ACT Math study guide · Chapter 38
Rates, Percentages, and Proportions
Percent change is the change in value divided by the original value, times 100: (new − old) / old × 100. This is different from a percentage point difference, which is the raw subtraction of two percentages. A tax rate rising from 10% to 15% increased by 5 percentage points, but the percent increase relative to the original rate is 50%. Reading which one a question asks for is the actual skill.
A proportion can represent a direct or an inverse relationship, and mixing them up is the most common error on this topic. A direct proportion, like a recipe scaled up, keeps a fixed ratio: more of one quantity means more of the other. An inverse proportion, like workers finishing a job, has one quantity increase while the other decreases: more workers means less time, at a fixed rate of work.
Recovering an original value from a percent increase or decrease means dividing by the growth factor, not applying the opposite percentage. A price increased 25% to reach $150 has an original value of 150 ÷ 1.25, not 150 × 0.75.
Where this appears as a model: a value that changes over time, stated in words instead of as a ready-made percent. A price that drops from $40 to $32 has decreased by 20%, found by dividing the $8 decrease by the original $40.
▸ Percentage points versus percent change
Worked Example A defect rate falls from 8% to 5%. Find both the percentage-point decrease and the percent decrease relative to the original rate.
These are two different questions with two different answers.
Percentage-point decrease: 8 − 5 = 3 percentage points
Percent decrease: (8 − 5)/8 × 100 = 37.5%
▸ Inverse proportions
Worked Example 4 machines can complete a production run in 9 hours. How many hours would 6 machines take, at the same rate?
More machines means less time, an inverse proportion.
4 × 9 = 6 × t
t = 36/6 = 6
▸ Practice
Question 1
- AA. 15%
- BB. 20%
- CC. 18.75%
- DD. 25%
Question 2
- FF. 20% decrease
- GG. 25% decrease
- HH. 30% decrease
- JJ. 20% increase
Question 3
- AA. 20
- BB. 25
- CC. 32
- DD. 16
Question 4
- FF. 50
- GG. 64
- HH. 60
- JJ. 70
Question 5
- AA. 50%
- BB. 6
- CC. 0.06
- DD. 12
Question 6
- FF. 10.67
- GG. 6.5
- HH. 5.33
- JJ. 6
Question 7
- AA. 117
- BB. 104
- CC. 130
- DD. 95
Question 8
- FF. 28%
- GG. 40%
- HH. 34%
- JJ. 45%
Question 9
- AA. 44
- BB. 36
- CC. 40
- DD. 48
Question 10
- FF. 13
- GG. 72.2
- HH. 72.2%
- JJ. 18
Question 11
- AA. 35
- BB. 40
- CC. 50
- DD. 45
Question 12
- FF. 25
- GG. 9
- HH. 10
- JJ. 12.5
Question 13
- AA. $134.40
- BB. $108
- CC. $120
- DD. $112
Question 14
- FF. $72
- GG. $80
- HH. $70.40
- JJ. $76.80
Question 15
- AA. 187
- BB. 170
- CC. 200
- DD. 165
Question 16
- FF. 25%
- GG. 18%
- HH. 20%
- JJ. 22%
Question 17
- AA. 58
- BB. 60
- CC. 62
- DD. 64
Question 18
- FF. 57
- GG. 13
- HH. 1.6
- JJ. 22
Question 19
- AA. 35
- BB. 40
- CC. 45
- DD. 30
Question 20
- FF. 32.4
- GG. 25
- HH. 10
- JJ. 9
▸ Answers and Explanations
Show the answers and explanations
1: D. (75−60)÷60×100 = 15÷60×100 = 25%. Choice A reports the raw dollar increase ($15) instead of converting it to a percent of the original value.
2: F. (120−150)÷150×100 = −30÷150×100 = −20%, a 20% decrease. Choice G divides by the new value (120) instead of the original value (150).
3: B. Cross-multiply: 8x=5(40)=200, so x=25. Choice A comes from a cross-multiplication slip instead of 8x=200.
4: H. 240 ÷ 4 = 60 miles per hour. Choice F comes from an arithmetic slip while dividing 240 by 4.
5: B. A percentage point difference is the raw subtraction: 18−12=6. Choice A computes the percent increase relative to the original value instead of the raw percentage-point difference.
6: J. More pumps means less time, an inverse proportion: 3×8=4×t, so t=24/4=6. Choice F sets up a direct proportion instead of recognizing that more pumps means less time, an inverse relationship.
7: A. 0.45×260=117. Choice B comes from an arithmetic slip while computing 0.45×260.
8: G. (119−85)÷85×100=40%. Choice F comes from an arithmetic slip while computing the percent increase.
9: A. 396÷9=44. Choice B comes from an arithmetic slip while dividing 396 by 9.
10: F. 31−18=13 percentage points. Choice G computes the percent increase relative to the original value instead of the raw percentage-point difference.
11: D. Cross-multiply: 14x=9(70)=630, so x=45. Choice A comes from a cross-multiplication slip instead of 14x=630.
12: G. More workers take less time, an inverse proportion: 6×15=10×t, so t=90/10=9. Choice F sets up a direct proportion instead of recognizing that more workers means less time, an inverse relationship.
13: C. 1.40×original=168, so original=168÷1.40=120. Choice A applies a percent reduction to the final price instead of dividing by 1.40 to recover the original.
14: J. Unit cost=32/5=6.40; 6.40×12=76.80. Choice F comes from an arithmetic slip while computing the unit cost.
15: A. 0.55×340=187. Choice B comes from an arithmetic slip while computing 0.55×340.
16: F. (90−72)÷72×100=25%. Choice G comes from an arithmetic slip while computing the percent increase.
17: C. 372÷6=62. Choice A comes from an arithmetic slip while dividing 372 by 6.
18: G. 35−22=13 percentage points. Choice F adds the two percentages instead of subtracting to find the point difference.
19: B. Cross-multiply: 15x=8(75)=600, so x=40. Choice A comes from a cross-multiplication slip instead of 15x=600.
20: H. More workers take less time, an inverse proportion: 5×18=9×t, so t=90/9=10. Choice F sets up a direct proportion instead of recognizing that more workers means less time, an inverse relationship.