SAT Math · Problem-Solving and Data Analysis

Percent change

Q: How do you calculate a percent increase quickly?

Multiply by 1 plus the percent as a decimal. A 15% increase on 40 is 40 × 1.15 = 46. This one step gives the new amount directly.

Q: How do you find the original price after a discount?

Divide the sale price by the multiplier. After a 20% discount the price is 0.8 times the original, so a $48 sale price means the original was 48 ÷ 0.8 = $60.

Q: Does a 20% increase then a 20% decrease cancel out?

No. The multipliers are 1.2 and 0.8, and 1.2 × 0.8 = 0.96, so you end up 4% below where you started. The decrease is taken from a larger amount.

Q: What is the difference between percent and percentage points?

Percentage points measure the gap between two percents. Going from 10% to 15% is 5 percentage points, but the relative change is 5 ÷ 10, a 50% increase.

Q: How can Aristotle help with SAT data analysis?

Aristotle is a voice AI SAT tutor. Your child talks through a percent question out loud, and Aristotle asks guiding questions instead of giving the answer, starting from the step where their reasoning broke. It tracks 113 SAT Math skills, including percentages.

Percent increase, decrease, and back-to-back changes, done with one multiplier instead of guesswork.

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Updated October 30, 2026

$\%\ \text{change} = \dfrac{\text{new} - \text{original}}{\text{original}}$

80 to 100 is a change of 20. $20 \div 80 = 0.25$ , so a 25% increase.

What is percent change?

\dfrac{\text{new} - \text{original}}{\text{original}} \times 100\%

It measures a change against where you started. Going from 80 to 100 is a change of 20 on a start of 80. Twenty is a quarter of eighty, so that is a 25% increase.

The extra 20 is a quarter of the original 80.

Same gap, different percent: the original is the base

\dfrac{20}{80} = 25\%

\dfrac{20}{100} = 20\%

Percent means per hundred of the original, so the bottom of the fraction is always the starting amount. The same 20 is a quarter of 80 but only a fifth of 100: up 25% from 80, down 20% from 100.

Drop 20 from 100 and the base is 100, so it is 20%, not 25%.

Changes in a row multiply

+50\% \text{ then } -50\%

1.5 \times 0.5 = 0.75

A 50% increase multiplies by 1.5; a 50% decrease multiplies by 0.5. The second change acts on the new, bigger amount, so up 50% then down 50% lands at 75, not 100. Percents in a row multiply. They never just add.

Up 50% then down 50% ends 25 below the start.

Why the multiplier works

Answer each question in your head, then scroll for the answer.

Think it through

Question 1
A price goes from 80 to 100. Divide the change by which number?
Answer
The original, 80: $20 \div 80$ = 25%.
Question 2
Now +20% then −20%. Are you back where you started?
Answer
No. $1.2 \times 0.8 = 0.96$ , so you are 4% lower.
Question 3
What undoes a 25% increase?
Answer
Divide by 1.25, which is a 20% decrease.

Try it yourself

Work it out first, then check your answer. SAT-style questions, not official College Board questions.

Question 1

A club had 240 members last year and 300 this year. By what percent did membership increase?

Question 2

After a 20% discount, a jacket costs $48. What was the original price?

Question 3

A stock rises 20% on Monday and falls 20% on Tuesday. What is the overall percent change?

Question 4

Which expression gives the total cost of a $50 item with 8% sales tax?

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The formulas

Percent change

$\dfrac{\text{new} - \text{original}}{\text{original}} \times 100\%$

The denominator is always the starting amount.

Increase by p%

$\text{new} = \text{original} \times \left(1 + \tfrac{p}{100}\right)$

A 15% raise is $\times 1.15$ . A 100% increase doubles the amount.

Decrease by p%

$\text{new} = \text{original} \times \left(1 - \tfrac{p}{100}\right)$

A 30% discount is $\times 0.70$ : you pay 70% of the price.

Finding the original

$\text{original} = \text{new} \div \text{multiplier}$

If a price after a 20% discount is $48, the original was $48 \div 0.8$ = $60.

Percent changes as multipliers
Change	Multiplier	100 becomes
+10%	1.10	110
+25%	1.25	125
+100%	2	200
−10%	0.90	90
−25%	0.75	75
−50%	0.50	50

Try it

Change it, then try to undo it

Set an amount and a change. The last line shows what undoes it.

Original amount80

Percent change+25%

Multiplier1 + 0.25 = 1.25

New amount80 × 1.25 = 100

To get back−20%, not −25%

Common mistakes

Dividing by the new amount

80 to 100 is $20 \div 80$ = 25%, not $20 \div 100$ = 20%.

Adding successive percents

Two 10% increases are $1.1 \times 1.1 = 1.21$ , a 21% increase, not 20%.

Undoing with the same percent

To undo a 20% discount, divide by 0.8. Adding 20% back falls short.

Confusing percent and percentage points

10% to 15% is 5 percentage points, but a 50% increase.

How it shows up on the SAT

Problem-Solving and Data Analysis is about 15% of SAT Math. Expect percent increase and decrease, working back to an original, and combined changes.

What Aristotle tracks in percentages

3 SAT Math skills, including:

Solving percent problems in real-world contexts
Solving problems with percent increases and decreases
Using percent change and growth factors

What is the formula for percent change?

Percent change = (new − original) ÷ original × 100%. Going from 80 to 100 is (100 − 80) ÷ 80 = 0.25, a 25% increase. A negative answer means a decrease.

How do you calculate a percent increase quickly?

How do you find the original price after a discount?

Does a 20% increase then a 20% decrease cancel out?

What is the difference between percent and percentage points?

How can Aristotle help with SAT data analysis?

More SAT concepts

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