ACT Math: Percentages
Percentages appear throughout the ACT Math test, from quick single-step calculations to multi-part word problems about discounts, tax, and repeated changes. This free, complete lesson covers percent basics (finding a percent of a number, and reverse percent problems), percent increase and decrease, discounts, markups, and tax, and multi-step successive percent problems, where more than one percent change is applied in sequence. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback, in the same four-answer-choice, calculator-allowed format as the current ACT. Everything here is free, and you can keep practicing afterward with the full ACT Math Question Bank linked below.
1. Percent Basics
"Percent" means "per hundred." To find a percent of a number, convert the percent to a decimal and multiply. To find what percent one number is of another, divide and convert to a percent. To find the original number when you know a percent of it, divide by the decimal form of that percent.
Worked Example: What is 35% of 80?
0.35 × 80 = 28
1. What is 40% of 150?
Explanation: 0.40 × 150 = 60.
2. 24 is what percent of 80?
Explanation: 24 ÷ 80 = 0.30 = 30%.
3. 45 is 15% of what number?
Explanation: 45 ÷ 0.15 = 300.
4. Convert 3/8 to a percent.
Explanation: 3 ÷ 8 = 0.375 = 37.5%.
5. What is 125% of 40?
Explanation: 1.25 × 40 = 50. A percent greater than 100% simply means the result is larger than the original number.
2. Percent Increase and Decrease
percent change = (new value − original value)/(original value) × 100%
The denominator is always the original value, not the new one, this matters more than it might seem.
Worked Example: A shirt's price increased from $40 to $52. What is the percent increase?
Increase: 52 − 40 = 12.
12 ÷ 40 = 0.30 = 30%
1. A population grows from 200 to 250. What is the percent increase?
Explanation: Increase: 250 − 200 = 50. Percent increase: 50 ÷ 200 = 0.25 = 25%.
2. A stock price falls from $80 to $68. What is the percent decrease?
Explanation: Decrease: 80 − 68 = 12. Percent decrease: 12 ÷ 80 = 0.15 = 15%.
3. If a value increases by 20% and the new value is 96, what was the original value?
Explanation: A 20% increase means the new value is 120% of the original: original × 1.20 = 96, so original = 96 ÷ 1.20 = 80.
4. A price decreases by 25% to $60. What was the original price?
Explanation: A 25% decrease means the new price is 75% of the original: original × 0.75 = 60, so original = 60 ÷ 0.75 = 80.
5. A quantity increases from 50 to 70, then later decreases back from 70 to 50. Are the percent increase and percent decrease equal?
Explanation: The percent increase is 20 ÷ 50 = 40%. The percent decrease is 20 ÷ 70 ≈ 28.6%. Since percent change always uses the original (starting) value as the denominator, and 70 is larger than 50, the decrease percentage comes out smaller.
3. Discounts, Markups, and Tax
A discount is a percent decrease from a listed price. A markup is a percent increase from a cost price. Sales tax is a percent added on top of a price. All three use the same percent increase/decrease relationships from the previous section.
Worked Example: A $60 jacket is on sale for 30% off. What is the sale price?
discount = 0.30 × 60 = 18
sale price = 60 − 18 = 42
1. A $150 television is discounted by 20%. What is the sale price?
Explanation: Discount: 0.20 × 150 = 30. Sale price: 150 − 30 = 120.
2. A store marks up the wholesale price of an item by 40%. If the wholesale price is $25, what is the retail price?
Explanation: Markup: 0.40 × 25 = 10. Retail price: 25 + 10 = 35.
3. A meal costs $45 before an 8% sales tax. What is the total cost including tax?
Explanation: Tax: 0.08 × 45 = 3.60. Total: 45 + 3.60 = 48.60.
4. A jacket originally priced at $80 is on sale for $56. What percent discount was applied?
Explanation: Discount amount: 80 − 56 = 24. Percent discount: 24 ÷ 80 = 0.30 = 30%.
5. An item's price after a 15% discount is $85. What was the original price?
Explanation: A 15% discount leaves 85% of the original price: original × 0.85 = 85, so original = 85 ÷ 0.85 = 100.
4. Multi-Step and Successive Percent Problems
When more than one percent change is applied in sequence, apply each one to the result of the previous step, never add the percents together and apply them all at once, since that gives the wrong answer.
Worked Example: A $100 item is discounted 20%, then an additional 10% is taken off the new (already discounted) price. What is the final price?
After first discount: 100 × 0.80 = 80
After second discount: 80 × 0.90 = 72
1. A $200 item is discounted 25%, then an additional 20% is taken off the new price. What is the final price?
Explanation: After the first discount: 200 × 0.75 = 150. After the second discount: 150 × 0.80 = 120.
2. A price increases by 10%, then decreases by 10%. Is the final price equal to the original price?
Explanation: Starting from x: x × 1.10 × 0.90 = 0.99x, which is 99% of the original, slightly less. The 10% decrease is applied to a larger number than the original, so it removes more than the 10% increase added.
3. A population increases by 20% one year and then by another 20% the next year. If the initial population was 500, what is the population after two years?
Explanation: After year one: 500 × 1.20 = 600. After year two: 600 × 1.20 = 720.
4. A worker's salary increases by 10%, and then the new salary is decreased by 10%. If the original salary was $50,000, what is the final salary?
Explanation: After the raise: 50,000 × 1.10 = 55,000. After the decrease: 55,000 × 0.90 = 49,500.
Common Mistakes to Avoid
- Using the wrong value as the base of a percent change. Percent increase or decrease is always calculated relative to the original (starting) value, not the new value.
- Adding successive percent changes together. A 20% discount followed by a 10% discount is not a 30% discount overall, since the second discount applies to an already-reduced price. Apply each step in sequence instead.
- Assuming an increase and an equal decrease cancel out. Increasing by 10% and then decreasing by 10% does not return you to the original value, because the second percent is applied to a different base amount.
- Confusing "percent of" with "percent increase/decrease from." "What is 30% of 80" is a different calculation from "80 increased by 30%," even though both involve the same numbers.
- Forgetting to divide (not multiply) when solving a reverse percent problem. If you know a percent of a number and want the original number back, divide by the decimal form of the percent, don't multiply.
Frequently Asked Questions
What's the fastest way to handle a percent increase or decrease?
Convert directly to a single multiplier: a 30% increase means multiplying by 1.30, and a 30% decrease means multiplying by 0.70. This skips the extra step of calculating the change amount separately and adding or subtracting it.
How do I know whether a percent problem needs multiple steps?
Watch for language like "then," "an additional," or a second discount or markup applied after the first. Any time a second percent change is based on a new, already-changed value, you're dealing with a successive percent problem rather than a single calculation.
Why doesn't a percent increase followed by an equal percent decrease return the original value?
Because the base amount changes between the two steps. The increase is calculated on the smaller, original number, while the decrease is calculated on the larger, already-increased number, so the decrease removes more in absolute terms than the increase added.
Where can I practice more problems like these?
The Percentages quizzes in the ACT Math Question Bank include additional original problems on this topic, along with quizzes covering every other topic tested on the ACT Math section.