© boardworks ltd 2005 1 of 56 n5 percentages ks4 mathematics

© Boardworks Ltd 2005 1 of 56

N5 Percentages

KS4 Mathematics


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N5.1 Fractions, decimals and percentages

Contents

N5 Percentages

N5.5 Reverse percentages

N5.6 Compound percentages

N5.2 Percentages of quantities

N5.4 Increasing and decreasing by a percentage

N5.3 Finding a percentage change


Writing percentages as fractions

‘Per cent’ means ‘out of 100’.

To write a percentage as a fraction we write it over a hundred.

For example,

46% =46

100Cancelling:

46100

=23

50

2350

180% =180100

Cancelling:180100

=

9

5

95

= 1 45

7.5% =7.5100

Cancelling:15

200

3

40

=340

=15

200


Writing percentages as decimals

We can write percentages as decimals by dividing by 100.

For example,

46% =46

100= 46 ÷ 100 = 0.46

7% =7

100= 7 ÷ 100 = 0.07

130% =130100

= 130 ÷ 100 = 1.3

0.2% =0.2

100= 0.2 ÷ 100 = 0.002


Percentages as fractions and decimals


Writing fractions as percentages

To write a fraction as a percentage, we can find an equivalent fraction with a denominator of 100.

85

For example,

=1720 100

× 5

× 5

and =10085

85%

1 725

= =3225

× 4

100

× 4

128and =

100128

128%


To write a fraction as a percentage you can also multiply it by 100%.

For example,38

=38

× 100%

=3 × 100%

8

25

2

=75%

2

= 3712%

Writing fractions as percentages

Remember, multiplying by 100% does not change the value of the number because it is equivalent to multiplying by 1.


Writing decimals as percentages

Decimals can also be converted to percentages by multiplying them by 100%.

For example,

0.08 = 0.08 × 100%

= 8%

1.375 = 1.375 × 100%

= 137.5%


Using a calculator

We can also convert fractions to decimals and percentages using a calculator.

For example,

516

= 5 ÷ 16 × 100% = 31.25%

47

= 4 ÷ 7 × 100% = 57.14% (to 2 d.p.)

58

=1 138

= 13 ÷ 8 × 100% = 162.5%


Table of equivalences


Ordering on a number line


One number as a percentage of another

There are 35 sweets in a bag. Four of the sweets are orange flavour.

What percentage of sweets are orange flavour?

Start by writing the proportion of orange sweets as a fraction.

4 out of 35 =435

Then convert the fraction to a percentage.

× 100% =435

4 × 100%35

20

7=

80%7

= 1137%


Petra put £32 into a bank account. After one year she received 80p interest.

To write 80p out of £32 as a fraction we must use the same units.

In pence, Petra gained 80p out of 3200p.

803200

=1

40

We then convert the fraction to a percentage.

140

× 100% = 100%

40

5

2

= 2.5%

One number as a percentage of another

What percentage interest rate did she receive?


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N5 Percentages







Calculating percentages using fractions

Remember, a percentage is a fraction out of 100.

15% of 90, means “15 hundredths of 90”

or

15100

× 90 =15 × 90

100

3

20

9

2

= 272

= 13 12

Find 15% of 90


Calculating percentages using decimals

We can also calculate percentages using an equivalent decimal operator.

4% of 9 = 0.04 × 9

= 4 × 9 ÷ 100

= 36 ÷ 100

= 0.36

What is 4% of 9?


Estimating percentages

We can find more difficult percentages using a calculator.

It is always sensible when using a calculator to start by making an estimate.

For example, estimate the value of:

19% of £82 20% of £80 = £16

27% of 38m 25% of 40m =10m

73% of 159g 75% of 160g = 120g


Using a calculator

One way to work out a percentage using a calculator is by writing the percentage as a decimal. For example,

What is 38% of £65?

38% = 0.38

So we key in:

0 . 3 8 × 6 5 =

The calculator will display the answer as 24.7.

We write the answer as £24.70


We can also work out a percentage using a calculator by converting the percentage to a fraction. For example,

What is 57% of £80?

57% = 57100

= 57 ÷ 100So we key in:

The calculator will display the answer as 45.6

We write the answer as £45.60

5 7 ÷ 1 0 0 × 8 0 =

Using a calculator


We can also work out percentages on a calculator by finding 1% first and then multiplying by the required percentage.

What is 37.5% of £59?

1% of £59 is £0.59 so, 37.5% of £59 is £0.59 × 37.5.

We key in:

And get an answer of 22.125

We write the answer as £22.13 (to the nearest penny).

0 . 5 9 × 3 7 . 5 =

Using a calculator


Calculating percentages


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N5 Percentages






Sometimes, we are given an original value and a new value and we are asked to find the percentage increase or decrease.

