Solutions for The Real ACT Prep Guide, 2nd Edition, practice test 1

Here are solutions for the first practice test in The Real ACT Prep Guide, 2nd Edition. While the book does provide solutions, the following solutions illustrate faster, less formal methods that may work better for you on the actual test. I haven't reprinted the questions here; see why do solutions pages not contain questions? for why they aren't here.

1. • Estimate the answer. 12⁄5 is 1.4, and 21⁄3 is 2.3, so the answer will be a bit more than 3.7.
   • Look at the possible answers. Out of the five choices, the only one larger than 3½ is E. 311⁄15, so that must be the answer.
2. Simplify the expression in your head or on paper:
     3x³ · 2x²y · 4x²y = (3 · 4 · 2)(x³ · x² · x²)(y · y) = 24x⁷y²
   Therefore, the answer must be H. 24x⁷y². You can also solve this problem by process of elimination: 3 · 2 · 4 = 24, so the answer can't be F. or G.. x³ · x² · x² = x⁷, so the answer can't be J. or K. either, leaving H. as the only possibility.
3. • Read the question carefully and determine what it is asking. You are asked to find (Mr. Dietz' daily earnings) − (substitute teacher's daily earnings)
   • Estimate the answer. Mr. Dietz makes $(22,570/185) per day. This is somewhat over $100 per day, but not too much more, definitely less than $150. So, the savings must be between between $20 and $70 per day, probably somewhere in the middle of that range (say, $45).
   • Look at the possible answers. The only answer that is in range is A. $42.
4. • Read the question carefully and determine what it is asking.
   • Estimate the answer. Since the student has two marks that are a fair bit below 80 and two marks only slightly above 80, the student will probably need a grade a fair bit above 80.
   • Look at the possible answers. There are two answers that may work: D. 99 and K. The student cannot earn an average of 80.
   • Guess and check. If the student earns 99 on the fifth test, his average will be (65 + 73 + 81 + 82 + 99) ÷ 5 = 80. Therefore, the answer must be J. 80.
5. • Read the question carefully and determine what it is asking. The question is asking you to find 7.3 ÷ 9.8.
   • Estimate the answer. 7.3 ÷ 10 is 73%. So, the answer to 7.3 ÷ 9.8 must be slightly larger than 73%.
   • Look at the possible answers. There are two answers (C. 73% and D. 74%) that are possible, so we need to perform the exact calculation.
   • Perform the required calculations: On your calculator, evaluate 7.3 ÷ 9.8. The result is 0.74489... . To the nearest percent, the answer is 74% (D.).
6. • Draw a diagram, and label it. The diagram might look something like the following:
     
   • Calculate the answer: The length of the fence must be 150 + 200 + 150 + 200 = 700. Therefore, the answer is J.
7. Carefully expand the expression:
   a[b + (c − d)]
   = ab + ac − ad
   The correct answer is A. ab + ac − ad.
8. • Carefully solve the equation:
     4x + 3 = 9x − 4 7 = 5x x = 7⁄5
     Therefore, the answer is F. 7/5.
9. There are several ways to solve this problem. One of them is demonstrated below:
   • Estimate the answer: Since 17 is around 20 and 41 is around 40, we might expect the first number to be about 1/3 of the way between 20 and 40, and the second number about 2/3 of the way. So, perhaps 26 and 33. You may also have noticed that, since the first and fourth terms are both odd, the second and third terms must also be odd. So a reasonable estimate might be 25 and 33 (or perhaps 27 and 33).
   • Look at the possible answers: Only A., C. and E. contain two odd numbers. The second term in A. appears to be too small. C. matches our estimate. E. is close to our estimate.
   • Guess and check: C. appears to be the best answer, so check whether it works. The difference between 25 and 17 is 8, the difference between 33 and 25 is 8, and the difference between 41 and 33 is 8. Therefore, the answer is C. 25, 33.
10. If x³ = 64 and x is a real number, than x must be equal to 4. Therefore, x² + √x = 16 + 2 = 18. Therefore, the answer is H.
11. • Read the problem and determine what it is asking: You are given the formula to find the volume of a sphere given a radius, and given the radius of a sphere.
