Fraction Reaction
Move fluently between fractions, decimals and percentages, then master the multiplier method: percentage change, reverse percentages, and the recurring-decimal proof that grade 9s never miss.
Get the method right under pressure
Free interactive practice on the steps that lose marks under exam pressure.
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Fraction Reaction
Fractions, decimals and percentages are three ways of writing the same thing: ¾ = 0.75 = 75%. Flipping between them at will is the key to the whole topic, and then one tool, the multiplier, turns percentage change and reverse percentages into one-step calculations. Three conversions to have on demand:
Match each fraction to its percentage
- 1/2
- 1/4
- 3/4
- 1/5
- 50%
- 25%
- 75%
- 20%
Fraction to decimal
Write 3/8 as a decimal (divide 3 by 8).
Percentages as operators
A percentage can act on an amount. To find a percentage of something, turn the percentage into a decimal and multiply: 15% of 240 = 0.15 × 240 = 36. This "×decimal" idea is the seed of the multiplier method you will use for every percentage-change question.
Percentage of an amount
Work out 15% of 240. (Multiply 240 by 0.15.)
The multiplier method
Find the multiplier once and it does both jobs. Increase by 15% → × 1.15 (100% + 15%); decrease by 20% → × 0.8 (100% − 20%); increase by 150% → × 2.5, since percentages over 100% are fine. What changes is the direction you use it in, and mixing those up is the commonest slip in this topic:
Increase by a percentage
Increase 80 by 15% using the multiplier 1.15 (work out 80 × 1.15).
Reverse percentage
After a 20% increase, a coat costs £60. What was the ORIGINAL price?
- £50
- £72
- £48
- £40
Recurring decimals
A recurring decimal such as 0.4545... is exactly equal to a fraction, and you prove it with algebra (a Higher "show that"): Let x = 0.4545... Two digits repeat, so multiply by 100: 100x = 45.4545... Subtracting removes the recurring tail: 100x − x = 45, so 99x = 45 and x = 45/99 = 5/11. Every algebraic line must be written down to score full marks.
Order the proof
Put the steps for writing 0.4545... as a fraction in order.
- Let x = 0.4545...
- Two digits repeat, so multiply by 100: 100x = 45.4545...
- Subtract: 100x − x = 45, so 99x = 45
- Solve and simplify: x = 45/99 = 5/11
Which way round?
Three percentage questions. For each, decide what to do with the multiplier.
- A £240 bike is reduced by 30% in a sale. Find the sale price.
- After a 30% reduction a bike costs £168. Find the price before the sale.
- A wage rose by 5% to £26 250. Find the wage before the rise.
Percentage rules
To increase an amount by 15%, multiply by _____. To reverse a percentage change and find the original amount, you _____ by the multiplier. To write a percentage as a fraction, put it over _____.