“Percent” means “per hundred.” 35% is 35 out of every 100, which is 0.35 as a decimal. Most percent problems become easy once you turn the percent into a decimal and treat “of” as multiplication.
Converting
- Percent to decimal: move the decimal point two places left. 35% = 0.35, 6% = 0.06, 150% = 1.5.
- Decimal to percent: move it two places right. 0.125 = 12.5%.
- Fraction to percent: divide, then convert. 3/8 = 0.375 = 37.5%.
The three kinds of percent problem
Every basic percent question is part = percent × whole, with one piece missing.
| Missing piece | Question | Work |
|---|---|---|
| The part | What is 20% of 80? | 0.20 × 80 = 16 |
| The percent | 18 is what percent of 72? | 18 ÷ 72 = 0.25 = 25% |
| The whole | 15 is 30% of what? | 15 ÷ 0.30 = 50 |
Percent change
Percent change = (new − old) / old
Always divide by the original value.
Example A price goes from $40 to $50. Find the percent change.
(50 − 40)/40 = 10/40 = 0.25, a 25% increase.
Example A price goes from $50 to $40. Find the percent change.
(40 − 50)/50 = −10/50 = −0.20, a 20% decrease.
The same $10 change is 25% one way and 20% the other, because the starting value is different.
Multipliers: the fast method
Increasing by r% is the same as multiplying by (1 + r). Decreasing by r% is multiplying by (1 − r).
- 15% off: × 0.85
- 6% tax: × 1.06
- 20% increase: × 1.20
Multipliers handle several changes in one line.
Example An $80 jacket is 25% off, and then 6% sales tax is added. What do you pay?
80 × 0.75 × 1.06 = 60 × 1.06 = $63.60.
Example A stock rises 20% and then falls 20%. Is it back where it started?
1.20 × 0.80 = 0.96. It ends at 96% of the original, a 4% loss.
Percent changes don’t add
A 20% rise followed by a 20% fall isn’t 0%. The second change is a percent of a different, bigger number. Multiply the multipliers.
Undoing a percent change
To find the original price before a change, divide by the multiplier.
Example After a 15% discount, a pair of shoes costs $68. What was the original price?
The sale price is 0.85 × original, so original = 68 / 0.85 = $80.
Taking 15% of $68 and adding it back gives $78.20, which is wrong, because the 15% was taken from the original price.
Percent vs. percentage points
If an interest rate goes from 10% to 12%, it rose by 2 percentage points, but by 20 percent (2 is 20% of 10). News stories mix these up all the time.
Interest
Simple interest is paid only on the original amount: I = Prt, where P is the principal, r the yearly rate and t the time in years.
Example $2,000 at 4% simple interest for 3 years
I = 2000 × 0.04 × 3 = $240. The total is $2,240.
Compound interest is paid on the interest too: A = P(1 + r)t when compounded yearly.
Example $2,000 at 4% compounded yearly for 3 years
A = 2000(1.04)³ = 2000 × 1.124864 ≈ $2,249.73.
The difference grows with time. To find how long money takes to grow to a target, you need logarithms; see Logarithms Explained.
Mental math tricks
- 10% is the number with the decimal moved one place: 10% of 64 is 6.4.
- 5% is half of 10%. 20% is double 10%.
- x% of y equals y% of x. 8% of 50 is the same as 50% of 8, which is 4.
Problems to try
What is 35% of 240?
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84. 0.35 × 240.
27 is what percent of 60?
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45%. 27 ÷ 60 = 0.45.
42 is 70% of what number?
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60. 42 ÷ 0.70.
A price rises from $25 to $32. What is the percent increase?
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28%. The change is 7, and 7/25 = 0.28.
After a 30% discount, an item costs $56. What was the original price?
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$80. 56 ÷ 0.70.
$500 is invested at 6% compounded yearly. How much is there after 2 years?
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$561.80. 500 × 1.06² = 500 × 1.1236.
A store marks a shirt up 50%, then puts it on sale for 50% off. How does the final price compare to the original?
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It’s 25% lower. 1.5 × 0.5 = 0.75.
Practice
Questions students ask
Can a percent be more than 100?
Yes. If a population grows from 100 to 250, it increased by 150%. What can’t happen is a decrease of more than 100%, since that would go below zero.
Is a percent decrease the same as a negative percent change?
Yes. A −20% change and a 20% decrease mean the same thing.
Where do percentages show up on the SAT?
In the Problem-Solving and Data Analysis questions: percent change, discounts, and reading percentages from tables. See the SAT Math guide for the full list.