Converting fractions, decimals and percentages

Knowing these makes non-calculator questions much faster.

• $\tfrac{3}{8}$: $3 \div 8 = 0.375$, so $37.5\%$.
• $0.45 = \tfrac{45}{100} = \tfrac{9}{20}$ (divide top and bottom by 5).
• $64\% = 0.64 = \tfrac{64}{100} = \tfrac{16}{25}$.
• $\tfrac{7}{20}$: make the bottom 100: $\tfrac{35}{100} = 0.35 = 35\%$.

If the denominator goes into 100 (2, 4, 5, 10, 20, 25, 50), scaling up to 100 is quicker than dividing. Simplify with the fraction calculator or how to simplify fractions.

Put in order, smallest first: $\tfrac{5}{8}$, 0.6, 62%.

Convert everything to decimals: $\tfrac{5}{8} = 0.625$, $0.6$, $62\% = 0.62$. So the order is 0.6, 62%, $\tfrac{5}{8}$. Writing 0.6 as 0.600 helps you compare digits in the same place.

$0.\dot{4} = 0.444\ldots$ Let $x = 0.444\ldots$. Then $10x = 4.444\ldots$. Subtract: $9x = 4$, so $x = \tfrac{4}{9}$.

For two repeating digits, multiply by 100: $x = 0.\dot{2}\dot{7}$, $100x = 27.2727\ldots$, $99x = 27$, $x = \tfrac{27}{99} = \tfrac{3}{11}$. This is a GCSE Higher skill.

• Writing 0.5% for one half (it's 50%).
• Writing $0.07$ as 7% correctly but $0.7$ as 7% (it's 70%).
• Dividing the bottom by the top.
• Ordering by the digits you see: 0.6 is bigger than 0.58, even though 58 > 6.

1. Write $\tfrac{9}{25}$ as a percentage.
2. Write $0.125$ as a fraction in its simplest form.
3. Write $85\%$ as a fraction in its simplest form.
4. Order, smallest first: $\tfrac{2}{3}$, 0.65, 66%.

Answers: 36%, $\tfrac{1}{8}$, $\tfrac{17}{20}$, then 0.65, 66%, $\tfrac{2}{3}$. Next, try percentage change.

The same equivalences make "of" questions easier. To find 35% of £60, you can use any form:

• Percentage: 10% is £6, 5% is £3, so 35% is three lots of £6 plus £3, which is £21.
• Decimal: $0.35 \times 60 = 21$.
• Fraction: $35\% = \tfrac{7}{20}$, and $\tfrac{7}{20} \times 60 = 21$.

Pick whichever is easiest for the numbers. Check with the percentage calculator.