Finding a percentage increase or decrease

We can do this using the following formulae:

Percentage increase =actual increase

original amount× 100%

Percentage decrease =actual decrease

original amount× 100%


Finding a percentage increase

The actual increase = 4.2 kg – 3.5 kg

= 0.7 kg

The percentage increase =0.73.5

× 100%

= 20%

A baby weighs 3.5 kg at birth. After 6 weeks the baby’s weight has increased to 4.2 kg.

What is the baby’s percentage increase in weight?


Finding a percentage decrease

All t-shirts were £25 now

only £17!

What is the percentage decrease?

The actual decrease = £25 – £17 = £8

The percentage decrease =825

× 100% = 32%


Finding a percentage profit

Her actual profit = 50p – 32p

= 18p

Her percentage profit =1832

× 100%

= 56.25%

A shopkeeper buys chocolate bars wholesale at a price of 32p per bar. She then sells the chocolate bar in her shop at 50p each.

What is her percentage profit?


Finding a percentage loss

Her actual loss = £3.68 – £3.22

= 46p

Her percentage loss =0.463.68

× 100%

= 12.5%

A share dealer buys a number of shares at £3.68 each. After a week the price of the shares has dropped to £3.22.

What is her percentage loss?

Make sure the units are the same.


Finding a percentage change


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N5 Percentages






Percentage increase

There are two methods to increase an amount by a given percentage.

The value of Frank’s house has gone up by 20% in three years. If the house was worth £150 000

three years ago, how much is it worth now?

Method 1

We can work out 20% of £150 000 and then add this to the original amount.

= 0.2 × £150 000= £30 000

The amount of the increase = 20% of £150 000

The new value = £150 000 + £30 000= £180 000


Percentage increase

We can represent the original amount as 100% like this:

100%

When we add on 20%,

20%

we have 120% of the original amount.

Finding 120% of the original amount is equivalent to finding 20% and adding it on.

Method 2

If we don’t need to know the actual value of the increase we can find the result in a single calculation.


Percentage increase

So, to increase £150 000 by 20% we need to find 120% of £150 000.

120% of £150 000 = 1.2 × £150 000

= £180 000

In general, if you start with a given amount (100%) and you increase it by x%, then you will end up with (100 + x)% of the original amount.

In general, if you start with a given amount (100%) and you increase it by x%, then you will end up with (100 + x)% of the original amount.

To convert (100 + x)% to a decimal multiplier we have to divide (100 + x) by 100. This is usually done mentally.


Here are some more examples using this method:

Increase £50 by 60%.

160% × £50 = 1.6 × £50

= £80

Increase £24 by 35%

135% × £24 = 1.35 × £24

= £32.40

Percentage increase

Increase £86 by 17.5%.

117.5% × £86 = 1.175 × £86

= £101.05

Increase £300 by 2.5%.

102.5% × £300 =1.025 × £300

= £307.50


Percentage decrease

There are two methods to decrease an amount by a given percentage.

A CD walkman originally costing £75 is reduced by 30% in a sale. What is the sale price?

Method 1We can work out 30% of £75 and then subtract this from the original amount.

= 0.3 × £75= £22.50

30% of £75 The amount taken off =

The sale price = £75 – £22.50= £52.50


Percentage decrease

100%

When we subtract 30%

30%

we have 70% of the original amount.

70%

Finding 70% of the original amount is equivalent to finding 30% and subtracting it.

We can represent the original amount as 100% like this:

Method 2

We can use this method to find the result of a percentage decrease in a single calculation.


Percentage decrease

So, to decrease £75 by 30% we need to find 70% of £75.

70% of £75 = 0.7 × £75

= £52.50

In general, if you start with a given amount (100%) and you decrease it by x%, then you will end up with (100 – x)% of the original amount.

In general, if you start with a given amount (100%) and you decrease it by x%, then you will end up with (100 – x)% of the original amount.

To convert (100 – x)% to a decimal multiplier we have to divide (100 – x) by 100. This is usually done mentally.


Here are some more examples using this method:

Percentage decrease

Decrease £320 by 3.5%.

96.5% × £320 = 0.965 × £320

= £308.80

Decrease £1570 by 95%.

5% × £1570 = 0.05 × £1570

= £78.50

Decrease £65 by 20%.

80% × £65 = 0.8 × £65

= £52

Decrease £56 by 34%

66% × £56 = 0.66 × £56

= £36.96


Percentage increase and decrease


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N5 Percentages







Reverse percentages

Sometimes, we are given the result of a given percentage increase or decrease and we have to find the original amount.

