Post on 13-Dec-2015
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KS3 Mathematics
N7 Percentages
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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
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Many words begin with ‘cent’:
Percentages
1900 - 20001900 - 2000
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“Percent” means . . .
Percentages
“out of a hundred”
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A percentage is just a special type of fraction.
1% means 1 part per hundred
or100
1= 0.01
Percentages
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A percentage is just a special type of fraction.
10% means 10 parts per hundred
or10010
=101
= 0.1
Percentages
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A percentage is just a special type of fraction.
25% means 25 parts per hundred
or10025
=41
= 0.25
Percentages
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A percentage is just a special type of fraction.
50% means 50 parts per hundred
or10050
=21
= 0.5
Percentages
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A percentage is just a special type of fraction.
100% means 100 parts per hundred
or100100
= 1
Percentages
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Percentages of shapes
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Estimating percentages
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Estimating percentages
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Equivalent fractions, decimals and percentages
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Writing percentages as fractions
‘Per cent’ means ‘out of 100’. ‘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
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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
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Percentages as fractions and decimals
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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%
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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
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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%
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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.)
13 ÷ 8 × 100% = 162.5%58
=1 =138
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Table of equivalences
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Table of equivalences
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Ordering on a number line
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Dominoes
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N7.2 Calculating percentages mentally
Contents
N7.1 Equivalent fractions, decimals and percentages
N7 Percentages
N7.3 Calculating percentages on paper
N7.4 Calculating percentages with a calculator
N7.5 Comparing proportions
N7.6 Percentage change
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Calculating percentages mentally
Some percentages are easy to work out mentally:
To find 1%1% Divide by 100Divide by 100
To find 10%10% Divide by 10Divide by 10
To find 25%25% Divide by 4Divide by 4
To find 50%50% Divide by 2Divide by 2
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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
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Spider diagram
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N7.3 Calculating percentages on paper
Contents
N7 Percentages
N7.1 Equivalent fractions, decimals and percentages
N7.2 Calculating percentages mentally
N7.4 Calculating percentages with a calculator
N7.5 Comparing proportions
N7.6 Percentage change
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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
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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
Calculating percentages using fractions
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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 fractions
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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?
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N7.4 Calculating percentages with a calculator
Contents
N7 Percentages
N7.1 Equivalent fractions, decimals and percentages
N7.2 Calculating percentages mentally
N7.3 Calculating percentages on paper
N7.5 Comparing proportions
N7.6 Percentage change
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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
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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.
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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:
And get an answer of 45.6.
We write the answer as £45.60.
5 7 ÷ 1 0 0 × 8 0 =
Using a calculator
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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:
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
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N7.5 Comparing proportions
Contents
N7 Percentages
N7.1 Equivalent fractions, decimals and percentages
N7.2 Calculating percentages mentally
N7.3 Calculating percentages on paper
N7.4 Calculating percentages with a calculator
N7.6 Percentage change
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One number as a percentage of another
There are 35 sweets in a bag. Four of the 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%
What percentage of sweets are orange flavour?
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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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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?
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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.
Using percentages to compare proportions
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Nutrition Information
ChocolateCookies
Typical Value Per 10g biscuit
EnergyProteinCarbohydrateFatFibreSodium
233kj0.6g6.7g2.2g0.2g
<0.05g
Nutrition Information
CheesyCrisps
Typical Value Per 23g bag
EnergyProteinCarbohydrateFatFibreSodium
504kj1.6g13g7g
0.3g0.2g
Which product contains the smallest percentage of carbohydrate?
Using percentages to compare proportions
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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.
Using percentages to compare proportions
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N7.6 Percentage change
Contents
N7 Percentages
N7.1 Equivalent fractions, decimals and percentages
N7.2 Calculating percentages mentally
N7.3 Calculating percentages on paper
N7.4 Calculating percentages with a calculator
N7.5 Comparing proportions
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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 month25% extra free!
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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% since last year. If the house was worth £150 000
last year 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
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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.
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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.
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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
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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
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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
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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
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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.
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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.
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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
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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%
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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?
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Finding a percentage decrease
All t-shirts were £24 now
only £18!
What is the percentage decrease?
The actual decrease = £24 – £18 = £6
The percentage decrease =624
× 100% = 25%1
4
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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
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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%
Finding the original amount
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Finding the original amount
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Finding the original amount
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.