    • Estimate the answer: Estimate the result of (4/3)π(1 1/4)³. (4/3)π is probably a bit more than 4, but what is (1 1/4)³? Well, (1 1/4)² is 25/16, which is somewhat less than 2, so (1 1/4)³ is probably somewhere around 2. So, our estimate is 4 × 2 = 8.
    • Look at the possible answers: There are two answers that are close to our estimate, B. 7 and C. 8.
    • Calculate the answer: V = (4/3)π(1.25)³, which is about 8.18. Therefore, the answer is C. 8.
12. • Estimate the answer: Since there are three colours of marbles and there are fewer white marbles than red ones, then the probability that the marble is white is less than 1/3. Therefore, the probability that the marble will not be white is more than 2/3.
    • Look at the possible answers: There are two possible answers, F. 3/4 and K. 7/10.
    • Perform the calculation: The probability that the marble will not be white is (8 + 6)/(8 + 6 + 6) = 14/20 = 7/10. Select K. 7/10.
13. You can solve the problem using matrix multiplication, but if the matrices intimidate you, don't worry about them. Just multiply the quantities together:
    40 · 0.3 + 60 · 0.4 + 80 · 0.2 + 80 · 0.5 = 92

    Therefore, the answer is D. 92.

14. • Read the question and determine what it is asking: You are asked to find (24 + 25 + 29) / 3.
    • Estimate the answer: The answer is definitely more than 25 but less than 29. A good estimate for the answer would be 26 or 27.
    • Look at the possible answers: H. 26 and J. 27 are the only reasonable answers.
    • Calculate the answer: (24 + 25 + 29) / 3 = 26. Therefore, the answer is H. 26.
15. • Read the question and determine what it is asking: You are asked to find which periods the number of students taking math is greater than 30 + 30 − 2 − 6 = 52.
    • Estimate the answer: It's hard to tell exactly just by looking at the table, but Period 4 (28 + 29) definitely looks like a good candidate.
    • Look at the answers: C. Period 4 only and D. Periods 3 and 4 only look like good candidates.
    • Perform the calculations: For Period 4, there are 29 + 28 = 57 calculators required. For Period 3, there are 25 + 23 = 48 calculators required, which is less than 52. Therefore, the answer is C. Period 4 only.
16. Looking at any completed row or column of the table, the sum of that row or column is 6x. Looking at the middle row of the table, its sum is 4x. Therefore, the value of the center cell must be H. 2x.
17. You may know the answer to this question offhand. If not:
    • Draw a diagram: It may be helpful to plot a few points on the diagram. For example, you may want to plot (2, 2), (2, −2), (−2, 2) and (−2, −2) on the diagram.
    • Calculate the answer: The point (2, −2) is in quadrant IV. The point (−2, 2) is in quadrant II. Therefore, the answer musts be E. Quadrant II or IV only.
18. • Read the question and determine what it is asking: To find how many distinct outfits can be selected, you need to multiply the given numbers together. So, you are asked to find 4 × 6 × 3.
    • Estimate or calculate the answer: The answer should probably be close to 5 × 5 × 3 = 75. Alternately, you may be able to quickly calculate 4 × 6 × 3 = 72.
    • Look at the answers: The only one anywhere near our estimate or calculation is J. 72. Therefore, select that answer.
19. It might be possible for you to simply evaluate the proportion in your head. If not:
    • Estimate the answer. It appears that somewhere between 1 and 2 (perhaps closer to 2) barrels of sand are required to produce one "barrel of a tarry material", whatever that happens to be. So, 3,000 barrels of this tarry material would require somewhere between 3,000 and 6,000 barrels of sand, perhaps around 5,000 barrels.
    • Look at the possible answers. A. 5,000 is the only answer in the above range, so the answer must be A.
20. • Draw a diagram. The diagram might resemble the diagram shown below. The length AC in the diagram is what we're looking for.
      
    • Estimate the answer. AC has to be longer than BC, so the answer must be greater than 72. AC has to be shorter than AB + BC (the shortest distance between two points is a straight line, or you can use the triangle inequality here), so the answer must be less than 54 + 72 = 126. Perhaps the answer might be somewhere around 100.
