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Percentages large and small

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Percentages

Percentages large and small

Percentages are not trapped between 0 and 100. Half a per cent is a real amount, and 175% is a perfectly ordinary answer.

100% is not a ceiling100%old price+75%new price = 175%…and 0.5% is half of ONE per cent, not a half50%0.5 · 1/25%0.05 · 1/200.5%0.005 · 1/200
Whatever the size, the rule is the same: divide by 100.

Percentages below 1%

The rule never changes: divide by 100. 50% = 0.5 = 50/100 = 1/2 5% = 0.05 = 5/100 = 1/20 0.5% = 0.005 = 0.5/100 = 1/200 To simplify 0.5/100, first multiply top and bottom by 10 to get whole numbers: 5/1000. Then divide both by 5: 1/200. 0.5% is far too small to show on a diagram — but it is still a real quantity.

Worked example: 17.5%

As a decimal: divide by 100. 17.5 ÷ 100 = 0.175 As a fraction: start with 17.5/100. Multiply top and bottom by 10 to clear the decimal: 175/1000. Divide by 5: 35/200. Divide by 5 again: 7/40. So 17.5% = 0.175 = 7/40.

Percentages above 100%

If a house price rises by 75%, the new price is the old price PLUS 75% of it: 100% + 75% = 175% So the new price is 175% of the old one. Anything over 100% simply means 'more than you started with'.

Using a percentage over 100%

A child's height is now 140% of what it was two years ago. Two years ago it was 70 cm. Using decimals: 140% = 1.4, so the height is 1.4 × 70 = 98 cm. Using fractions: 140% = 140/100 = 7/5, so 7/5 × 70 = 70 ÷ 5 × 7 = 14 × 7 = 98 cm. Both give 98 cm. You can even write it with algebra: n = 1.4t, where n is the height now and t the height then.

Percentages of a total

Arun has $20. He gives $5 to Marcus and $13 to Sofia. Marcus gets 5/20 = 1/4 = 25%. Sofia gets 13/20 = 65/100 = 65%. What is left is $20 − $5 − $13 = $2, which is 2/20 = 1/10 = 10%. Check: 25% + 65% + 10% = 100%. The parts of a whole must always total 100%.

Worked examples

Write 0.5% as a decimal and a fraction

  1. Divide by 100: 0.5 ÷ 100 = 0.005.
  2. 0.5/100 → ×10 gives 5/1000 → ÷5 gives 1/200.

Answer: 0.005 and 1/200

Write 17.5% as a fraction

  1. Start with 17.5/100.
  2. ×10 to clear the decimal: 175/1000.
  3. ÷5 twice: 35/200 then 7/40.

Answer: 7/40

A price rises by 75% — what percentage is the new price?

  1. The old price is 100%.
  2. 100% + 75% = 175%.

Answer: 175% of the old price

A height is 140% of 70 cm

  1. 140% = 1.4.
  2. 1.4 × 70 = 98.

Answer: 98 cm

The same, using fractions

  1. 140% = 140/100 = 7/5.
  2. 70 ÷ 5 = 14, then 14 × 7 = 98.

Answer: 98 cm — the same answer

Arun has $20 and gives $5 to Marcus

  1. 5/20 = 1/4.
  2. 1/4 = 25/100.

Answer: Marcus got 25%

Where you see this in real life

  • A bank interest rate of 0.5% sounds like nothing, but on £20 000 it is still £100 a year.
  • 'Sales are 130% of last year' means they have grown — anything above 100% is more than you started with.

Key words

Common factor
a number that divides into both the top and the bottom
Mixed number
a whole number and a fraction together
Simplest form
a fraction that will not cancel any further
Per cent
out of 100
Decimal
a number written with a decimal point
Increase
make bigger by a given amount
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