A percent is a fraction with a denominator of 100. Saying 35% means 35 out of every 100, or $latex \frac{35}{100} = 0.35$. Every percentage problem has the same three pieces: a part, a whole and a rate (the percent). You know two of them and you want the third.
This guide covers the percent of a number, what percent one number is of another, increases and discounts, two discounts in a row, and percent change. Three interactive tools let you move the numbers and watch the bars change.
What a percent really is
To turn a percent into a decimal, divide by 100: $latex 35\% = 0.35$. To go back, multiply by 100: $latex 0.08 = 8\%$. A few percents are worth knowing by heart:
| Percent | Fraction | Decimal | Quick way |
|---|---|---|---|
| 50% | $latex \frac{1}{2}$ | 0.5 | divide by 2 |
| 25% | $latex \frac{1}{4}$ | 0.25 | divide by 4 |
| 20% | $latex \frac{1}{5}$ | 0.2 | divide by 5 |
| 10% | $latex \frac{1}{10}$ | 0.1 | move the decimal point one place left |
| 5% | $latex \frac{1}{20}$ | 0.05 | half of 10% |
| 1% | $latex \frac{1}{100}$ | 0.01 | move the decimal point two places left |
These shortcuts build any other percent. For 15% of 80, take 10% (8) and add 5% (4): 12.
The three basic percent problems
1. Find a percent of a number
$$ \text{part} = \frac{p}{100} \times \text{whole} $$
Example. What is 15% of 80?
$$ \frac{15}{100} \times 80 = 0.15 \times 80 = 12 $$
2. Find what percent one number is of another
$$ p = \frac{\text{part}}{\text{whole}} \times 100 $$
Example. You got 18 questions right out of 24. What percent is that?
$$ \frac{18}{24} \times 100 = 0.75 \times 100 = 75\% $$
3. Find the whole when you know the part
$$ \text{whole} = \frac{\text{part}}{p/100} $$
Example. 30 is 12% of what number?
$$ \frac{30}{0.12} = 250 $$
Check it: 12% of 250 is 30. Going backward like this is the quickest way to catch a slip.
Increases and discounts
Adding p% to a price means you pay 100% + p% of it, which is $latex 1 + \frac{p}{100}$ times the price. Taking p% off means you pay $latex 1 - \frac{p}{100}$ times the price. One multiplication does the whole job:
$$ \text{new price} = \text{price} \times \left(1 \pm \frac{p}{100}\right) $$
Example: a discount. A jacket costs $80 and is 25% off. The factor is $latex 1 - 0.25 = 0.75$, so you pay $latex 80 \times 0.75 = 60$, which is $60.
Example: an increase. A restaurant bill is $64 and you leave a 15% tip. The factor is 1.15, so the total is $latex 64 \times 1.15 = 73.6$, or $73.60.
Example: finding the original price. A game costs $64 after a 20% discount. The sale price is 80% of the original, so divide:
$$ \frac{64}{0.80} = 80 $$
The original price was $80. Adding 20% to the sale price would give $76.80 instead, which is wrong because the 20% was taken from the original, not from $64.
Two discounts in a row
A store takes 20% off and then another 10% off the new price. Many people add them and expect 30% off. The second discount is applied to a smaller price, so it removes fewer dollars. Multiply the factors instead:
$$ 0.80 \times 0.90 = 0.72 $$
You pay 72% of the price, so the total discount is 28%, not 30%.
In general, for two discounts $latex d_1$ and $latex d_2$ written as decimals, the total discount is $latex 1 - (1 - d_1)(1 - d_2)$. The order does not matter, and the same idea works for any chain of increases and decreases. The slider below shows how far the real price sits from the “add them up” price.
A related trap: raise a price by 20%, then lower it by 20%. The factors are 1.20 and 0.80, and $latex 1.20 \times 0.80 = 0.96$. The price ends 4% below where it started.
Percent change
Percent change compares a new value to an old one. The old value is always the base:
$$ \text{percent change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100 $$
A positive result is an increase and a negative result is a decrease. Rent that goes from 50 to 75 changes by $latex \frac{75 - 50}{50} \times 100 = 50\%$. Going back from 75 to 50 is $latex \frac{50 - 75}{75} \times 100 \approx -33.33\%$. Same gap of 25, different starting point, different percent.
Common mistakes
- Adding successive percents. 20% off and then 10% off is 28% off, not 30%.
- Undoing a percent with the same percent. A 20% increase is not cancelled by a 20% decrease, because the two are taken from different bases.
- Dividing by the wrong number in percent change. Always divide by the old value, even when the new one looks like the natural base.
- Mixing percent and percentage points. A rate that rises from 4% to 5% rose by 1 point and by 25%.
- Skipping the conversion to a decimal. 35% of 240 is 0.35 × 240, not 35 × 240.
Practice problems
Problem 1
What is 35% of 240?
Answer
$latex 0.35 \times 240 = 84$.
Answer: 84.
Problem 2
Shoes cost $120. They are 30% off, and then 8% sales tax is added to the sale price. What is the final price?
Answer
Discount: $latex 120 \times 0.70 = 84$. Tax: $latex 84 \times 1.08 = 90.72$.
Answer: $90.72.
Problem 3
A phone costs $450 after a 10% discount. What was the original price?
Answer
The sale price is 90% of the original: $latex \frac{450}{0.90} = 500$.
Answer: $500.
Problem 4
Rent rises from $800 to $868. What is the percent increase?
Answer
$$ \frac{868 - 800}{800} \times 100 = \frac{68}{800} \times 100 = 8.5\% $$
Answer: 8.5%.
Problem 5
A store gives two discounts in a row, 25% and then 20%. What is the total discount?
Answer
Factors: $latex 0.75 \times 0.80 = 0.6$. You pay 60% of the price.
Answer: a total discount of 40%.
Problem 6
A price rises 20% and later falls 20%. Is it back to the original? Give the net change.
Answer
$latex 1.20 \times 0.80 = 0.96$. The final price is 96% of the original.
Answer: no. The net change is -4%.
Frequently asked questions
How do I find 20% of a number in my head?
Take 10% by moving the decimal point one place left, then double it. For 20% of 70, 10% is 7, so 20% is 14.
How do I find the original price after a discount?
Divide the sale price by the fraction you still pay. After a 25% discount you pay 75%, so original price = sale price ÷ 0.75.
Can a percent be larger than 100%?
Yes. 150% of a number is one and a half times that number. A value that triples has increased by 200%.
What is the difference between percent and percentage points?
A percent compares a change to a starting value. Percentage points are the plain difference between two percents. From 4% to 5% is 1 percentage point, or a 25% increase.
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- Percentage calculator (discount, increase and percent change)
- Compound interest calculator
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