    • Look at the possible answers. H. 90 is the only answer within the above range, so the answer must be H. 90.
21. Solution 1:
    • Guess and check: Have a look at each answer and see if it evaluates to x/y. Answer A. does, if you cancel the z from numerator and denominator. Select A. (x · z)/(y · z).
    Solution 2:
    • Try a special case: Let x = 2, y = 3, z = 4. Then x/y = 2/3.
    • Look at the answers: Answer A. works out to 2/3, answer B. to 4/9, answer C. to 1, answer D. to 2, and answer E. to 6/7. Select the only answer that works out to 2/3, A. (x · z)/(y · z).
22. Add y to both sides of the equation 8x − y − 6 = 0; this results in 8x − 6 = y. Therefore, the answer is H. y = 8x − 6.
23. Solution 1:
    • Solve the equation:
      x² − 36x = 0
      x(x − 36) = 0
      x = 0 or x = 36
    • Look at the answers: None of the answers are 0. Select B. 36.
    Solution 2: Guess and check: Substitute the value of each answer in the given equation, and determine if any gives a result of 0. If x = 36 (B.), then x² − 36x = 0. Select B. 36.
24. The tangent is opposite/adjacent. From angle R, the opposite side is r and the adjacent side is t. Therefore, the answer is G. r/t.
25. • Read the question and understand what it is asking: You are asked to find the hypotenuse of the right triangle in the diagram. You may want to write a "?" next to the hypotenuse on the diagram.
    • Draw a diagram: The diagram is already given, but it can help to label the bottom side of the triangle. The length of the bottom side of the triangle is ½ of the length of the chord (24), or 12.
    • Estimate the answer: The length must be greater than 12, but less than 17.
    • Look at the answers: There are two plausible answers: C. 16.9 and D. 13.0.
    • Calculate the answer: Use the Pythagorean theorem:
      r² = 5² + 12²
      r² = 169
      r = 13
      Select D. 13.0.
26. • Read the problem carefully and determine what it is asking: You are asked to solve 0.18 = (2/3)F + 0.03
    • Estimate the answer: F must be greater than 0.15, but less than 0.25. Perhaps the answer is somewhere around 0.2.
    • Look at the possible answers: The only one in range is H. 0.225. Select that answer.
27. • Draw a diagram: You may find it useful to draw the football field as seen from directly overhead.
    • Ask "what if...": What if the field were 100 yards by 100 yards? Then the field would be covered to a depth of 1 yard. What if the field were 200 yards by 50 yards? Then the field would also be covered to a depth of 1 yard. What if the field were half that size, 100 yards by 50 yards? Then the field would be covered to a depth of 2 yards. Since the field is somewhat longer and wider, but not significantly longer and wider, the snow on the field must be between 1 and 2 yards deep. Therefore, the answer must be C. Between 1 and 2.
28. • Ask "what if...", or estimate the answer: If PR were 15, QR would be 9. Since PR is 16, QR must be slightly larger than 9.
    • Look at the answers: The only answer slightly larger than 9 is G. 9.6. Select G. 9.6.
29. The pictograph depicts 2.5 cars for Coupeville, and 10 cars in total. Therefore, the fraction of the cars assembled in Coupeville is 2.5/10 = 1/4. Therefore, the answer is B. ¼. Note that we don't have to convert the cars in the pictograph to the number of cars assembled.
30. Solution:
    • Estimate the answer: The diagram isn't drawn to scale, but the answer appears to be less than 1/2 of AC and around 1/3 of BD.
    • Look at the answers: G. seems reasonable, although the diagram wasn't drawn to scale to it may be useful to perform the calculation.
    • Calculate the answer: BC = AC + BD − AD = 16 + 20 − 30 = 6. Therefore, the answer is G. 6.
31. Solution 1:
    • Read the question and determine what it is asking: You are being asked, in essence, to solve 2x + 6 = 3x + 4.
    • Solve the equation:
      2x + 6 = 3x + 4
      x = 2
      Therefore, the answer is B. 2.
