Fractions, decimals and percentages

Maths · Number. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

Fractions, decimals and percentages are three ways of writing the same number: 3/4 = 0.75 = 75%. You should be able to move between them quickly and choose the easiest one for the question.

A terminating decimal stops, so you can read it as tenths, hundredths or thousandths. A recurring decimal repeats for ever, but it is still an exact fraction. You find that fraction with algebra, by multiplying by a power of 10 and subtracting so the repeating part disappears.

A fraction or percentage can also act as an operator: "3/5 of" and "60% of" both mean multiply by 0.6. In a ratio, the parts added together make the whole.

Rules to remember

Terminating decimal to fraction: use place value, then simplify, e.g. 0.36 = 36/100 = 9/25.
Fraction to decimal: divide the top by the bottom, e.g. 5/8 = 5 ÷ 8 = 0.625.
Recurring decimal: let x = the decimal. Multiply by 10 if one digit repeats, by 100 if two digits repeat, then subtract to remove the repeating part.
Quick check: one repeating digit is that digit over 9 (0.777… = 7/9). Two repeating digits are those digits over 99 (0.4545… = 45/99 = 5/11).
Ratio a : b means the two parts are a/(a + b) and b/(a + b) of the whole.
"Of" means multiply: 2/3 of 60 = 2/3 × 60 = 40, and 35% of 60 = 0.35 × 60 = 21.
If you know a percentage of a number, divide to find 1% (or 10%) first, then multiply up.

Common mistake

Turning the ratio 3 : 5 into the fraction 3/5. The whole is 3 + 5 = 8 parts, so the fractions are 3/8 and 5/8.

Worked example

Write 0.41666… (only the 6 recurring) as a fraction in its simplest form.

Let x = 0.41666… Then 100x = 41.666… and 1000x = 416.666…
Subtract: 1000x − 100x = 416.666… − 41.666…, so 900x = 375.
x = 375/900. Divide top and bottom by 75: x = 5/12.
Check: 5 ÷ 12 = 0.41666…

Answer: 5/12

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