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KS4 Mathematics

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KS4 Mathematics

N6 Ratio and proportion

N6 Ratio and proportion

- A

Contents

N6.2 Dividing in a given ratio

- A

N6.3 Direct proportion

- A

N6.1 Ratio

N6.4 Inverse proportion

- A

N6.5 Proportionality to powers

- A

N6.6 Graphs of proportional relationships

- A

Comparing lengths of line segments

What is the ratio of red counters to blue counters?

A ratio compares the sizes of parts or quantities to each other.

For example,

Ratio

red : blue

= 9 : 3

= 3 : 1

For every three red counters there is one blue counter.

For example,

What is the ratio of blue counters to red counters?

A ratio compares the sizes of parts or quantities to each other.

Ratio

The ratio of blue counters to red counters is not the same as the ratio of red counters to blue counters.

blue : red

= 3 : 9

= 1 : 3

For every blue counter there are three red counters.

What is the ratio of red counters to yellow counters to blue counters?

Ratio

red : yellow : blue

= 12 : 4 : 8

= 3 : 1 : 2

For every three red counters there is one yellow counter and two blue counters.

÷ 7

÷ 7

÷ 3

÷ 3

Ratios can be simplified like fractions by dividing each part by the highest common factor.

For example,

Simplifying ratios

21 : 35

= 3 : 5

For a three-part ratio all three parts must be divided by the same number.

For example,

6 : 12 : 9

= 2 : 4 : 3

Simplifythe ratio 90p : £3

÷ 30

÷ 30

When a ratio is expressed in different units, we must write the ratio in the same units before simplifying.

First, write the ratio using the same units.

90p : 300p

When the units are the same we don’t need to write them in the ratio.

90 : 300

= 3 : 10

Simplify the ratio 0.6 m : 30 cm : 450 mm

÷ 15

÷ 15

First, write the ratio using the same units.

60 cm : 30 cm : 45 cm

60 : 30 : 45

= 4 :

2 :

3

Simplify the ratio 0.8 : 2

× 10

× 10

÷ 4

÷ 4

When a ratio is expressed using fractions or decimals we can simplify it by writing it in whole-number form.

We can write this ratio in whole-number form by multiplying both parts by 10.

0.8 : 2

= 8 : 20

= 2 : 5

Simplify the ratio : 4

2

: 4

3

× 3

× 3

÷ 2

÷ 2

2

3

We can write this ratio in whole-number form by multiplying both parts by 3.

= 2 : 12

= 1 : 6

÷ 5

÷ 5

÷ 8

÷ 8

We can compare ratios by writing them in the form 1 : morm: 1, wheremis any number.

For example, the ratio 5 : 8 can be written in the form 1 : m by dividing both parts of the ratio by 5.

Comparing ratios

5 : 8

= 1 : 1.6

The ratio 5 : 8 can be written in the form m : 1 by dividing both parts of the ratio by 8.

5 : 8

= 0.625 : 1

The ratio of boys to girls in class 9P is 4:5.

The ratio of boys to girls in class 9G is 5:7.

Which class has the higher proportion of girls?

The ratio of boys to girls in 9P is

4 : 5

÷ 4

÷ 5

÷ 4

÷ 5

The ratio of boys to girls in 9G is

5 : 7

Comparing ratios

= 1 : 1.25

= 1 : 1.4

9G has a higher proportion of girls.

θ

However in this context we write the ratio as .

opposite

adjacent

In some situations a ratio can be given as a single fraction.

For example, suppose we are investigating the lengths of the sides in a right angled triangle:

Writing ratios as fractions

We could write the ratio of the length of the opposite side to the length of the adjacent side as

This is the side opposite the angle θ.

This is the side adjacent to the angle θ.

opposite : adjacent

This ratio is called the tangent of the angle θ.

What is the ratio of the height to the width of the photograph

a) using ratio notation

b) as a fraction?

÷ 2.5

÷ 2.5

height

7.5

b)

=

width

12.5

3

3

=

5

5

We could say that the height is of the width.

Writing ratios as fractions

a) height : width

7.5 : 12.5

3 : 5

7.5 cm

12.5 cm

Suppose the picture is reduced in size so that its width is 7.5 cm. What is the height of the reduced picture?

Finding the missing number in a ratio

We have established that the ratio of the height to the width is 3 : 5.

?

The ratio of the height to the width must remain the same or the picture will be distorted.

7.5 cm

We must therefore find a ratio equivalent to 3 : 5 but with the second part equal to 7.5.

3 : 5

? : 7.5

× 1.5

× 1.5

To find the missing number in the ratio we have to work out what we have multiplied 5 by to get 7.5:

Finding the missing number in a ratio

3 : 5

… so the 3 must be multiplied by 1.5.

The 5 is multiplied by 1.5 …

? : 7.5

4.5

To do this divide 7.5 by 5.

7.5 ÷ 5 = 1.5

So when the width of the rectangle is 7.5 cm this height is 4.5 cm.

× 12

× 12

The ratio of boys to girls in year 10 of a particular school is 6 : 7. If there are 72 boys, how many girls are there?