    Solution 2: Draw a diagram: On your graphing calculator, plot the two lines. You should be able to see that the x-coordinate where the two lines intersect is 2; therefore, the answer is B. 2.
    Solution 3: Guess and check: Substitute each answer into the two equations and see whether the y-coordinate is the same in both equations. If it is, that means that that x-coordinate represents the point of intersection. After a few tries, you should find that B. 2 is the correct answer.
32. Solution 1: Solve the equation:
    M = 3V + 6
    M − 6 = 3V
    V = (M − 6) / 3
    Therefore, the answer is J. (M − 6) / 3.
    Solution 2:
    • Try a special case: If V = 1, M = 9.
    • Guess and check: Substitute M = 9 into each answer and see which answer is equal to 1. Answer F. is equal to −3. Answer G. is equal to 9. Answer H. is equal to 21. Answer J. is equal to 1. Answer K. is equal to 5. Select answer J. (M − 6) / 3.
33. The area of a parallelogram is bh, where b is the length of the base and h is the perpendicular height. Therefore, the area is 4(3 + 6) = 36. So, the answer is B. 36.
34. Solution 1:
    • Look at the answers: The fourth power of a number could never be negative, so F. and G. are wrong. Given the simple numbers in the problem, the answer is not likely to be J., as 8 is not a fourth power of an integer. Given the 2 in the original question and the fact that −16 is a distractor, the answer is probably K.
    • Calculate the answer:
      (b − a)⁴
      = (b − (b + 2)⁴
      = (−2)⁴
      = 16
      Therefore, the answer is K. 16.
35. • Draw a diagram: The diagram is already given, but the position of the fountain is not. Draw the position of the fountain on the diagram.
    • Estimate the answer: Since D is 7 blocks to the east, the fountain has to be more than 7 blocks to the east. Since D is less than 12 blocks north, the fountain has to be less than 6 blocks north.
    • Look at the possible answers: The only answer in range of our estimate is D. 8½ blocks east and 4½ blocks north.
36. • Look at the answers: We'd expect the answer to have a 3x term in it, so F. and G. are probably wrong.
    • Convert the problem to equations: If we let y be the larger number for now, the first sentence can be written as (and be careful here) y − 2x = 8, or y = 2x + 8. The second sentence can be written as 2y + 3x = 65. Since y = 2x + 8, the second sentence can also be written as 2(2x + 8) + 3x = 65. Therefore, the answer is J. 2(2x + 8) + 3x = 65.
37. • Read the question carefully and determine what it is asking.
    • Draw a diagram: Your diagram may look something like the following. You are asked to find how far up the building the ladder reaches, which I've labelled x in the diagram.
      
    • Estimate the answer: x has to be greater than 20, due to the triangle inequality. On the other hand, x has to be less than 30, since the hypotenuse of a right-angled triangle has to be the largest side.
    • Look at the possible answers: The only answer in the range we've estimated is C. 28. Therefore, the answer must be C. 28.
38. • Estimate the answer: The area of the circle is π(7)² = 49π, which is about equal to 150. Therefore, the area of the square must be somewhat larger than that, perhaps around 200.
    • Look at the answers: The only answer larger than 49π or 150 is K. 196. Select K. 196.
39. Calculate the answer: The answer is equal to (8/12) · 15 or 10. Therefore, the answer is B. 10.0.
40. • Read the question and understand what it is asking: A trapezoid is a polygon with four sides, two of which are parallel. Looking at the diagram, BC and AD are the parallel sides. BD is a line that intersects the two parallel lines.
    • Draw a diagram: A diagram is already given. However, it may be useful to extend BC, BD, and/or AD for you to see that the problem involves a line intersecting parallel lines. Since BD intersects parallel lines, angle BDA must equal angle CBD, or 30°. Fill that in in the diagram.
    • Calculate the answer: Angle BCD = 180° − 105° − 35 ° = 45°. Therefore, the answer is K. 45°.
41. • Draw a diagram: You might notice that the perimeter of the given figure is
      
      Doing a few more steps like this, the perimeter is equal to:
    • Calculate the answer: The perimeter is equal to 2 · (26 + 4 + 6 + 4) = 80. Therefore, the answer is E. 80.
