SAT Percent Change: Calculating Growth, Decay, and Repeated Percentage Changes

Students often assume that applying a percentage twice equals double that percentage. If a value grows by 5% in year 1 and 5% in year 2, the total growth is not 10%—it is 10.25%—because the second year's 5% applies to the already-grown value, not the original. This compound effect appears throughout SAT word problems involving investments (compound interest), population growth, inflation, and depreciation. Understanding why repeated percentages compound prevents the most common error on percentage problems: treating them as simple addition instead of multiplication.

The formula for repeated percentage change is: Final value = Initial value × (1 + rate)^n, where rate is the decimal form of the percentage and n is the number of times it is applied. A $100 investment growing at 5% annually for two years: Final = 100 × (1.05)^2 = 100 × 1.1025 = $110.25. Not $110. This distinction matters on the SAT because wrong answers typically show the simple-addition result ($110), trapping students who did not understand compounding.

For repeated percentage changes, convert the percentage to a decimal multiplier: 5% growth = ×1.05, 5% decay = ×0.95, 10% growth = ×1.10, 20% decay = ×0.80. Then multiply the multiplier by itself (or raise it to a power) for the number of applications. Example: A population of 10,000 grows 3% annually for 4 years. Multiplier = 1.03. Final population = 10,000 × (1.03)^4 = 10,000 × 1.1255 ≈ 11,255. (Not 10,000 × 1.12 = 11,200, which is the wrong answer you will find in wrong options.) Step-by-step: 10,000 → 10,300 (after year 1) → 10,609 (after year 2) → 10,927 (after year 3) → 11,255 (after year 4). Each year grows from the previous year's total, not the original.

The error prevention routine: (1) Identify the percentage change and convert to a multiplier. (2) Identify how many times it applies (n). (3) Calculate: Final = Initial × (multiplier)^n. (4) Verify: Does the final value make sense? Should it be larger (growth) or smaller (decay) than the initial? Is the magnitude reasonable? (A 3% annual growth over 4 years should roughly quadruple is wrong; it should grow roughly 12%, matching our 10,000 → 11,255.)

Example 1 - Investment Growth: $5,000 grows at 6% annually for 3 years. Multiplier = 1.06. Final = 5,000 × (1.06)^3 = 5,000 × 1.1910 ≈ $5,955. (Not $5,900, which is 6% × 3 = 18% simple addition.) Example 2 - Population Decline: A town of 50,000 loses 2% annually for 5 years. Multiplier = 0.98. Final = 50,000 × (0.98)^5 = 50,000 × 0.9039 ≈ 45,195. (Not 50,000 - (0.02 × 50,000 × 5) = 45,000, which is simple-addition thinking.) Example 3 - Mixed Rates: Value starts at $100. Year 1 grows 10%, year 2 grows 5%. Multipliers = 1.10 and 1.05. Final = 100 × 1.10 × 1.05 = 100 × 1.155 = $115.50. (Not $115, which is 10% + 5% = 15% thinking, and overlooks that the 5% applies to $110, not $100.)

All three examples show the same pattern: percentage changes compound multiplicatively, not additively. Missing this leads to wrong answers that are suspiciously close to your solution.

Strengthen compound percentage recognition by solving pairs of problems: one with repeated percentages, one with simple addition, and explicitly comparing the answers. For each day's three problems, solve one compound percentage problem, calculate what the wrong answer would be if you used simple addition, and verify that the wrong answer appears in the multiple choice options. This practice trains your instinct to avoid the trap. Once you realize "oh, the wrong answer is always the simple-addition result," you guard against it automatically.

Create a reference card showing the multiplier method and sample calculations. Before test day, review this card for one minute to reinforce the concept. On test day, when you see a repeated percentage problem, your first instinct will be to create the multiplier and raise it to a power rather than multiply percentages together.