Percentages, Fractions and Decimals: Converting in Your Head
Percentages, fractions and decimals are three ways of writing the same quantity. Discounts, tips, test scores, interest rates and recipes all switch between them, so being able to convert quickly is one of the most useful everyday math skills. Here is a compact toolkit that handles almost every case without a calculator.
One idea: they are all "parts of a whole"
A percentage is a fraction with a denominator of 100: "37%" means 37⁄100. A decimal is a fraction whose denominator is a power of ten: 0.37 is also 37⁄100. Converting between the forms is just rewriting the same number, and the easiest route is nearly always through the hundredths.
- Percentage → decimal: divide by 100, which moves the decimal point two places left. 37% = 0.37, and 5% = 0.05.
- Decimal → percentage: multiply by 100, which moves the point two places right. 0.6 = 60%.
- Fraction → decimal: divide the numerator by the denominator. 3⁄8 = 3 ÷ 8 = 0.375.
- Fraction → percentage: convert to a decimal, then multiply by 100: 3⁄8 = 0.375 = 37.5%.
The benchmark table to memorize
Fewer than a dozen conversions come up over and over. Learn them as a set, and you can build most other values from them.
| Fraction | Decimal | Percentage |
|---|---|---|
| 1⁄2 | 0.5 | 50% |
| 1⁄4 | 0.25 | 25% |
| 3⁄4 | 0.75 | 75% |
| 1⁄5 | 0.2 | 20% |
| 1⁄8 | 0.125 | 12.5% |
| 1⁄10 | 0.1 | 10% |
| 1⁄3 | 0.333… | 33.3% |
| 2⁄3 | 0.666… | 66.7% |
From these you can build others: 3⁄8 is three one-eighths, so 3 × 12.5% = 37.5%. And 1⁄6 is half of 1⁄3, so about 16.7%.
Finding a percentage of a number
The most useful technique is to start from 10% and 1%. Finding 10% means moving the decimal point one place to the left; finding 1% means moving it two places. Everything else is built from those two.
Example: 35% of 80
10% of 80 = 8, so 30% = 24. 5% is half of 10%, so 4. Add: 24 + 4 = 28.
Example: 15% tip on a $46 bill
10% = 4.60 and 5% = 2.30. Together: $6.90.
A neat symmetry helps here: x% of y equals y% of x. If 4% of 75 seems awkward, flip it: 75% of 4 is 3. Both are the same number.
Discounts, increases and "what percent is this?"
Percentage off
To take 20% off a price, find 10%, double it, and subtract. A $65 jacket with 20% off: 10% is 6.50, so 20% is 13, and the sale price is 65 − 13 = $52. A shortcut: paying 80% of the price means multiplying by 0.8, so 65 × 0.8 = 52.
Percentage increase
To add 15% to a number, multiply by 1.15. A rent of $800 rising by 15% becomes 800 × 1.15 = 920. In your head, build the increase from 10% and 5%: 10% of 800 is 80 and 5% is 40, so the increase is 120 and the new rent is 800 + 120 = $920.
What percent is A of B?
Write the fraction A⁄B and convert it. 18 out of 24 is 18⁄24 = 3⁄4 = 75%. Simplify first: dividing top and bottom by their common factor makes the fraction one of the benchmark fractions you already know.
Percent change is not symmetric
A 50% rise followed by a 50% fall does not bring you back to the start. Take 100: up 50% is 150, and down 50% of 150 is 75, not 100. The reason is that the second percentage applies to a different (larger) base. To undo a 25% increase you need a 20% decrease, not 25%, since 100 → 125 → 100 requires removing 25 out of 125, which is 20%.
Common mistakes to avoid
- Adding percentages of different bases. A 10% discount followed by another 10% discount is 19% off, not 20%, because the second 10% is of the reduced price.
- Confusing percent and percentage points. Moving from 20% to 25% is a rise of 5 percentage points, but a 25% relative increase.
- Misplacing the decimal point. 5% is 0.05, not 0.5. Write the decimal form once when you are unsure.