Percentages
Problem-Solving and Data Analysis is about 15% of SAT Math (5–7 of 44 questions), and percentages appear in almost every test as a story about prices, populations, discounts or survey results. They also hide inside other topics, such as exponential growth.
Part, whole and percent
Every basic percent problem has three pieces: part = percent × whole. Change the percent to a decimal by dividing by 100. 35% of 120 is 0.35 × 120 = 42. If the part and the whole are known, the percent is part ÷ whole × 100. If the part and the percent are known, the whole is part ÷ percent. The word “of” means multiply, and the word “is” means equals.
Percent change
Percent change = (new − old) ÷ old × 100. The bottom of the fraction is always the old (starting) amount. A rise from 250 to 310 is a change of 60, and 60 ÷ 250 = 0.24, a 24% increase. Dividing by the new amount is the most common mistake. A negative result means a decrease.
Multipliers and successive changes
A multiplier does a percent change in one step. Increase by 20%: multiply by 1 + 0.20 = 1.20. Decrease by 20%: multiply by 1 − 0.20 = 0.80. To apply two changes in a row, multiply the multipliers. A 20% rise followed by a 20% fall gives 1.20 × 0.80 = 0.96, so the final amount is 4% less than the start. You cannot just add or subtract the percents, because the second change is taken from a new amount.
| Change | Multiplier |
|---|---|
| +25% | 1.25 |
| +8% | 1.08 |
| −10% | 0.90 |
| −35% | 0.65 |
| +100% | 2.00 |
Working backward: finding the original
If an amount is the result of a percent change, divide by the multiplier to get back to the start. A coat costs $48 after a 20% discount. The multiplier is 0.80, so the old price was 48 ÷ 0.80 = $60. Do not take 20% of 48 and add it back: 20% of the new price is not 20% of the old price. This also works for taxes. In Uzbekistan the VAT rate is 12%, so a price with VAT is the price without VAT times 1.12.
Percent versus percentage points
When a percent itself changes, two different questions can be asked. If a rate goes from 8% to 6%, it fell by 2 percentage points (6 − 8 = −2). Its percent change is −2 ÷ 8 = −25%. Percentage points are a plain subtraction of the two percents; percent change compares the difference to the starting percent. Read the words carefully: “percentage points” or “percent”?
- Use part = percent × whole: 84 = 0.35 × whole.
- whole = 84 ÷ 0.35 = 240.
- Check: 0.35 × 240 = 84 ✓
- Change: 14,000 − 12,500 = 1,500.
- Divide by the old amount: 1,500 ÷ 12,500 = 0.12.
- 0.12 × 100 = 12%.
- 3% higher
- 17% higher
- 20% higher
- 40% higher
- Multipliers: 1.30 for the rise and 0.90 for the discount.
- Multiply: 1.30 × 0.90 = 1.17.
- 1.17 means 117% of the original, which is a 17% increase.
- Price with VAT = price without VAT × 1.12.
- Price without VAT = 3,360,000 ÷ 1.12 = 3,000,000.
- Check: 3,000,000 × 1.12 = 3,360,000 ✓
- Dividing by the new amountPercent change always divides by the old (starting) amount.
- Adding successive percents+30% then −10% is not +20%. Multiply the multipliers: 1.30 × 0.90 = 1.17.
- “Add back” instead of dividingTo undo a 20% discount, divide by 0.80. Adding 20% of the sale price is wrong.
- Percent and percentage points8% to 6% is 2 percentage points (subtract) and a 25% decrease (divide by 8).
- Forgetting to change the percent to a decimal35% is 0.35 in a calculation. Typing 35 × 120 gives 4,200 instead of 42.
- Typing % or a comma in the answerIn a typed answer, write 24, not 24% or 24.0 percent. Only digits, a decimal point and a minus sign.
Type a percent equation with x for the unknown and Desmos draws a vertical line at the solution. For successive changes, type the product of the multipliers and read the result. You can also use a slider: type a = 20 and then 100 × (1 + a/100) × (1 − a/100), and move a to see how a rise and a fall of the same percent never cancel.
- Type 0.35x = 84
- A vertical line appears at x = 240
- Type 1.3 × 0.9 to see 1.17
Use Desmos for awkward divisions such as 84 ÷ 0.35 or for checking multipliers. Choosing the right base (old amount) is something you must do yourself.
Open Desmos ↗