42. • Estimate the answer: If around 2/5 are going to college and around 1/4 of those are going to a state university, then about (2/5)(1/4) = (1/10) of the entire student body is going to a state university. Since there are 804 students, about 80 are going to a state university.
    • Look at the answers: Select the only reasonable answer, F. 80.
43. This question is a little easier if you've had some practice with Diophantine equations, which unfortunately most high school students haven't. However, the question yields to some logical thinking. Assuming x and y have no factors in common, the greatest common factor of x²y² and xy³ is xy². Now, 45 = 5 · 3². Since x and y are positive integers, the only wy that xy² could equal 5 · 3² is if y = 3. So, the answer must be E. 3.
44. • Estimate the answer: Perhaps the answer is about 2/3 of 460, or around 300.
    • Look at the answers: F. 280, G. 300 and H. 320 seem to be plausible.
    • Calculate the answer:
      1.15x = 460
      x = 460/1.15
      0.75x = 0.75(460/1.15) = 300
      Therefore, the answer is G. 300.
45. • Draw a diagram:
    • Estimate the answer: The answer has to be greater than 5, but less than 6.
    • Look at the possible answers: The only answer between 5 and 6 is E. √26. Select that answer.
46. Try a special case: Assume that the radius of the first circle is 4 and the radius of the second circle is 9. Then, the circumference of the first circle is 2π(4) = 8π, and the circumference of the second circle is 2π(9) = 18π. The ratio between the two is 8π:18π = 4:9. Therefore, the answer is G. 4:9.
47. • Read the question and determine what it is asking: You are asked to find the equation of a circle that passes through (5,0) and (0,5).
    • Guess and check: See which answers are satisfied by both (5,0) and (0,5). Both B. and D. meet this condition.
    • Draw a diagram: It can be helpful to draw a diagram here. This would show that the circle is centered at (5, 5). Therefore, the answer is D.
48. Don't freak out if you haven't seen complex numbers before. Just perform the calculation and keep in mind that i² = −1:
    1/(1 + i) · (1 − i) / (1 − i)
    = (1 − i) / (1 + i)(1 − i)
    = (1 − i) / (1 − i²)
    = (1 − i) / (1 − (− 1))
    = (1 − i) / 2
    Therefore, the answer is J. (1 − i) / 2.
49. Solution 1:
    • Try a special case: When n = 1, the number of dots in the first n rows of the triangle is 1.
    • Look at the possible answers: If there is 1 dot in the first row, then answers A., B., and C. are all incorrect. Answers D. and E. could be correct; we need more information.
    • Try another special case: You can continue trying further cases until you are satisfied that the correct answer is D. This total is equal to the square of the number of rows.
    Solution 2:
    • Read the problem carefully and understand what it is asking: In essence, you are being asked to find the value of 1 + 3 + 5 + ... + n.
    • Calculate the answer: The formula for the sum of an arithmetic sequence is S_n = ½n(2a + (n − 1)d, where a is the first term (1 in this case) and d the common difference between the terms (2 in this case). So, we get:
      S_n = ½n(2 + 2n − 2)
      S_n = ½n(2n)
      S_n = n²
      Therefore, the answer is D. This total is equal to the square of the number of rows.
50. • Read the question carefully and determine what it is asking.
    • Draw a diagram: Draw a diagram representing 20 students, indicate somehow that 8 of them play guitar and 9 play piano. Since we're asked to find the minimum number of people who have to play both, choose different people as far as possible. We could do that by selecting people from opposite sides of the diagram. The diagram might look somewhat like the following:
      
      The diagram shows 0 students who play both.
    • Look at the answers: Since we're looking for a minimum, start with the smallest answer. The smallest answer is 0, and we've shown that it is possible for 0 students to play both. Therefore, the answer must be F. 0.
51. Solution 1:
    • Subtract x from both sides of the inequality. You are left with 3 > 5, which is false. Therefore, no values of x satisfy the inequality and the answer is A. The empty set.
    Solution 2:
    • Try a special case: If x = 0, then the inequality becomes 3 > 5, which is false. If x = 0 does not satisfy the inequality, then B., D., and E. are incorrect, leaving just A. and C.
