Percentages & Percent Reasoning
A complete, test-focused breakdown of every Percentages concept on the ACT — from the three core question types to percent change, successive percents, reverse percent problems, compound interest, and real-world percent word problems — with worked examples and strategy notes for every question type you'll encounter.
In This Lesson
- 01 Percent Foundations & Conversions
- 02 The Three Core Percent Questions
- 03 Percent Change (Increase & Decrease)
- 04 The Multiplier Method
- 05 Successive Percent Changes
- 06 Reverse Percent Problems
- 07 Percent of a Total & Part/Whole
- 08 Tax, Tip, Discount & Markup
- 09 Simple & Compound Interest
- 10 Percent Word Problems
- 11 Test Day Strategy
- → Practice Quizzes
Percent Foundations & Conversions
Percent means "per hundred" — it is simply a ratio with a denominator of 100. Every percentage problem ultimately involves converting between percent, decimal, and fraction form.
Decimal ↔ Percent: Multiply by 100 (move decimal 2 places RIGHT)
Percent ↔ Fraction: P% = P/100 (then simplify)
// 45% = 0.45 = 45/100 = 9/20
Must-Know Conversions
Mental math shortcut — 10% trick: To find 10% of any number, move the decimal one place left. Then use it to build other percentages: 5% = half of 10%; 20% = double 10%; 15% = 10% + 5%; 30% = 3 × 10%. This is faster than multiplying on most ACT problems.
The Three Core Percent Questions
Every basic percent problem on the ACT is one of three types. Identify the type, plug into the formula, solve. All three come from the same master equation: Part = Percent × Whole.
// Rearrange to solve for any of the three unknowns
What is 35% of 80?
= 0.35 × 80 = 28
18 is what % of 72?
= (18/72) × 100 = 25%
24 is 60% of what?
= 24 / 0.60 = 40
Universal setup — "IS over OF": Any percent problem can be set up as:
(IS / OF) = (Percent / 100)
"IS" is the part you're looking at. "OF" is the whole. Cross-multiply to solve for whichever is unknown.
Percent Change (Increase & Decrease)
Percent change measures how much a quantity has changed relative to its original value. The original value is always the denominator — this is where most errors occur.
// Positive result → increase. Negative result → decrease.
// Always divide by the ORIGINAL (starting) value — never the new value.
Original vs. New — the most common ACT trap: "A price increased from $50 to $60, what percent of the NEW price is the increase?" → 10/60 = 16.7%. "What percent INCREASE?" → 10/50 = 20%. The denominator changes the answer completely. Always identify what the base is before dividing.
The Multiplier Method
The multiplier method is the fastest way to apply a percent increase or decrease — instead of calculating the change and adding it, you multiply by a single number. This is especially powerful for multi-step percent problems.
Increase by P%: multiply by (1 + P/100)
Decrease by P%: multiply by (1 − P/100)
The multiplier captures the original 100% plus or minus the change — in one step.
Common Multipliers to Memorize
The multiplier method is faster than the two-step approach on the ACT. Instead of finding 15% of $240 ($36) and then subtracting ($240 − $36 = $204), you multiply once: $240 × 0.85 = $204. Save 10 seconds per problem — that adds up across the test.
Successive Percent Changes
When a value undergoes multiple percent changes in sequence, the changes compound — each one applies to the result of the previous change, not the original value. This makes successive percents one of the most heavily tested and most misunderstood ACT topics.
Chain-multiply the multipliers:
Final Value = Original × (Multiplier₁) × (Multiplier₂) × (Multiplier₃) × …
Never add or subtract the percentages directly. A 30% increase followed by a 30% decrease does NOT return to the original — you lose money.
Why P% up then P% down always loses money: The increase applies to a smaller base, but the decrease applies to a larger base. After a 25% increase on $100, you have $125. A 25% decrease on $125 removes $31.25 — bringing you to $93.75, not back to $100. The loss is always (P/100)² of the original.
Reverse Percent Problems
A reverse percent problem gives you the value after a percent change and asks for the value before. This is Type C thinking applied to percent change — and the most common ACT percent trap.
If a price is $80 after a 20% discount, the original is NOT $80 + 20% of $80 = $96.
The 20% was taken off the original, not the sale price. You must divide by the multiplier:
Original = Final Value ÷ Multiplier
The wrong approach everyone tries: "Price is $56 after 20% off, so add back 20% of $56 = $11.20, giving $67.20." This is wrong because 20% of $56 ≠ 20% of the original. Always divide by the multiplier — never add a percent of the sale price back.
Percent of a Total & Part/Whole Reasoning
Many ACT percent problems involve identifying what is the "part" and what is the "whole" in a real-world context. Getting this wrong is the single biggest source of errors on percentage problems.
