Fractions are exact in a way decimals are not. One third is 0.333… forever as a decimal, but 1/3 is precise, and three of them make exactly one. That exactness is why recipes, construction measurements, music notation and probability all stay in fractions rather than converting to decimals and accumulating rounding error.
The rules differ by operation in a way that catches people out: multiplication and division are straightforward, while addition and subtraction require a common denominator first.
The four operations
- Addition and subtraction: convert to a common denominator, then add or subtract the numerators only. 1/4 + 1/6 becomes 3/12 + 2/12 = 5/12.
- Multiplication: multiply across the top and across the bottom. 2/3 × 3/4 = 6/12, which simplifies to 1/2.
- Division: invert the second fraction and multiply. 1/2 ÷ 1/4 = 1/2 × 4/1 = 2.
- Simplifying: divide both parts by their greatest common factor. 18/24 with a GCF of 6 gives 3/4.
Common fractions as decimals and percentages
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.333… | 33.33% |
| 1/4 | 0.25 | 25% |
| 1/5 | 0.2 | 20% |
| 1/8 | 0.125 | 12.5% |
| 2/3 | 0.666… | 66.67% |
| 3/4 | 0.75 | 75% |
| 5/8 | 0.625 | 62.5% |
Mixed numbers and improper fractions
A mixed number such as 2 3/4 is easier to picture; the improper fraction 11/4 is easier to compute with. Convert to improper form before doing arithmetic and back afterwards. To convert: multiply the whole number by the denominator and add the numerator, so 2 × 4 + 3 = 11. Going back, divide 11 by 4 to get 2 remainder 3.
Where fractions beat decimals
- Cooking: halving 3/4 cup gives 3/8 cup exactly; in decimals you are estimating 0.375 of a cup with no measure to match.
- Construction: imperial tape measures are marked in sixteenths, so fractional arithmetic maps directly onto the tool.
- Music: note durations are literally fractions of a bar.
- Probability: a 1/6 chance stays exact through repeated multiplication in a way 0.1667 does not.