Last updated July 07, 2026 · Reviewed by the AnvayaPrep team
Introduction
Percentages is the unit covering all forms of percent problems tested on the SAT, from foundational percent calculations through complex multi-step and successive percent changes. The unit's 30 topics span the fundamentals (percent basics, percent of a number, decimal-percent conversion, fraction-percent conversion, ratio-percent conversion), core calculations (percent increase, percent decrease, percent change, finding original value after a percent change, reverse percentage), applied contexts (discounts, markup, sales tax, simple interest, compound interest basics, interest), multi-step and compound percent problems (successive percent changes, multi-step percentage, conditional percentages, percent comparison, percent vs. percentage points, percent points, percent error), data contexts (data percent questions, probability-percent, probability with percentages, mixture percent problems), connections to growth/decay (growth factor, decay factor), and SAT-specific traps.
Learning Objectives
- Convert fluently between percents, decimals, and fractions
- Calculate percent of a number, percent increase, percent decrease, and percent change using the appropriate formula
- Work backward from a changed value to find the original value before a percent change
- Calculate discounts, markups, sales tax, and simple and compound interest using percent methods
- Solve successive percent change problems correctly, recognizing that two percent changes applied sequentially are NOT additive
- Distinguish between percent change (relative, expressed as a percent) and percentage point change (absolute difference in percent values)
- Solve multi-step percent problems that combine percent of a number with percent change
- Interpret percent in data contexts: probability as percent, conditional percents from two-way tables
High-Yield Concepts
| Concept | What It Tests | Key Formula | ||
|---|---|---|---|---|
| Percent of a number | Calculating a portion of a whole | percent/100 × whole = part; or decimal × whole = part | ||
| Percent change | Expressing a change as a percent of the original | percent change = (new - original)/original × 100 | ||
| Finding original value | Working backward from a changed amount | original = new value / (1 + percent change as decimal) | ||
| Successive percent changes | Applying two percent changes sequentially | Multiply the growth/decay factors: final = original × (1 + r1) × (1 + r2) | ||
| Percent vs. percentage points | Distinguishing relative from absolute | Percent change is relative to the original; percentage point difference is just subtraction | ||
| Growth/decay factor | Multiplying by (1 + r) or (1 - r) | Growth factor for p% increase: multiply by (1 + p/100); decay: (1 - p/100) | ||
| Compound interest | Applying percent change repeatedly | A = P(1 + r)^n for annual compounding; r is rate as decimal, n is periods | ||
| Reverse percentage | Finding original from a percentage of it | If "the price after a 20% discount is $80," then 0.8 × original = 80 | ||
| Percent error | Comparing estimate to actual value | percent error = | estimated - actual | /actual × 100 |
| SAT percent traps | Avoiding common percent misconceptions | Two percent discounts applied sequentially are not the same as one combined discount |
Study Strategy
Start with the foundational conversions: percent to decimal (divide by 100 or move decimal left two places), decimal to percent (multiply by 100 or move decimal right two places), and fraction to percent (divide numerator by denominator and multiply by 100). These conversions are prerequisites for everything else.
Then study the core calculation types: percent of a number, percent increase, percent decrease, and percent change. Know both the formula approach and the multiplier approach (which is more efficient for multi-step problems).
The multiplier approach is key: a 30% increase means multiplying by 1.3; a 20% decrease means multiplying by 0.8. After mastering the multiplier for single changes, study successive percent changes, which chain these multipliers together.
Study finding original value (reverse percentage) as its own topic. This is conceptually distinct from percent change and appears frequently on the SAT.
Finish with compound interest, percent vs. percentage points (a conceptual distinction with major practical implications), and multi-step percent word problems.
Common Mistakes
Adding successive percent changes instead of multiplying growth factors. A 10% increase followed by a 10% decrease is NOT the same as 0% change. The result is 1.1 × 0.9 = 0.99, a 1% decrease. Students who simply add 10% - 10% = 0% get the wrong answer.
Applying percent change to the wrong base. Percent change is always calculated relative to the ORIGINAL (starting) value. A $100 item that increases to $130 has a 30% increase because (30/100) × 100 = 30%. Dividing by the new value (30/130) instead of the original gives the wrong percentage.
Confusing percent increase with percentage point increase. If a rate goes from 20% to 30%, that is a 10 percentage point increase (absolute), but a 50% percent increase (relative to the original 20%). The SAT tests this distinction and wrong answers exploit it.
Finding the percent of the new value when trying to find the original. If a price after a 25% discount is $60, the original is NOT 60 × 1.25 = $75. Actually 0.75 × original = 60, so original = 60/0.75 = $80. Adding back a percent of the new value is incorrect; divide the new value by the remaining decimal.
Misidentifying the whole in "percent of" problems. In percent problems, "percent of" always means "percent of the specified whole." Read carefully which quantity is the whole before setting up the calculation.
Exam Tips
Use the multiplier approach for all percent increase and decrease problems. A p% increase multiplies by (1 + p/100), also written as (100 + p)/100. A p% decrease multiplies by (1 - p/100), also written as (100 - p)/100. For multi-step percent changes, multiply all the factors: original × multiplier1 × multiplier2 × ... = final. This approach is faster and less error-prone than recalculating the percent amount at each step.
"Percent of" always refers to the original or base quantity, not the new quantity. When a question says "Item A is 30% more expensive than Item B," the 30% is calculated on Item B's price (the base), not Item A's. Applying the percent to the wrong quantity is the most frequently exploited error in SAT percent problems.
For finding the original price after a discount: the original price IS 100%. If there was a 30% discount, you paid 70% (100% - 30%). So if the sale price is $140 and that represents 70%, the original is $140 / 0.7 = $200. The formula: original = sale price / (1 - discount rate as decimal). Equivalently: original = sale price / (remaining percent as decimal).
When encountering "percent vs. percentage points" questions: percent CHANGE is always relative to the original value (computed as (new - old)/old × 100%). PERCENTAGE POINT difference is just the arithmetic difference between two percent values. If the unemployment rate rose from 4% to 6%, that is a 2 percentage point increase, but a 50% increase in the unemployment rate (because 2/4 × 100 = 50%). The SAT may use either framing, so read carefully.
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