, so the slope of the two points given is "Use the Answers" or cross–multiply to solve for p: p = 11.
  15. H   Solve the equation for y to get y = . The slope is – and a line parallel to this line will have the same slope.
  16. B   The graph shows that Reisner won 25 tournaments. Remember that half a symbol is 5 tournaments, not half a tournament. There were a total of 100 tournaments, so Reisner won of them.
  17. K   He has been to 10 classes, so simply plug in 10 for c to get 3(10) + 8 = 38. Remember that for a function question, P(10) does not mean P times 10—it means "P of (10)" or "the value of P when c is 10."
  18. C   Use the distance formula:
  19. J   Did you freak out and guess? It might look impossible—unless you use "Make It Real"! Then, it's easy! Just choose a number for x, say 3. When x = 3, y = . To see what happens when x gets larger, let's try x = 10: . And finally, let's try x = 1000 to see what happens when it gets really big: y = . So, clearly, as x gets very large, y approaches zero. Another case cracked by "Make It Real"!
  20. C   Since the square requires 80 feet of fence, each side must be 20 feet long. Then the area equals (side)(side) = (20)(20) = 400.
  21. H   Fun question. Remember that the area of a shaded region equals area big guy minus area donut hole. The area of the whole rectangle is 6 × 9 = 54, and the area of the unshaded triangle is (this is the cool part) Area = 0.5bh = 0.5(5)(6) = 15. It looks like the height wasn't given, but it was—it's 6, it's as tall as the rectangle. So the shaded region = 54 – 15 = 39.
  22. D   "Numerator" is just a superfancy word for the top of a fraction, and "denominator" is just the bottom. So if the numerator is 4 more than the denominator and the fraction equals , we can represent this as and crossmultiply to solve for d. Now (d + 4)(3) = (d)(5) which equals 3d + 12 = 5d. Collect like terms and 2d = 12 so d = 6. Notice that the question does not ask for d, but for the numerator which is d + 4; the numerator is 10. Careless Error Buster: Always check to make sure that you've finished the question and didn't stop in a preliminary step.
  23. J   First multiply the 2 in the second matrix to get . Then subtract the matrices by simply subtracting the numbers in each corresponding spot:
  24. D   When a picture is described, but not shown, draw it! Don't be a hero and try to do this one without a diagram. A diagram helps so much. It shows you what to do next, and it avoids careless errors. As soon as you draw a decent diagram, you immediately see that to be a right triangle, b must equal 5.
  25. F   If the shortest side is x and the lengths of the sides are consecutive integers, then the longer sides can be called (x + 1) and (x + 2). Now, we just plug these into a2 + b2 = c2 to get (x)2 + (x + 1)2 = (x + 2)2.
  26. B   I love this question. It's rated "medium" only because most kids do not know special right triangles. If you take five minutes right now and memorize them, just the info on the page for Skill 25, then this "medium" becomes easy and you gain points! Once you know about special right triangles, choice B is obviously the answer. It follows the pattern x, x, x. If, for some crazy reason, you have not memorized the pattern, first of all shame on you, you silly slacker, but second of all, you could draw a diagram for each choice and see which look 45–45–90ish.
  27. H   When a picture is described, draw it. This shows you that we have two triangles, which are similar. Set up a proportion: . When you see a proportion, cross–multiply: 5 = 3x. And divide by 3 to get 1.66 ≈ 1.7.
  28. B   Translate English to math. "Is" means = and "what" means x, so 71 = x(1420). Divide both sides by 1420 to get x = 0.05, or 5%.
  29. F   Translate English to math. "Of" means multiply, so (4.3)(1.10) + (0.5)(0.82) = 5.14.
  30. B   Springboard! When you see x2y2, factor it. So (xy)(x + y) = 72, and we are told that xy = 8, so 8(x + y) = 72 and x + y = 9.
  31. H   Great exponents review! "Product" means multiply, and when you multiply, you add the exponents. But the bases must match. The bases, m and b, do not match, so we cannot combine them to get mb8; instead, 3m4 × 2b4 = 6m4b4 = 6(mb)4.
  32. B   Since both terms have an x2, they are matching terms. To add them, we add their coefficients. "Coefficient" is just the fancy vocab term for the number in front of the variable. So 4x2 + 5x2 = 9x2.
  33. J   Brian spends 3 minutes per posture, so that number goes next to the p. Brian spends 7 minutes on warm–ups, and this number does not change with the number of yoga postures, so that number goes alone. y = 3x + 7.