I bought some jeans in a sale. They had 15% off and I only paid £25.50 for them.

What is the original price of the jeans?

We can solve this using inverse operations.

Let p be the original price of the jeans.

p × 0.85 = £25.50 so p = £25.50 ÷ 0.85 = £30


Sometimes, we are given the result of a given percentage increase or decrease and we have to find the original amount.

I bought some jeans in a sale. They had 15% off and I only paid £25.50 for them.

What is the original price of the jeans?

We can show this using a diagram:

Price before discount.

× 0.85%Price after discount.

÷ 0.85%

Reverse percentages


Reverse percentages


Reverse percentages

We can also use a unitary method to solve these type of percentage problems. For example,

Christopher’s monthly salary after a 5% pay rise is £1312.50. What was his original salary?

The new salary represents 105% of the original salary.

105% of the original salary = £1312.50

1% of the original salary = £1312.50 ÷ 105

100% of the original salary = £1312.50 ÷ 105 × 100

= £1250This method has more steps involved but may be easier to remember.


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AN5.6 Compound percentages

Contents


N5 Percentages






A jacket is reduced by 20% in a sale.

Compound percentages

Two weeks later the shop reduces the price by a further 10%.

What is the total percentage discount?

When a percentage change is followed by another percentage change do not add the percentages together to find the total percentage change.

The second percentage change is found on a new amount and not on the original amount.

It is not 30%!



To find a 10% decrease we multiply by 90% or 0.9.

A 20% discount followed by a 10% discount is equivalent to multiplying the original price by 0.8 and then by 0.9.


original price × 0.8 × 0.9 = original price × 0.72






This is equivalent to a 28% discount.

The sale price is 72% of the original price.

A 20% discount followed by a 10% discount

A 28% discount

A 20% discount followed by a 10% discount

A 28% discount






After a 20% discount it costs 0.8 × £100 = £80

Suppose the original price of the jacket is £100.

After an other 10% discount it costs 0.9 × £80 = £72

£72 is 72% of £100.

72% of £100 is equivalent to a 28% discount altogether.





Jenna invests in some shares.


After one week the value goes up by 10%.

The following week they go down by 10%.

Has Jenna made a loss, a gain or is she back to her original investment?

To find a 10% increase we multiply by 110% or 1.1.


original amount × 1.1 × 0.9 = original amount × 0.99

Fiona has 99% of her original investment and has therefore made a 1% loss.


Jack puts £500 into a savings account with an annual compound interest rate of 6%.

Compound interest

How much will he have in the account at the end of 4 years if he doesn’t add or withdraw any money?

At the end of each year interest is added to the total amount in the account. This means that each year 5% of an ever larger amount is added to the account.

To increase the amount in the account by 5% we need to multiply it by 105% or 1.05.

We can do this for each year that the money is in the account.


At the end of year 1 Jack has £500 × 1.05 = £525

Compound interest

At the end of year 2 Jack has £525 × 1.05 = £551.25

At the end of year 3 Jack has £ 551.25 × 1.05 = £578.81

At the end of year 4 Jack has £578.81 × 1.05 = £607.75

(These amounts are written to the nearest penny.)

We can write this in a single calculation as

£500 × 1.05 × 1.05 × 1.05 × 1.05 = £607.75

Or using index notation as

£500 × 1.054 = £607.75


How much would Jack have after 10 years?

Compound interest

After 10 years the investment would be worth

£500 × 1.0510 = £814.45 (to the nearest 1p)

How long would it take for the money to double?

£500 × 1.0514 = £989.97 (to the nearest 1p)

£500 × 1.0515 = £1039.46 (to the nearest 1p)

Using trial and improvement,

It would take 15 years for the money to double.


Compound interest


We can use powers to help solve many problems involving repeated percentage increase and decrease. For example,

Repeated percentage change

The population of a village increases by 2% each year.If the current population is 2345, what will it be in 5 years?

To increase the population by 2% we multiply it by 1.02.

After 5 years the population will be

2345 × 1.025 = 2589 (to the nearest whole)

What will the population be after 10 years?

After 5 years the population will be

2345 × 1.0210 = 2859 (to the nearest whole)


Repeated percentage change

The car costs £24 000 in 2005. How much will it be worth in 2013?

To decrease the value by 15% we multiply it by 0.85.

After 8 years the value of the car will be

£24 000 × 0.858 = £6540 (to the nearest pound)

The value of a new car depreciates at a rate of 15% a year.

There are 8 years between 2005 and 2013.

© boardworks ltd 2005 1 of 56 n5 percentages ks4 mathematics

Documents

percentages decimals

percentage n5

writing percentages

reverse percentages

compound percentages

percentages of quantities

percentage change slide

percentage of sweets