Fractions, Decimals & Percents

These are three ways to write the same thing. 3/4 = 0.75 = 75%.

Fraction to decimal: divide top by bottom. 3 ÷ 4 = 0.75.
Decimal to percent: multiply by 100, or move the point two places right. 0.75 → 75%.
Percent to decimal: move the point two places left. 20% → 0.20.
Decimal to fraction: say it out loud. 0.35 is "thirty-five hundredths," so 35/100 = 7/20.

Worth memorising: 1/2 = 50%, 1/3 ≈ 33.3%, 1/4 = 25%, 1/5 = 20%, 1/8 = 12.5%.

"Percent" is a unit meaning hundredths, so p% = p/100 by definition. Conversions are multiplication or division by 100, which in base ten is a two-place shift of the decimal point.

Fraction-to-decimal is long division, and the result is always either terminating or eventually repeating. Which one you get is decided by the denominator's prime factorization in lowest terms: it terminates exactly when the denominator's only prime factors are 2 and 5, because those are the primes of base ten. So 7/20 terminates (20 = 2²·5) but 1/3 and 1/6 repeat.

Converting a repeating decimal back uses algebra. For x = 0.overline{27}, multiply by 100 to get 100x = 27.overline{27}, subtract to get 99x = 27, so x = 3/11. That subtraction trick is the standard proof that every repeating decimal is rational.