© boardworks ltd 2004 1 of 58 ks3 mathematics n7 percentages

© Boardworks Ltd 20041 of 58

KS3 Mathematics

N7 Percentages


N7.1 Equivalent fractions, decimals and percentages

Contents

N7 Percentages

N7.2 Calculating percentages mentally

N7.3 Calculating percentages on paper

N7.4 Calculating percentages with a calculator

N7.5 Comparing proportions

N7.6 Percentage change


Many words begin with ‘cent’:

Percentages

1900 - 20001900 - 2000


Percent means . . .

Percentages


A percentage is just a special type of fraction.

1%1% means 1 part per hundred1 part per hundred

or100100

11= 0.010.01

Percentages



10%10% means 10 parts per hundred10 parts per hundred

or1001001010

=101011

= 0.10.1

Percentages




or1001002525

=4411

= 0.250.25

Percentages




or1001005050

=2211

= 0.50.5

Percentages




or100100100100

= 11

Percentages


Percentages of shapes


Estimating percentages


Equivalent fractions, decimals and percentages


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


Writing decimals as percentages

To write a decimal as a percentage you can multiply it 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


Dominoes



Contents


N7 Percentages






Calculating percentages mentally

Some percentages are easy to work out mentally:

To find 1% Divide by 100





We can use percentages that we know to find other percentages.

Suggest ways to work out:

20%30%

60%

15%

2%75%

150%

49%

11%0.5 % 17.5%

Calculating percentages mentally


Spider diagram



Contents

N7 Percentages







Calculating percentages using fractions

Remember, a percentage is a fraction out of 100.

16% of 90, means “16 hundredths of 90”

or

16100

× 90 =16 × 90

100

4

25

18

5

= 725

= 14 25


What is 23% of 57?

We can use fractions:

23% of 57 =23

100× 57

=23 × 57

100

Working

× 20 3

50

7

1000 150

140 21

1150

+ 161

11

1

31= 1311

100

= 13 11100



What is 87% of 28?

Using fractions again:

87% of 28 =87

100× 28

=87 × 28

100

7

25

Working

87× 7

94

60

= 60925

= 24 925



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?



Contents

N7 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

By writing a percentage as a decimal, we can work out a percentage using a calculator.

Suppose we want to work out 38% of £65.

38% = 0.38

So we key in:

0 . 3 8 × 6 5 =

And get an answer of 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.

Suppose we want to work out 57% of £80.

57% = 57100

= 57 ÷ 100

So we key in:

5 7 ÷ 1 0 0 × 8 0


We write the answer as £45.60

=

Using a calculator


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

Suppose we want to work out 37.5% of £59.

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

We key in:


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

0 . 5 9 × 3 7 . 5 =

Using a calculator



Contents

N7 Percentages







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.What percentage interest rate did she receive?

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


Using percentages to compare proportions

To compare the marks we can write each fraction as a percentage.

Matthew sat tests in English, Maths and Science.

His results were:

ScienceMathsEnglish

7480

1720

6670

Which test did he do best in?


English

7480

=7480

× 100% = 74 ÷ 80 × 100% = 92.5%

Maths

1720

=1720

× 100% = 17 ÷ 20 × 100% = 85%

Science

6670

=6670

× 100% = 66 ÷ 70 × 100% = 94.3% (to 1 d.p.)

We can see that Matthew did best in his Science test.



Nutrition Information

ChocolateCookies

Typical Value Per 10g biscuit

EnergyProteinCarbohydrateFatFibreSodium

233kj0.6g6.7g2.2g0.2g

<0.05g

Nutrition Information

CheesyCrisps

Typical Value Per 23 g bag

EnergyProteinCarbohydrateFatFibreSodium

504kj1.6g13g7g

0.3g0.2g

Which product contains the smallest percentage of carbohydrate?



The chocolate cookies contain 6.7g of carbohydrate for every 10g of biscuits.

6.7g out of 10g =6.710

× 100% = 6.7 ÷ 10 × 100% = 67%

The cheesy crisps contain 13g of carbohydrate for every 23g of crisps.

13g out of 23g =1323

× 100% = 13 ÷ 23 × 100%

= 56.5% (to 1 d.p)

The cheesy crisps contain a smaller percentage of carbohydrate.




Contents

N7 Percentages







Percentage increase and decrease

Factory workers demand 15% pay increase

SALE20% off all

marked prices!

Bus fares set to rise by 30%

PC now only

£568 Plus 17 % VAT1

2

House prices predicted to fall by 2% next year25% extra free!


Percentage increase

To increase an amount by a 20%, for example, we can find 20% of the amount and then add it on to the original amount.

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.


For example,

Increase £50 by 60%.

160% × £50 = 1.6 × £50

= £80

Increase £20 by 35%

135% × £20 = 1.35 × £20

= £27

Percentage increase


What happens if we increase an amount by 100%?

We take the original amount

100%

and we add on 100%.

100%

We now have 200% of the original amount.

This is equivalent to 2 times the original amount.

Percentage increase


What happens if we increase an amount by 200%?

We take the original amount

100%

and we add on 200%.

200%

We now have 300% of the original amount.

This is equivalent to 3 times the original amount.

Percentage increase


Percentage decrease

To decrease an amount by 30%, for example, we can find 30% of the amount and then subtract it from the original amount.

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

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.


For example,

Decrease £75 by 20%.

80% × £75 = 0.8 × £75

= £60

Decrease £56 by 34%

66% × £56 = 0.66 × £56

= £36.96

Percentage decrease


Jason bought £200 worth of shares.

In the first week the shares went up 12%.

In the second week, however, the shares went down 12%.

“Oh well,” said Jason, “at least I’m back to the amount I started with.”

Jason is wrong. Why?

Shares problem


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

All t-shirts were £25 now

only £17!

Finding a percentage change

What is the percentage decrease?

The decrease is £25 – £17 = £8

The percentage decrease is825

=32100

= 32%


Finding the original amount

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%


© boardworks ltd 2004 1 of 58 ks3 mathematics n7 percentages

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