Percentages - Emaths



Percentages

Percent means ‘out of 100’

Expressing an amount as a percentage of another:

Non-calculator method - write as a fraction and change into hundredths using equivalent fractions

e.g. express 12 out of 20 as a percentage [pic]

Calculator method – if the fraction can not be easily changed into hundredths, then write the fraction as a decimal then multiply by 100

e.g. express 13 out of 18 as a percentage [pic]

Finding a percentage of an amount

Non-calculator method – it is easy to find 10% of an amount by dividing by 10

(10x10%=100%)

5% is half of 10%, 2.5% is half of 5% .

e.g. Find 35% of £60 10% of £60 = £60 ÷ 10 = £6

30% of £60 = 3 x 10% = 3 x £6 = £18

5% of £60 = 10% ÷ 2 = £6 ÷ 2 = £3

so £35% of £60 = 30% of £60 + 5% of £60

= £18 + £3

= £21

Calculator method – write the percentage as a decimal and multiply

e.g. find 28% of £460 28% of £460 = 0.28 x £460

= £128.80

I am sure that you will agree that this is easy but only if you bring a calculator.

Percentage profit/loss/increase/decrease

Percentage profit/loss/increase/decrease = actual profit/loss/increase/decrease x 100%

starting (original) value

e.g. A car is bought for £4000 and then sold on later for £4800. Calculate the percentage profit

actual profit = £4800 - £4000 = £800

Percentage profit = 800 x 100%

4000

= 800 ÷ 4000 x 100% = 20%

Multipliers and Compound Interest

If a question tells you that you save 30% of the cost of an item in a sale and then asks you to ‘calculate the sale price’, you can just find £30% and then subtract this from the original price. Alternatively you might realise that if you are saving 30% then your answer will be 70% of the original price. If you were to multiply the original price by 0.7 (the multiplier which corresponds to a reduction of 30%), you would have the ‘sale price’

If you have to add V.A.T. (17.5%) onto an amount, you could multiply by 1.175 (the multiplier which corresponds to an increase of 17.5%) to get your answer.

This is very useful when answering questions about compound interest.

For compound interest:

new value = starting value x (percentage change as a multiplier) no. of years, months etc

e.g. A man puts £2500 in a bank account which pays 4% interest per year. How much money will he have after 3 years?

starting value = £2500 multiplier = 1.04 no. of years = 3

new value = 2500 x 1.043

= £2812.16

N.B. This is NOT the same as finding 4% of £2500 and then multiplying by the number of years because as each year goes by you are finding 4% of the initial value plus the previous years interest.

Finding the Original Price

A coat costs £72 in a sale after it has been reduced by 20%. Find the cost of the coat before this reduction.

Coat now costs 80% of it’s ‘pre-sale’ value (multiplier = 0.8)

0.8 x pre-sale value = £72

pre-sale value = £72 ÷ 0.8 = £90

A house is now worth 30% more than it was purchased for.

If it is now worth £156000, what was the original purchase price?

House is worth 130% of original purchase price (multiplier = 1.3)

1.3 x original purchase price = £156000

original purchase price = £156000 ÷ 1.3

= £120000

Past Paper Questions

[pic]

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[pic]

10% of £140 = £140 ÷ 10 = £14

5% of £140 = 10% of £140 ÷ 2 = £14 ÷ 2 = £7

V.A.T. = £7

[pic]

Total amount = £140 + £7 = £147

[pic]

% increase = INCREASE x 100

ORIGINAL VALUE

= 0.7 x 100

3.20

= 21.875%

[pic]

After the reduction we have 80% (100% - 20%) of the original amount and we need to find 100%.

80% = 120 so 10% = 80% ÷ 8 = 120 ÷ 8 = 15

100% = 10% x 10 = 15 x 10 = 150

150 tissues in the box before the reduction.

[pic]

10% of 80 = 8

5% of 80 = 4

30% of 80 = 24

35% of 80 = 24 + 4 = 28

Number of employees = 80 + 28 = 108

[pic]

12% of €2.80 = 0.12 x €2.80 = €0.34

[pic]

A 6% decrease written as a multiplier is 0.94.

There are 4 lots of 3 months in a year so:

Value on 1st January 2004 = £800 x 0.944

= £624.60

[pic]

We know has 140% of the original number of books so we need to find 100%

140% = 70 so 10% = 70 ÷ 14 = 5

100% = 10 x 10% = 10 x 5 = 50

John originally had 50 books

And know for some Higher level Questions

[pic]

Let the original cost of the dresses be x and the original number of dresses sold be y.

Original takings = xy

After 1 month, dresses cost 0.7x but they sell 1.6y dresses

takings = 0.7x x 1.6y

= 1.12xy

Percentage Increase = INCREASE x 100

ORIGINAL TAKINGS

= 1.12xy – xy x 100

xy

= 0.12xy x 100

xy

= 0.12 x 100 = 12%

[pic]

% saving = saving x 100

‘full price’

= 1.15 x 100 = 20%

5.75

[pic]

If we call the cost of each item, x

% saving = x x 100 = 33.3%

3x

[pic]

20% increase as a multiplier = 1.2

If 1st windmill produces 1.7 MW then 2nd windmill produces 1.7 x 1.2 = 2.04 MW

[pic]

3rd windmill produces 1.7 x 1.23 = 2.9376 MW

4th windmill produces 1.7 x 1.24 = 3.52512 MW

Every windmill after windmill 3 produces at least 3MW which is 7 windmills.

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