ACT Math: Set Up Percentage Problems Using the Percent Equation

Every percentage problem reduces to one formula: Part=Percent (as decimal)×Whole. Example: "What is 15% of 200?" Part=0.15×200=30. Another: "30 is what percent of 200?" 30=x×200, so x=30/200=0.15=15%. Another: "30 is 15% of what number?" 30=0.15×x, so x=30/0.15=200. These three questions use the same formula rearranged; learn one format and you have all three.

Translation guide: "What is P% of W?" means Part=P%×W. "X is P% of what?" means X=P%×W, solve for W. "X is what % of W?" means X=%×W, solve for %. Percentage increase/decrease: New=(100%±change%)×Old. Example: "A price of $50 increases by 20%." New=120%×50=1.2×50=$60. Example: "A price decreases by 20%." New=80%×50=0.8×50=$40. The equation stays the same; only the percentage changes.

Trap 1: Forgetting to convert percentage to decimal. 15% means 0.15, not 15. Trap 2: Confusing successive percentages. "A 20% discount, then a 10% discount" is not 30% off total. First discount: 0.80×price. Second: 0.90×(0.80×price)=0.72×price, or 28% off. Trap 3: Mixing up increase and decrease. A 50% increase means ×1.5; a 50% decrease means ×0.5. Trap 4: Percentage vs. percentage point. "Unemployment rose from 5% to 7%" is a 2 percentage point increase, but a 40% relative increase (2/5=0.4). ACT usually asks for the latter. Always re-read the question to confirm whether it asks for absolute change or relative percent change.

Example Trap 2: A shirt costs $100. Sale: 40% off. New price=$100×0.6=$60. Then 10% off the sale price: $60×0.9=$54. Total discount: ($100-$54)/$100=46%, not 50%.

Problem 1: What is 25% of 80? Problem 2: 15 is 30% of what number? Problem 3: 45 is what percent of 150? Problem 4: A price of $120 increases by 15%. New price? Problem 5: A $200 item is discounted by 20%, then by 10% off the sale price. Final price? Problem 6: Sales rose from $50,000 to $60,000. Percent increase? Problem 7: A test score of 72 is 90% of the maximum. Maximum score? Problem 8: Tax adds 8% to a $50 purchase. Total cost? For each, (1) identify the equation form (Part=Percent×Whole), (2) rearrange if needed, (3) solve, (4) check. Do this drill without a calculator first, then verify with one.

Answers: 1) 0.25×80=20. 2) 15=0.30×x, x=50. 3) 45=x×150, x=0.30=30%. 4) 1.15×120=$138. 5) 0.80×0.90×200=$144. 6) (60000-50000)/50000=20%. 7) 72=0.90×x, x=80. 8) 1.08×50=$54.

Percentage problems appear on every ACT Math section in word-problem, data-interpretation, and algebra contexts. The formula Part=Percent×Whole solves them all instantly. One hour spent drilling this formula will earn you 2-3 points on test day because percentage competency transfers across topics.

Commit to the eight-problem drill twice this week. By test day, you will recognize percentage problems instantly and solve them in 30 seconds without guessing.