    • Try another special case: To see whether C. is correct, try x = −1. The inequality now becomes 2 > 4, which is still incorrect. Therefore, the answer is A. The empty set.
52. It can be tempting to select F. 8 here, but that's not the correct answer. There are a few ways to find the correct answer.
    Solution 1:
    • Draw a diagram: The octagon is already drawn, but not any of the diagonals. Draw all diagonals, counting them as you go. You should end up with a total of 20. Therefore, the answer is H. 20 (Even if you miss one or two, you should realize that there are more than 16 but well under 30, so you would probably guess answer H. anyway).
    Solution 2:
    • Draw a diagram: Select any one vertex of the octagon, and draw all diagonals from it. There are 5 diagonals in total. Each of the 8 vertices will also have five diagonals, for a total of 40. Don't stop here though! Since the diagonal from point A to point B is the same diagonal as the one from point B to point A, so we'll need to divide by 2 to get the correct answer. The correct answer is H. 20.
53. • Read the question and determine what it is asking: You are asked to find 155% × 360°.
    • Draw a diagram: It can be helpful to draw a diagram here:
    • Estimate the answer: 25% of 360° is 90°, so the answer should probably be greater than 45° but less than 60°.
    • Look at the answers: The only answer in range is B. 54°. Select this answer.
54. • Draw a diagram: It can be helpful to draw a diagram here.
    • Calculate the answer: In the third quadrant, tan is positive. So, the answer must be J. ¾.
55. Look at the answers: Use process of elimination. The equations must be connected with an "and", otherwise quite a lot of area would be shaded. This eliminates B. and D. One of the equations must be x ≥ 3, so that eliminates E. The other must be y ≤ −2x, so that eliminates C. Therefore, the answer must be A. y ≤ −2x and x ≥ 3.
56. Don't panic if you haven't taken calculus. Just do what the question asks:
    f(x + h) = (x + h)² − 2
    = x² + 2xh + h² − 2
    Therefore, the answer is K. x² + 2xh + h² − 2.
57. One way of getting the answer is to simplify the equation, resulting in y = 2x + 1 where x ≠ 0. This corresponds to answer A.
58. • Draw a diagram. The question doesn't specify what type of triangle ABC is, or where the points are located. So, you can draw a diagram with the points located anywhere.
    • Try a special case: Say that c = 2 and d = 2 (you can, of course, choose any numbers). A′ would then have an x-coordinate of 2 and a y-coordinate of −2. This corresponds to answer F.. Thinking about this, you can see that this answer will always be true. Therefore, the answer is F. (c, −d).
59. • Try a special case: When t = 4, x = −1 and y = 1.
    • Guess and check: Since y = 1 when x = −1, check all answers, looking for one that gives y = 1 when x = −1. The only answer that fits is A. y = (1 − x)/2.
60. Solution 1:
    • Calculate the decimal value of sin(π/12) on your calculator. The result is 0.2588...
    • Look at the possible answers: F. and G. are obviously not equal to 0.2588... H. is negative, so that can't be the answer either.
    • Guess and check: Evaluate J. and K. to determine which is equal to 0.2588... The correct answer is K. (√6 − √2)/4.
    Solution 2:
    • Estimate the answer: There are a few ways of doing this:
      • Calculate the decimal value of the answer as in solution 1.
      • Since sin 0 = 0 and sin π/6 = ½, sin π/12 must be somewhere between the two, say around ¼.
      • You may remember that sin x = x for small values of x, so the value of sin π/12 is around π/12, or around ¼.
    • Look at the answers: F. ¼ and K. (√6 − √2)/4 appear to be the only reasonable answers.
    • Calculate the answer: The question provides the information needed to calculate the answer. Let α = π/3 and β = π/4 and substitute into the equation given:
      sin(π/12) = sin(π/3 − π/4) = (sin π/3)(cos π/4) − (cos π/3)(sin π/4)
      = (√3/2)(√2/2) − (½)(√2/2)
      = (√6 − √2)/4
      Therefore, the answer is K. (√6 − √2)/4.

Next test in this book: test 2.