# Percentages on the SAT

Canonical: https://duckyhelper.com/learn/sat-math/percentages/
Updated: 2026-10-01

Percent means "per hundred", so 35% is 0.35. On the SAT, turn every percent change into a multiplier: an increase of p% multiplies by \(1 + \frac{p}{100}\), and a decrease multiplies by \(1 - \frac{p}{100}\). Multiply the multipliers for back-to-back changes, and divide by the multiplier to find an original value. Most percent traps come from using the wrong whole.

## The key idea

Every percent question has a part, a whole and a percent. "Of" usually points to the whole. A percent change always compares to the starting value:

$$
\text{percent change} = \frac{\text{new} - \text{original}}{\text{original}} \times 100
$$

Multipliers turn every percent problem into one multiplication:

**Percent changes as multipliers**

| Change | Multiplier | Example on $80 |
| --- | --- | --- |
| Increase by 20% | 1.20 | \(80 \times 1.2 = 96\) |
| Decrease by 15% | 0.85 | \(80 \times 0.85 = 68\) |
| Add 6% tax | 1.06 | \(80 \times 1.06 = 84.80\) |
| Increase 10%, then decrease 10% | \(1.1 \times 0.9 = 0.99\) | \(80 \times 0.99 = 79.20\), a 1% loss |

## Worked examples

**Example 1: percent of a number**

Problem: What is 18% of 250?

1. Write 18% as a decimal and multiply.

   $$
   0.18 \times 250 = 45
   $$

Answer: 45

**Example 2: find the original price**

Problem: After a 20% discount, a jacket costs $68. What was the price before the discount?

1. A 20% discount leaves 80% of the price, so the multiplier is 0.8. Call the original price p.

   $$
   0.8p = 68
   $$
2. Divide by the multiplier, do not add 20% back.

   $$
   p = \frac{68}{0.8} = 85
   $$
3. Check: 20% of $85 is $17, and \(85 - 17 = 68\). Adding 20% to $68 would give $81.60, which is wrong because that 20% is taken of the wrong whole.

Answer: $85

**Example 3: back-to-back changes**

Problem: A stock price rises 25% in one month and falls 20% the next month. What is the overall percent change?

1. Multiply the two multipliers.

   $$
   1.25 \times 0.80 = 1
   $$
2. A total multiplier of 1 means the price is back where it started. The 20% drop is taken of a bigger number, so it cancels the 25% rise exactly.

Answer: 0%: no overall change.

**Example 4 (SAT-hard): percent of a percent**

Problem: The number x is 150% of y, and y is 40% of z. The number x is what percent of z?

1. Write both facts as multiplications.

   $$
   x = 1.5y
   $$
2. Second fact:

   $$
   y = 0.4z
   $$
3. Substitute and multiply.

   $$
   x = 1.5(0.4z) = 0.6z
   $$
4. 0.6 is 60%.

Answer: 60%

## Common mistakes

- **Adding the percent back to undo a discount.** \(68 \times 1.2\) is not the original price. Fix: divide by the multiplier, \(68 \div 0.8\).
- **Adding percents for back-to-back changes.** Up 25% then down 20% is not up 5%. Fix: multiply \(1.25 \times 0.8\).
- **Dividing by the new value for percent change.** From $40 to $52 is \(\frac{12}{40} = 30\%\), not \(\frac{12}{52}\). Fix: the original value always goes on the bottom.
- **Mixing up percent and percentage points.** Going from 40% to 50% is an increase of 10 percentage points, but a 25% increase. Fix: read which one the question asks for.
- **Typing the percent sign.** On a student-produced response, enter 30, not 30%. College Board's directions say not to enter symbols like %, $ or commas.

## Quick method

> **Tip: Pick 100**
>
> When a question gives no starting number, start with 100. Up 30% then down 30%: \(100 \to 130 \to 91\), a 9% loss. This is exact for percent questions because percents scale with the starting value.

## Practice

**5 SAT-style questions**

1. A price goes from $40 to $52. What is the percent increase?
   A. 12%
   B. 23%
   C. 30%
   D. 130%

   Answer: 30%. The change is $12, and it is compared to the original $40: \(\frac{12}{40} = 0.30\). 23% divides by the new price. 130% is the new price as a percent of the old one, not the increase.

2. A town of 8,000 people shrinks by 15%. What is the new population?
   A. 1,200
   B. 6,800
   C. 7,985
   D. 9,200

   Answer: 6,800. Multiply by 0.85: \(8000 \times 0.85 = 6800\). 1,200 is the number of people lost, and 7,985 subtracts 15 people instead of 15%.

3. A restaurant bill including 6% sales tax is $95.40. What was the bill before tax?
   A. $89.68
   B. $90.00
   C. $94.00
   D. $101.12

   Answer: $90.00. Before tax times 1.06 is 95.40, so divide: \(95.40 \div 1.06 = 90\). $89.68 subtracts 6% of the total, which takes 6% of the wrong whole.

4. The number a is 25% greater than b. The number b is what percent of a?
   A. 75%
   B. 80%
   C. 120%
   D. 125%

   Answer: 80%. \(a = 1.25b\), so \(b = \frac{a}{1.25} = 0.8a\). Try numbers: if \(b = 100\), then \(a = 125\), and \(\frac{100}{125} = 80\%\). 75% assumes the percents are symmetric, and they are not.

5. Student-produced response: a shirt's price is cut by 30%, then the sale price is cut by another 10%. The final price is what percent of the original price?

   Answer: 63. Multiply the multipliers: \(0.7 \times 0.9 = 0.63\), so 63%. Enter 63 without the percent sign. Two cuts of 30% and 10% are not a 40% cut.

## Frequently asked questions

### How do I find the original number after a percent change?

Write the change as a multiplier and divide by it. If a price after a 20% discount is $68, the multiplier is 0.8, and the original is \(68 \div 0.8 = 85\). Never add the percent back onto the new number.

### Why isn't up 10% then down 10% back to the start?

The second 10% is taken of a bigger number. \(1.1 \times 0.9 = 0.99\), so you end 1% below where you started. Back-to-back percent changes multiply, they never just add.

### How should I enter a percent answer on the SAT?

For a student-produced response, type the number only. College Board's directions say not to enter symbols such as a percent sign, a comma or a dollar sign. If the question asks for a percent, 63% is entered as 63.

## Sources

- [College Board: SAT Math, Problem-Solving and Data Analysis skills](https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types/problem-solving), accessed 2026-10-01
- [College Board: SAT Bluebook test directions (student-produced responses)](https://satsuite.collegeboard.org/media/pdf/english-sat-test-directions-bb.pdf), accessed 2026-10-01

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- [SAT Math study guides](https://duckyhelper.com/learn/sat-math/)

## Try asking Ducky

- "Why do I divide by 0.8 instead of adding 20% back?"
- "Is this asking for percent or percentage points?"
- "Give me five quick percent change questions to warm up."

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