SAT percent problems: multipliers, change, reverse

Q: 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.

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?

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?

A 20% discount leaves 80% of the price, so the multiplier is 0.8. Call the original price p.
$0.8p = 68$
Divide by the multiplier, do not add 20% back.
$p = \frac{68}{0.8} = 85$
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?

Multiply the two multipliers.
$1.25 \times 0.80 = 1$
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?

Write both facts as multiplications.
$x = 1.5y$
Second fact:
$y = 0.4z$
Substitute and multiply.
$x = 1.5(0.4z) = 0.6z$
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

Practice

5 SAT-style questions

A price goes from $40 to $52. What is the percent increase?
1. 12%
2. 23%
3. 30%
4. 130%
Show answer
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.
A town of 8,000 people shrinks by 15%. What is the new population?
1. 1,200
2. 6,800
3. 7,985
4. 9,200
Show answer
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%.
A restaurant bill including 6% sales tax is $95.40. What was the bill before tax?
1. $89.68
2. $90.00
3. $94.00
4. $101.12
Show answer
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.
The number a is 25% greater than b. The number b is what percent of a?
1. 75%
2. 80%
3. 120%
4. 125%
Show answer
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.
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?
Show answer
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, accessed October 1, 2026
College Board: SAT Bluebook test directions (student-produced responses), accessed October 1, 2026

Percentages on the SAT

The key idea

Worked examples

Common mistakes

Quick method

Practice

Frequently asked questions

Sources

Try asking Ducky

The key idea

Worked examples

Common mistakes

Quick method

Practice

Frequently asked questions

Sources

Related

Try asking Ducky