Percentages
Percentages are one of the most-tested topics on the digital SAT — and one of the most missed, because the SAT writes percent questions specifically to catch students who rush. This note covers the four flavors that appear: basic percent of, percent change, multi-step percent (percent of a percent), and exponential percent change. The recurring SAT trap: the test deliberately writes wrong-answer choices that match what you get if you set up the problem one step incorrectly.
The percent translation
The single most useful technique for percentage problems is translating English directly into algebra. Three keywords always mean the same thing:
| English | Algebra |
|---|---|
| "of" | × (multiply) |
| "is" / "are" / "equals" | = |
| "what" / "what number" | x (the unknown) |
| "percent" or "%" | ÷ 100 (divide by 100) |
Read the problem from left to right, replacing each keyword with its algebraic equivalent. The result is an equation you can solve.
Percent change
Percent change measures how much a quantity grew or shrank, relative to its starting value. The denominator is always the original (old) value, not the new one. This is the most-missed setup error.
A stock price rose from $40 to $50. What is the percent increase?
A jacket originally priced at $80 is on sale for $60. What is the percent decrease?
Instead of computing the change and adding/subtracting, multiply the original by a single number:
To increase by N%: multiply by (1 + N/100). To increase by 25%: × 1.25.
To decrease by N%: multiply by (1 − N/100). To decrease by 25%: × 0.75.
This is faster on the SAT and unlocks the multi-step problems below.
Multi-step percent problems
The SAT's hardest percent problems chain two or more percent changes together. The trap: percent changes do NOT simply add or subtract.
A 20% increase followed by a 20% decrease is NOT 0% change.
Always multiply the changes; never add them.
For multiple percent changes applied in sequence, multiply all the multipliers together (in order). Each multiplier is applied to the result of the previous step, not the original.
A computer originally costs $1,200. The price is increased by 10%, then later decreased by 15% from the new price. What is the final price?
In a class, 60% of students are juniors. Of those juniors, 25% are taking calculus. What percent of the entire class are juniors taking calculus?
Exponential percent change
When a quantity grows or shrinks by the same percent each period (each year, each month, etc.), it grows exponentially, not linearly.
Where P is the starting amount, r is the percent change as a decimal, and t is the number of time periods.
A savings account starts with $2,000 and earns 5% interest per year, compounded annually. How much is in the account after 3 years?
A car worth $20,000 loses 12% of its value each year. Write an equation for its value V after t years.
Watch the wording carefully:
"Increases BY 5% each year" → exponential. The base for next year's increase is the new total. Multiplier method, raise to the t.
"Increases by $5 each year" → linear. Same dollar amount added every year. Add 5t to the starting value.
The SAT writes wrong-answer choices for both interpretations to catch students who skip this distinction.
Sample SAT-style problems
If 35% of a number is 91, what is 50% of the same number?
- Find the number first:
0.35x = 91 x = 260
- Then 50% of 260:
0.50 × 260 = 130
A store's revenue increased by 20% in 2024 and then decreased by 25% in 2025. If the revenue in 2023 was $400,000, what was the revenue in 2025?
- Apply each multiplier in sequence:
400000 × 1.20 × 0.75 = 360000
The population of a town is currently 8,000 and is increasing by 3% each year. Which equation gives the population P after t years?
- (A) P = 8000 + 0.03t
- (B) P = 8000 × 0.03^t
- (C) P = 8000 × (1.03)^t
- (D) P = 8000 × (1 + 3)^t
- "Increasing by 3% each year" → exponential growth.
- Multiplier is (1 + r) = (1 + 0.03) = 1.03.
- Equation: P = 8000 × (1.03)^t.
After a 25% discount, a coat sells for $90. What was its original price?
- Let original = x. After a 25% discount, the price is 75% of x:
x × 0.75 = 90
- Solve:
x = 90 / 0.75 x = 120
Common trap: Students who calculate "$90 + 25% = $112.50" get a wrong answer that often appears as a choice. The 25% was taken off the original $120, not added to the final $90.
1. Wrong denominator on percent change. Always divide by the ORIGINAL value, never the new one.
2. Adding percent changes instead of multiplying. A +20% then −20% is a 4% net loss, not 0%.
3. Confusing "percent of" with "percent increase." "20% of 80" = 16. "80 increased by 20%" = 96. Same number, different operations.