"Per cent" literally means "per hundred". A percentage is a fraction with the denominator fixed at 100 — which is exactly what makes percentages so easy to compare.
Percent means out of one hundred
Once every part is measured against the same total of one hundred, comparing amounts becomes effortless: \( 25\% \) is simply \( 25 \) out of \( 100 \). Written three different ways, the same amount looks like this:
\[ 25\% = \frac{25}{100} = \frac{1}{4} = 0.25 \]The same amount wears three outfits — percent, fraction, decimal. They are not three different quantities; they are three costumes on one idea. Slide the value below and watch the shaded squares be the percent:
Converting between percent, decimals and fractions
Because a percentage is just hundredths, moving between the three forms is a matter of where the decimal point sits.
- Percent → decimal: divide by 100, which moves the point two places to the left. So \( 25\% = 0.25 \).
- Decimal → percent: multiply by 100, moving the point two places to the right. So \( 0.4 = 40\% \).
- Fraction → percent: rewrite the fraction over 100 (or divide, then multiply by 100). So \( \frac{3}{5} = \frac{60}{100} = 60\% \).
In words A percentage is simply hundredths in disguise. So \( 47\% \) is just another way to write \( \frac{47}{100} \) or \( 0.47 \) — pick whichever form makes the next step easiest.
Finding a percent of a number
To find a percent of something, turn the percent into a decimal (or fraction) and multiply — because in mathematics the word "of" means multiply. For example, \( 20\% \) of \( 80 \) is:
\[ 20\% \text{ of } 80 = 0.2 \times 80 = 16 \]You can do exactly the same with the fraction form, and you land on the same answer:
\[ \frac{20}{100} \times 80 = 16 \]Tip The word "of" means multiply. To find a percent of an amount, write the percent as a decimal and multiply. Once that clicks, every percent problem becomes a single multiplication.
Increases and discounts
A percentage increase adds that percent on top of what you started with. A \( 10\% \) rise on \( 50 \) adds \( 5 \), giving \( 55 \):
\[ 50 + (10\% \text{ of } 50) = 50 + 5 = 55 \]A discount works the other way — it subtracts. A \( 30\% \) discount on \( 40 \) removes \( 12 \), giving \( 28 \):
\[ 40 - (30\% \text{ of } 40) = 40 - 12 = 28 \]Benchmarks worth memorising
A handful of percentages come up so often that it pays to know them by heart:
- \( 50\% = \frac{1}{2} \) — exactly half.
- \( 25\% = \frac{1}{4} \) — one quarter.
- \( 10\% = \frac{1}{10} \) — one tenth.
- \( 1\% = \frac{1}{100} \) — one hundredth.
These also unlock a fast mental trick. To find \( 10\% \), just move the decimal point one place to the left. From there you can scale: \( 30\% \) is three lots of \( 10\% \), so find \( 10\% \) and triple it.
- Turn the percent into a decimal: \( 15\% = 0.15 \).
- The word "of" means multiply, so work out \( 0.15 \times 60 \).
- That gives \( 0.15 \times 60 = 9 \).
So \( 15\% \) of \( 60 = \mathbf{9} \).
- First find the discount: \( 25\% \) of \( 80 = 0.25 \times 80 = 20 \).
- A discount is subtracted, so take that \$20 off the original price.
- That leaves \( 80 - 20 = 60 \).
So the new price is \( 80 - 20 = \mathbf{\$60} \).
Practice
Try each one yourself, then reveal the full solution.
1. Write \( 40\% \) as a fraction in simplest form.
A percentage is hundredths, so \( 40\% = \frac{40}{100} \).
Both numbers divide by \( 20 \), which simplifies the fraction: \( \frac{40}{100} = \frac{2}{5} \).
So \( 40\% = \frac{40}{100} = \mathbf{\frac{2}{5}} \).
2. What is \( 30\% \) of \( 50 \)?
Turn the percent into a decimal: \( 30\% = 0.3 \).
The word "of" means multiply, so work out \( 0.3 \times 50 \).
So \( 30\% \) of \( 50 = 0.3 \times 50 = \mathbf{15} \).
3. A \$25 item rises in price by \( 20\% \). What is the new price?
First find the increase: \( 20\% \) of \( 25 = 0.2 \times 25 = 5 \).
A rise is added on top, so add that \$5 to the original price.
So the new price is \( 25 + 5 = \mathbf{\$30} \).