ACT Math study guide · Chapter 41
Multi-Step Problems
A multi-step chain combines several ordinary skills in sequence: a percent change followed by another percent change, a recipe scaled up and then converted to different units, an average recomputed after a new value is added. No single step is difficult; the actual skill is completing every step in the chain instead of stopping after the first one.
Two sequential percent changes do not combine into a single simple percentage. A 20% increase followed by a 10% decrease is not a net 10% increase; each percentage applies to the new value from the previous step, not the original value. The two multipliers, 1.20 and 0.90, must be applied one after the other.
When a question asks for a value in different units than the ones given, that conversion is a required final step, not an optional afterthought. A chain that scales a recipe and then asks for the answer in a different unit, or computes a distance in miles and asks for it in kilometers, is not finished until that last conversion is applied.
▸ Sequential percent changes
Worked Example A $200 item is increased 15%, then decreased 20% from the new price. Find the final price.
Apply each percentage to the result of the previous step, not to the original price.
After the increase: 200 × 1.15 = 230
After the decrease: 230 × 0.80 = 184
▸ Scaling and converting in the same chain
Worked Example A recipe for 6 servings needs 3 cups of sugar. Scaled to 15 servings, find the amount of sugar needed in tablespoons. (1 cup = 16 tablespoons)
Scale the recipe first, then convert the result to the requested unit.
Scaled sugar: 3 × (15/6) = 7.5 cups
Converted: 7.5 × 16 = 120 tablespoons
▸ Practice
1. A $50 item is increased 20%, then decreased 10% from the new price. Find the final price.
A) $54 B) $55 C) $50 D) $60
2. A cyclist holds a pace of 8 minutes per mile for 3.5 miles. Find the total time in minutes.
F) 24.5 G) 28 H) 31.5 J) 3.5
3. Five quiz scores average 82. A sixth score of 94 is added. Find the new average of all six scores.
A) 86 B) 88 C) 82 D) 84
4. A recipe for 4 servings needs 2 cups of flour. Scaled to 10 servings, how many fluid ounces of flour are needed? (1 cup = 8 fluid ounces)
F) 40 G) 13 H) 5 J) 64
5. A trip covers 150 miles at 60 miles per hour, then 90 miles at 45 miles per hour. Find the average speed for the entire trip.
A) 52.5 B) 53.33 C) 50 D) 55
6. A store marks up its cost by 40% to set a retail price of $84. Find the original cost.
F) $60 G) $50.40 H) $58.80 J) $117.60
7. A $90 item is increased 30%, then decreased 10% from the new price. Find the final price.
A) $117 B) $105.30 C) $99 D) $90
8. A cyclist holds a pace of 6.5 minutes per mile for 5.2 miles. Find the total time in minutes.
F) 31.5 G) 37.5 H) 26 J) 33.8
9. Six exams average 78. A seventh exam score of 95 is added. Find the new average of all seven exams, rounded to the nearest hundredth.
A) 78 B) 85 C) 80.43 D) 83
10. A recipe for 8 servings needs 4 cups of flour. Scaled to 20 servings, how many tablespoons of flour are needed? (1 cup = 16 tablespoons)
F) 10 G) 160 H) 80 J) 320
11. A store marks up its cost by 45% to set a retail price, then discounts 20% from that retail price. If the original cost was $80, find the final price.
A) $116 B) $92.80 C) $80 D) $76.80
12. Three tests are weighted 20% each for the first two and 60% for the third. The scores are 75, 88, and 84. Find the weighted average.
F) 80 G) 86 H) 78 J) 83
13. A $150 item is discounted 25%, then 8% sales tax is added to the discounted price. Find the final price.
A) $121.50 B) $112.50 C) $108 D) $162
14. A car travels at 60 miles per hour for 2.5 hours. Find the distance traveled in kilometers, rounded to the nearest tenth. (1 mile ≈ 1.60934 kilometers)
F) 241.4 G) 150.0 H) 96.6 J) 200.0
15. A $120 item is increased 15%, then decreased 25% from the new price. Find the final price.
A) $117 B) $99 C) $103.50 D) $90
16. A runner holds a pace of 7.5 minutes per mile for 4 miles. Find the total time in minutes.
F) 26.5 G) 30 H) 32.5 J) 27.5
17. Five exams average 85. A sixth exam score of 97 is added. Find the new average of all six exams.
A) 85 B) 90 C) 82 D) 87
18. A recipe for 10 servings needs 5 cups of flour. Scaled to 25 servings, how many cups of flour are needed?
F) 10 G) 15 H) 12.5 J) 20
19. A store marks up its cost by 35% to set a retail price, then discounts 15% from that retail price. If the original cost was $90, find the final price.