Identifying Part and Whole
"30% of the budget was spent" → budget = whole
"The amount spent" → amount = part
Nested percent shortcut: When you need "X% of Y% of the whole," you can multiply the percents directly: 60% × 45% = 27% of the whole. This works because percentage multiplication is associative — no need to find the intermediate group size.
Tax, Tip, Discount & Markup
These are the most common real-world applications of percentages on the ACT. Each follows the same multiplier logic — know the formula for each and you'll never be stuck.
| Situation | Formula | Multiplier | Example |
|---|---|---|---|
| Tax | Final = Original × (1 + tax rate) | 1 + r | $50 + 8% tax = $50 × 1.08 = $54 |
| Tip | Total = Bill × (1 + tip rate) | 1 + r | $80 bill + 20% tip = $80 × 1.20 = $96 |
| Discount | Sale = Original × (1 − discount rate) | 1 − r | $120 − 30% off = $120 × 0.70 = $84 |
| Markup | Selling Price = Cost × (1 + markup rate) | 1 + r | Cost $60, markup 40% → $60 × 1.40 = $84 |
Order matters for stacked operations! A 25% discount followed by 8% tax gives a different answer than 8% tax followed by a 25% discount on the taxed amount. On the ACT, always apply operations in the order stated in the problem.
Simple & Compound Interest
Simple Interest
Simple interest is calculated on the original principal only — interest earned in earlier periods does not earn additional interest.
Total Amount (A) = P + I = P(1 + rt)
// P = principal, r = annual interest rate (decimal), t = time (years)
Compound Interest
Compound interest earns interest on both the principal and previously earned interest. It grows faster than simple interest because each period's interest becomes part of the next period's base.
// P = principal, r = annual rate (decimal), n = compounding periods per year, t = years
// Annual: n=1 · Semi-annual: n=2 · Quarterly: n=4 · Monthly: n=12 · Daily: n=365
Compound Interest — Year by Year (Simple Version)
For small numbers of years and annual compounding, the ACT may expect you to build a year-by-year table. Each year's amount = previous year × (1 + r).
$1,000 at 10% annual compound interest for 3 years = $1,331. Simple interest would give only $1,300.
Simple interest: $1,000 × 10% × 3 years = $300 interest → $1,300 total.
Compound interest: $1,000 × (1.10)³ = $1,331 total → $331 interest.
Compound interest earns $31 more because interest earned in years 1 and 2 also earns interest in later years.
Percent Word Problems
Strategy: Label Everything Before Calculating
On percent word problems, identify three things before touching the calculator: (1) What is the whole? (2) What is the part? (3) What is the percent? Then plug into Part = Percent × Whole.
Test Day Strategy
Top ACT Percent Traps
- Wrong base for percent change: always divide by the ORIGINAL, never the new value.
- Successive percents: 20% up + 20% down ≠ 0% — always chain-multiply multipliers.
- Reverse percent: don't add a% of the sale price to find the original — divide by the multiplier.
- Average of percents: if group sizes differ, you can't average percent rates directly.
- Percent vs. percentage points: "rose from 20% to 25%" is a 5 percentage point increase, but a 25% relative increase.
- Compound vs. simple interest: only compound interest earns interest on interest — know the difference.
Decision Framework
- Find P% of W → multiply: W × (P/100)
- X is what % of W? → divide: (X/W) × 100
- X is P% of what? → divide: X ÷ (P/100)
- Apply percent change → use multiplier method: × (1 ± r)
- Multiple changes → chain-multiply all multipliers
- Find original from final → divide final by multiplier
- Interest problem → identify simple (I=Prt) vs. compound (A=P(1+r/n)^nt)
Complete Formula Reference
| Concept | Formula | Watch-out |
|---|---|---|
| Part of a whole | Part = (P/100) × Whole | Identify what the "whole" is first |
| Percent of a total | % = (Part / Whole) × 100 | The "OF" value is the denominator |
| Reverse: find whole | Whole = Part ÷ (P/100) | Divide by the percent, don't subtract |
| Percent change | % = [(New − Old)/Old] × 100 | Old (original) in denominator |
| New value after increase | New = Old × (1 + r) | r is decimal form of rate |
| New value after decrease | New = Old × (1 − r) | r is decimal form of rate |
| Successive changes | Final = Original × m₁ × m₂ × … | Never add/subtract the percents |
| Reverse percent | Original = Final ÷ Multiplier | Don't apply % to the wrong value |
| Nested percents | P% of Q% of W = (P×Q)/10000 × W | Multiply decimal forms directly |
| Simple interest | I = Prt ; A = P(1 + rt) | Interest on principal only |
| Compound interest | A = P(1 + r/n)^(nt) | n = compounds per year |
Put It to the Test
You've mastered every percentage concept the ACT tests. Now build speed and accuracy with 5 quizzes of ACT-style timed questions. Work through each quiz in order — difficulty increases progressively.