Converting Between Fractions, Decimals and Percentages

Grade 3

Why you need all three forms

Fractions, decimals and percentages are three ways of writing the same thing. A quarter, 0.25, and 25% are identical in value — the form just changes. Exams will swap between them constantly, so being able to move in any direction is essential.

The good news: there are just a handful of rules to learn, and they always work.

Percentage ↔ Decimal

This is the easiest pair. A percentage is always "out of 100", so:

Percentage → Decimal: divide by 100 (shift the decimal point two places left)
Decimal → Percentage: multiply by 100 (shift the decimal point two places right)
Dividing by 100 moves the decimal point two places to the left. 45% → 0.45. Multiplying by 100 moves it two places to the right. 0.07 → 7%.

Fraction ↔ Decimal

To turn a fraction into a decimal, divide the top number (numerator) by the bottom number (denominator). Think of the fraction bar as a division sign.

Fraction → Decimal: top ÷ bottom

To go the other way, write the decimal as a fraction over the appropriate power of 10 (10, 100, 1000…), then simplify.

EXAMPLE
Convert ³⁄₈ to a decimal.
Divide top by bottom: 3 ÷ 8 = 0.375. So ³⁄₈ = 0.375.

Fraction ↔ Percentage

The cleanest route is to go via the decimal: fraction → decimal → percentage. Or, if the denominator divides neatly into 100, you can scale the fraction so its denominator is 100 — the numerator then gives you the percentage directly.

EXAMPLE
Write ¹³⁄²⁰ as a percentage.
20 goes into 100 five times, so multiply top and bottom by 5: 13 × 5 = 65, 20 × 5 = 100. So ¹³⁄²⁰ = ⁶⁵⁄₁₀₀ = 65%.
If the denominator doesn't divide neatly into 100 (e.g. ³⁄₇), use the decimal route instead: 3 ÷ 7 = 0.4286…, then × 100 = 42.86%.
Nova
Try this one

In a sale, a television is reduced by 18% to £516.

(a) Work out the original price of the television. [2]

After the sale, the price is increased by 18% from the sale price.

(b) Explain why the price after the increase is not the same as the original price, and find the new price. [2]

Nova's hint:

Part (a) is a reverse percentage; part (b) explores why percentage changes are not symmetrical.

Ask yourself: 'What percentage of the original price is £516, and what do I divide by to reverse a percentage reduction?' Think about whether 18% of £516 is the same value as 18% of the original price.

Try to Solve it with Nova?