… A fraction is a part of a whole.
… As long as you multiply the top (the numerator) and the bottom (the denominator) by the same number, you will keep the fraction the same. It will become an equivalent fraction. For example 2/5 = 6/15 (multiplying top and bottom by 3).
… Always see if you can divide the top and bottom of a fraction by a number in order to simplify the fraction into its ‘lowest form’.
… When adding or subtracting fractions, you must first find a common …?… It is easiest to find the LCM (lowest common multiple) of each denominator.
For example:
3/4 + 5/6 … the LCM is 12 so you would need to multiply the first fraction (top and bottom) by …?… and the second fraction by …?…
Note that you could also use 6 x 4 = 24 as the common denominator.
… To multiply fractions, first make sure you have cancelled numerators and denominators as far as possible. Then multiply the numerators together and the denominators together.
… To divide fractions, flip the second fraction on its head and change the ÷ to a ×.
Percentages
… A percentage is like a fraction because it is also a part of a whole:
Percentage = (the part or change / starting amount) x 100
- Start with the fraction (part / total) and then multiply the top and bottom to make the denominator 100, for example:
3/20 = 15/100 = 15% (this is useful for the non-calculator paper).
OR, for more awkward fractions by multiplying the fraction by 100%, for example:
3/20 × 100% = 15% (this is useful for the calculator paper, but you here we can do this by cancelling)
… Finding the percentage of an amount, e.g. find 17.5% of £60 by splitting up and adding the percentages, i.e. finding 10%, 5% and 2.5%
… Finding the percentage of an amount using a calculator method. For example:
Find 7% of £450
7/100 × 450
= £31.50
… Calculating percentage increase and decrease using a calculator method , for example,
increase £30 by 12%
= 112% x 30
= 112/100 x 30
= 1.12 x 30 = 33.6
Decrease 240 by 12%
= 88% x 240
= 88/100 x 240
= 221.2
… Reverse percentage questions, e.g. finding the ‘original price’ after a percentage reduction or increase. An easy way of doing this is to us ratios. For example, a car is priced at £3200 in a sale with 20% off. What was its original price?
100 – 20 = 80%
80% = £3200
(÷ 80) (÷ 80)
1% = £40
(× 100 ) (× 100)
100% = £4000
… Reverse percentage questions. E.g. Finding the ‘Original price’ after a percentage reduction. Remember to call the original price ‘x’ and form an equation using your percentage increase or decrease method.
… If a shop has a 20% off sale and a computer now costs £540, what was the original price of the computer before the sale?
… Compound interest calculation. For example, if your investment pays 4% each year, then if you start with £1000:
End of Year 1 = 1000 x 104/100 = 1040 … Interest = £40
End of year 2 = 1040 x 104/100 = 1081.6 … Interest = £41.6
Compound interest means that your money grows faster and faster each year – so it’s best to start saving from a young age!
… The faster way of calculating compound interest is to use the formula:
Final Amount = Starting amount x (100% + interest%)^n,
where n = the number of years.
For example, in the above case:
Final amount = £1000 x (100% + 4%)^2 = 1000(1.04)^2 = £1081.60
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