A) $103.28 B) $90 C) $121.50 D) $76.50
20. A car travels at 55 miles per hour for 3 hours. Find the distance traveled in kilometers, rounded to the nearest tenth. (1 mile ≈ 1.60934 kilometers)
F) 165 G) 88.5 H) 265.5 J) 200
▸ Answers and Explanations
1: A. After the increase: 50×1.20=60; after the decrease: 60×0.90=54. Choice B comes from an arithmetic slip in the two-step chain.
2: G. 8 × 3.5 = 28 minutes. Choice F comes from an arithmetic slip while multiplying the pace by the distance.
3: D. Total of the first five: 82×5=410; new total: 410+94=504; new average: 504÷6=84. Choice A comes from an arithmetic slip while recomputing the average with the new score.
4: F. Scaled flour: 2×(10/4)=5 cups; converted to fluid ounces: 5×8=40. Choice G stops after scaling the recipe, forgetting to complete the unit conversion to fluid ounces.
5: B. Time: 150/60=2.5 hr and 90/45=2 hr; average speed = 240/4.5 ≈ 53.33. Choice A averages the two speeds directly instead of using total distance divided by total time.
6: F. cost × 1.40 = 84, so cost = 84 ÷ 1.40 = 60. Choice G applies a percent reduction to the retail price instead of dividing by the markup factor to recover the cost.
7: B. After the increase: 90×1.30=117; after the decrease: 117×0.90=105.30. Choice A reports the price after only the first step, forgetting to apply the second.
8: J. 6.5×5.2=33.8 minutes. Choice F comes from an arithmetic slip while multiplying the pace by the distance.
9: C. Total of the first six: 78×6=468; new total: 468+95=563; new average: 563÷7≈80.43. Choice A reports the original average, forgetting to recompute after adding the new score.
10: G. Scaled flour: 4×(20/8)=10 cups; converted to tablespoons: 10×16=160. Choice F stops after scaling the recipe, forgetting to complete the unit conversion to tablespoons.
11: B. Retail: 80×1.45=116; after the discount: 116×0.80=$92.80. Choice A reports the price after only the first step (the markup), forgetting to apply the discount.
12: J. 0.20(75)+0.20(88)+0.60(84)=15+17.6+50.4=83. Choice F comes from an arithmetic slip while combining the three weighted scores.
13: A. After the discount: 150×0.75=112.50; after tax: 112.50×1.08=$121.50. Choice B reports the price after only the discount, forgetting to add the sales tax.
14: F. Distance in miles: 60×2.5=150; converted to kilometers: 150×1.60934≈241.4. Choice G reports the distance in miles, forgetting to convert to kilometers.
15: C. After the increase: 120×1.15=138; after the decrease: 138×0.75=$103.50. Choice A reports the price after only the first step, forgetting to apply the second.
16: G. 7.5×4=30 minutes. Choice F comes from an arithmetic slip while multiplying the pace by the distance.
17: D. Total of the first five: 85×5=425; new total: 425+97=522; new average: 522÷6=87. Choice A reports the original average, forgetting to recompute after adding the new score.
18: H. Scaled flour: 5×(25/10)=12.5 cups. Choice F comes from an arithmetic slip while scaling the recipe to the new serving size.
19: A. Retail: 90×1.35=121.50; after the discount: 121.50×0.85=$103.28. Choice B reports the original price, forgetting to apply either the markup or the discount.
20: H. Distance in miles: 55×3=165; converted to kilometers: 165×1.60934≈265.5. Choice F reports the distance in miles, forgetting to convert to kilometers.