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MathA half is 0.5 is 50%. Same value, three costumes — and every field dresses numbers in its preferred outfit: recipes use fractions, calculators use decimals, news headlines use percentages. Fluency in all three directions is one of the highest-leverage math skills there is.
This guide covers every conversion path with the two-direction rules, a table of the values worth memorizing, and the repeating-decimal quirk explained.
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The two-direction rules
Every conversion runs through the decimal — it's the hub between the other two:
Fraction → decimal: divide numerator by denominator (3/4 → 3 ÷ 4 = 0.75)
Decimal → fraction: write over 1, 10, 100… then simplify (0.75 → 75/100 → 3/4)
Decimal → percent: × 100 (0.75 → 75%)
Percent → decimal: ÷ 100 (75% → 0.75)
Decimal → fraction: write over 1, 10, 100… then simplify (0.75 → 75/100 → 3/4)
Decimal → percent: × 100 (0.75 → 75%)
Percent → decimal: ÷ 100 (75% → 0.75)
So 3/4 → 0.75 → 75%, and backwards: 75% → 0.75 → 75/100 → 3/4. Chain the rules and any conversion is two steps.
The values worth memorizing
| Fraction | Decimal | Percent |
|---|---|---|
| 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% |
| 1/6 | 0.1666… | 16.7% |
| 5/8 | 0.625 | 62.5% |
The eighths (12.5%, 37.5%, 62.5%, 87.5%) are the sleeper hit of this table — they appear constantly in measurements, finance, and statistics, and most people can't place them.
Worked examples
- 3/8 to percent: 3 ÷ 8 = 0.375 → × 100 = 37.5%
- 0.42 to fraction: 42/100 → simplify ÷ 2 → 21/50
- 60% to fraction: 60/100 → ÷ 20 → 3/5
- 7/12 to percent: 7 ÷ 12 = 0.5833… → 58.3%
The repeating-decimal quirk
Some fractions never terminate: 1/3 = 0.333…, 1/6 = 0.1666…, 5/11 = 0.4545…. This isn't an error — a fraction terminates only when its denominator's prime factors are all 2s and 5s (like 8 = 2³ or 20 = 2²×5). Any other factor produces a repeating tail. Round to a sensible precision (two decimal places is standard) and note that the fraction itself remains the exact value — 1/3 is exact, 0.333 is an approximation.
Quick practice set
Convert each value into all three forms — answers in parentheses:
- 5/8 → 0.625 → 62.5%
- 0.04 → 4/100 = 1/25 → 4%
- 150% → 1.5 → 3/2
- 7/20 → 0.35 → 35%
If the third one felt odd — 150% as 3/2 — that's the whole point of the exercise: percentages above 100% are just fractions bigger than one, wearing different clothes.
Which form for which job
- Decimals for arithmetic — calculators, spreadsheets, and formulas want decimals.
- Fractions for exactness — 1/3 is exact; no decimal is.
- Percentages for communication — "up 20%" lands instantly; "up 0.2×" doesn't.
For the percentage deep-dive, see what a percentage really is.
Frequently asked questions
Divide the numerator by the denominator, then multiply by 100. 3/8 = 0.375, and 0.375 × 100 = 37.5%. Two steps: fraction → decimal → percent.
Write the percentage over 100 and simplify. 40% = 40/100 = 2/5. 12.5% = 12.5/100 = 125/1000 = 1/8 after simplifying.
1/3 = 0.333… (repeating) = 33.33…%, usually rounded to 33.3%. Fractions whose denominators have prime factors other than 2 and 5 produce repeating decimals.
Decimals for arithmetic (calculators want decimals), fractions for exactness (1/3 is exact; 0.333 is not), and percentages for communication. Convert to whatever the job needs.
Disclaimer: Calculator content is for education only — examples are illustrative and rounded; double-check figures used in financial or contractual decisions.