  34. D   When you see an arrangement question, draw a blank for each position, fill in the # of possibilities to fill each position, and multiply. So we have six seats to fill with six students: 6 × 5 × 4 × 3 × 2 × 1 = 720. Six students were available to fill the first seat, and once 1 sat down, there were only 5 available to fill the next seat, and so on.
  35. G   SohCahToa! Sin L means opposite over hypotenuse, so sin L .
  36. A   Great beyond SohCahToa question! All you have to do is to follow the directions. Even if you barely know trig, you can get this one if you follow the directions in the question! We are told that the law of cosines states that for any triangle with vertices A, B, and C and the sides opposite those vertices with lengths a, b, and c, respectively, c2 = a2 + b2 – 2ab cos C. So, to get the length of the side marked with the question mark, we call that side c, which makes its opposite angle C. We assign a and b to the other two sides. Then we plug in the appropriate measures to get c2 = a2 + b2 – 2ab cos C = 412 + 322 – 2(41)(32) cos 67.
  37. J   Probability = , so the probability of NOT selecting a white button is . This question requires that we reduce the .
  38. No problem. Stop looking at me like that, reducing is easy; just divide the top and bottom of the fraction by the same number. Or, if you really don't like reducing, use your calculator. Divide 12 by 16 to get 0.75. Then divide each answer choice (top by the bottom) to get decimals and see which one matches 0.75. Nice.

  39. A   Factor x2 + 2x – 24 = (x – 4)(x + 6) = 0. Either (x – 4) or (x + 6) must equal zero, so the solutions are 4 and –6, and the product (multiply) is (4)(–6) = –24. This question is a nice review of factoring, of "anything times zero is zero," and of the vocab word "product."
  40. F   Rewrite y – 5 = (x – 4)2 so it matches the form y = (xh)2 + k, by just adding 5 to both sides to get y = (x – 4)2 + 5. In this form, the vertex or minimum/maximum point is (h, k), so (4, 5) is the minimum point.
  41. D   We just need to apply the info we are given to the equation for a circle, which is (xh)2 + (yk)2 = r2. Since the center is (4, 9) and the radius is 9, h = 4, k = 9, and r = 9. So D is the answer. Notice that we can easily get this question by the process of elimination. For example, since r = 9, the number to the right of the equals sign, r2, must equal 81, not 9. Also, watch out for choice C, which looks good except that we need a plus sign, not a minus sign, between the parentheses. Moral of the story, "Memorize the equation of a circle."
  42. G   Since the ratio of their radii is 2 : 3, we can just use these as potential radii. So their areas are 4π and 9π, and the ratio of their areas is 4π:9π, which reduces to 4: 9.
  43. E   Great "Use the Answers" review! Try each answer choice in the equation. Choice E is correct, since |15 – 5| = |10| = 10 and |–5 – 5| = |–10| = 10.
  44. J   A geometric sequence is just a superfancy term for a list of numbers where you multiply each member by the same number to arrive at the next member on the list. To find that number, just notice what it is or divide a term by the one before it. –6/3 = –2. So –2 is the number you multiply each term by. The next term is 12 × –2 = –24.
  45. A   You can do the algebra or just "Use the Answers," whichever you prefer. To use the answers, plug in 137 for F, and then try each answer for C, to see which choice works. To do the algebra, plug in 137 for F and solve for C:
  46. G   Careless Error Buster: Remember to distribute the negative sign! (5x + 2) – (8x – 5) = 5x + 2 – 8x + 5 = –3x + 7
  47. D   When x = –3, then y = 2(–3)2 + 4 = 22. Careless Error Buster: PEMDAS, remember to square –3 and then multiply by 2.
  48. H   "Anything times zero is zero." To multiply to get zero, either (x – 2) = 0 or (x + 3) = 0. So x = 2 or –3. You could also just "Use the Answers" and try each choice to see which one(s) work(s).
  49. D   log6 216 = x means 6x = 216. To solve, just "Use the Answers." Using your calculator, it takes only a few seconds; just try each answer in the equation to see which one works. Choice D is correct since 63 = 216.
  50. G   "The product of 3 and (2i2 + x) equals –9" translates to 3(2i2 + x) = –9. Distribute the 3 to get 6i2 + 3x= –9. And since i2 = –1, we can plug that into the equation for i2, to get 6(–1) + 3x = –9, which simplifies to 3x = –3, and x = –1.
  51. C   The line through the points (–6, 3) and (5, 8) has slope . Since parallel lines have equal slopes, the slope of the line parallel will also be .
  52. J   Great review of weird number behavior and "Make It Real." Plug m = –2 and a value for n, say n – 3, into each answer choice, and look for the greatest result. Choice J is correct because the even exponent cancels out the negative sign, whereas the odd exponent in choice K keeps the negative.