Why fractions beat decimals more often than you think
Decimals feel modern and fractions feel like school. But a third has no exact decimal, and that gap explains recipes, spanners and rounding errors.
Most people leave school glad to be done with fractions. Decimals seem strictly better — they line up in columns, calculators accept them, money uses them. Then you meet a third. Written as a decimal it is 0.333… and it never finishes, which means every decimal you write down is slightly wrong. As a fraction it is exact, forever, in three characters.
Some numbers have no decimal form
A decimal can only terminate if the denominator divides a power of ten — that is, if its only prime factors are 2 and 5. Halves, quarters, fifths and eighths are fine. Thirds, sixths, sevenths and ninths are not, and never will be.
| Fraction | As a decimal | Exact? |
|---|---|---|
| 1/2 | 0.5 | Yes |
| 3/8 | 0.375 | Yes |
| 1/10 | 0.1 | Yes |
| 1/3 | 0.333… | No — repeats forever |
| 2/3 | 0.666… | No — repeats forever |
| 1/7 | 0.142857… | No — repeats in a 6-digit cycle |
This is not a quirk of notation. Add 1/3 three times and you get exactly 1. Add 0.333 three times and you get 0.999 — close, but wrong, and the error compounds every time you reuse it.
Where the difference actually bites
- Recipes. A third of a cup is a real, measurable quantity. 0.333 cups is not something any measuring jug will help you with.
- Tools and timber. Imperial sizes are built on halves, quarters, eighths and sixteenths. A 3/8 spanner is 3/8 — writing 0.375 helps nobody at the bench.
- Music. Note lengths are fractions by definition: a quarter note is a quarter of a bar. Rhythm is fraction arithmetic performed in real time.
- Odds and probability. A 1/6 chance of rolling a six is exact. 0.1667 is an approximation of it.
The rules, and why they look different
Multiplying and dividing fractions is easier than the decimal equivalent. Multiply straight across: 3/4 × 2/5 = 6/20, which reduces to 3/10. To divide, flip the second fraction and multiply: 7/8 ÷ 1/4 = 7/8 × 4/1 = 28/8 = 7/2, or 3½.
Adding is the one that needs work, and the reason is worth understanding. You cannot add halves to thirds any more than you can add metres to pounds — the pieces are different sizes. So you rewrite both over a common denominator first: 1/2 + 1/3 becomes 3/6 + 2/6 = 5/6. The denominator is not being added; it is being agreed on.
Always finish by simplifying
A fraction is in lowest terms when the numerator and denominator share no common factor. Divide both by their greatest common divisor and you are done: 6/20 has a GCD of 2, giving 3/10. This is the same GCD that squares up ratios and reduces 1920:1080 to 16:9 — one idea doing several jobs.
Converting back when you need to
Terminating decimals convert directly. 0.375 has three decimal places, so it is 375/1000, which simplifies to 3/8. Repeating decimals need a different approach — feed 0.66667 to a converter and it recognises 2/3, because it searches for the simplest fraction that lands close enough rather than taking the digits literally.
The practical rule: use decimals when you are measuring or reporting, and fractions when you are dividing things exactly. Money is decimal because it stops at cents. A recipe, a rhythm and a spanner set are fractional because thirds and eighths are real quantities that decimals can only approximate.
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