Fractions, Decimals and Percentages

Fractions, decimals and percentages are three ways of writing one value. A half can be written 1/2, 0.5 or 50%. Each form suits a different job: fractions are exact and easy to picture, decimals suit calculators and measuring, and percentages make comparisons easy because everything is 'out of 100'.

The word percent means 'per hundred', so 50% is 50/100. That single fact links percentages to the other two forms.

A handful of equivalents come up again and again: halves, quarters, fifths, tenths, eighths and thirds. Knowing them instantly saves time and lets you build others. If 1/8 = 0.125, then 3/8 is 3 × 0.125 = 0.375 and 5/8 is 0.625. If 1/5 = 20%, then 4/5 = 80%. It also helps you spot when a calculator answer looks wrong, and it makes mental calculations like '25% off' (take off a quarter) quick.

A fraction is a division, so divide the numerator by the denominator. 3/8 = 3 ÷ 8 = 0.375.

If the denominator multiplies easily to 10, 100 or 1,000, use an equivalent fraction instead: 7/20 = 35/100 = 0.35, and 3/25 = 12/100 = 0.12.

Use place value. The last digit of the decimal tells you the denominator: tenths, hundredths or thousandths.

• 0.8 is 8 tenths: 8/10 = 4/5.
• 0.35 is 35 hundredths: 35/100 = 7/20 (divide by 5).
• 0.125 is 125 thousandths: 125/1000 = 1/8 (divide by 125).

Always simplify at the end.

• Decimal to percentage: multiply by 100. 0.07 × 100 = 7%, and 0.375 × 100 = 37.5%.
• Percentage to decimal: divide by 100. 45% = 0.45, and 4% = 0.04 (not 0.4).
• Percentage to fraction: write it over 100 and simplify. 45% = 45/100 = 9/20.
• Fraction to percentage: make the denominator 100 if you can (9/20 = 45/100 = 45%), or divide and multiply by 100 (5/8 = 0.625 = 62.5%).

Some fractions give decimals that never end. 1 ÷ 3 = 0.333…, with the 3s repeating for ever, so 1/3 is about 33.3% (exactly 33 1/3%). 2/3 = 0.666…, which rounds to 0.667 to three decimal places, or about 66.7%. These are called recurring decimals; the dots show that the digits carry on.

There is a quick test. Simplify the fraction, then look at the prime factors of the denominator. If they are only 2s and 5s, the decimal terminates (ends): 3/40 = 0.075, because 40 = 2 × 2 × 2 × 5. If any other prime appears, the decimal recurs: 1/6 = 0.1666…, because 6 = 2 × 3.

To put 3/5, 0.62 and 61% in order, write them all in the same form. As decimals they are 0.6, 0.62 and 0.61, so the order from smallest is 3/5, 61%, 0.62. Decimals are usually the easiest common form for comparing; percentages work just as well.