Again we can work this out by finding the missing number in the ratio.

Finding the missing number in a ratio

6 : 7

The 6 is multiplied by 12 …

… so the 7 must be multiplied by 12.

72 : ?

84

To do this divide 72 by 6.

72 ÷ 6 = 12

If there are 72 boys there must be 84 girls.

N6 Ratio and proportion

N6.1 Ratio

- A

Contents

- A

N6.3 Direct proportion

- A

N6.2 Dividing in a given ratio

N6.4 Inverse proportion

- A

N6.5 Proportionality to powers

- A

N6.6 Graphs of proportional relationships

- A

Dividing a length in a given ratio

Divide £40 in the ratio 2 : 3.

A ratio is made up of parts.

Dividing in a given ratio

We can write the ratio 2 : 3 as

2 parts : 3 parts

The total number of parts is

2 parts + 3 parts = 5 parts

We need to divide £40 by the total number of parts.

£40 ÷ 5 = £8

Divide £40 in the ratio 2 : 3.

and

3 parts =

Each part is worth £8 so

Dividing in a given ratio

2 parts =

2 × £8 =

£16

3 × £8 =

£24

£40 divided in the ratio 2 : 3 is

£16 : £24

Always check that the parts add up to the original amount.

£16 + £24 =

£40

A citrus twist cocktail contains orange juice, lemon juice and lime juice in the ratio 6 : 3 : 1.

How much of each type of juice is contained in 750 ml of the cocktail?

Dividing in a given ratio

First, find the total number of parts in the ratio.

6 parts + 3 parts + 1 part =

10 parts

Next, divide 750 ml by the total number of parts.

750 ml ÷ 10 =

75 ml

A citrus twist cocktail contains orange juice, lemon juice and lime juice in the ratio 6 : 3 : 1.

How much of each type of juice is contained in 750 ml of the cocktail?

Dividing in a given ratio

Each part is worth 75 ml so,

6 parts of orange juice = 6 × 75 ml =

450 ml

3 parts of lemon juice = 3 × 75 ml =

225 ml

1 part of lime juice = 75 ml

Check that the parts add up to 750 ml.

450 ml + 225 ml + 75 ml =

750 ml

Dividing in a given ratio spider diagram

N6 Ratio and proportion

N6.1 Ratio

- A

Contents

N6.2 Dividing in a given ratio

- A

- A

N6.3 Direct proportion

N6.4 Inverse proportion

- A

N6.5 Proportionality to powers

- A

N6.6 Graphs of proportional relationships

- A

Two quantities are said to be in direct proportion if they increase and decrease at the same rate. That is, if the ratio between the two quantities is always the same.

Direct proportion

For example, the speed that a car travels is directly proportional to the distance it covers.

If the car doubles its speed it will cover double the distance in the same time.

If the car halves its speed it will cover half the distance in the same time.

If the car is at rest it won’t cover any distance. That is, if its speed is zero the distance covered is zero.

Are the following directly proportional?

3 packets of crisps weigh 84 g.

How much do 12 packets weigh?

Direct proportion problems

3 packets weigh 84 g.

× 4

× 4

12 packets weigh

336 g.

If we multiply the number of packets by four then we have to multiply the weight by four.

If all the packets weigh the same then the ratio between the number of packets and the weight is constant.

3 packets of crisps weigh 84 g.

How much does 1 packet weigh?

Direct proportion problems

3 packets weigh 84 g.

÷ 3

÷ 3

1 packet weighs

28 g.

We divide the number of packets by three and divide the weight by three.

Once we know the weight of one packet we can work out the weight of any number of packets.

3 packets of crisps weigh 84 g.

How much do 7 packets weigh?

÷ 3

÷ 3

× 7

× 7

Direct proportion problems

3 packets weigh 84 g.

1 packet weighs 28 g.

7 packets weigh

196 g.

This is called using a unitary method.

3 packets of crisps weigh 84 g.

How much do 7 packets weigh?

7

7

×

×

3

3

7

3

Direct proportion problems

We could also work this out in a single step as follows,

3 packets weigh 84 g.

7 packets weigh

196 g.

What do we multiply 3 by to get 7?

.

To work this out we divide 7 by 3 to get

3 packets of crisps weigh 84 g.

How much do 7 packets weigh?

× 28

× 28

Direct proportion problems

Alternatively, we could scale from 3 to 84 by multiplying by 28.

3 packets weigh 84 g.

7 packets weigh

196 g.

£8 is worth 13 euros.

How much is £2 worth?

1

To scale from £8 to £2 we

or × 0.25

×

4

1

1

×

×

4

4

or × 0.25

or × 0.25

Direct proportion problems

£8 is worth 13€

£2 is worth

(13 ÷ 4)€

= 3.25€

£8 is worth 13 euros.

How much is £2 worth?

13

Alternatively, to scale from 8 to 13 we

or × 1.625

×

8

13

13

or × 1.625

or × 1.625

×

×

8

8

Direct proportion problems

£8 is worth 13€

£2 is worth

(2 × 1.625)€ =

3.25€

£8 is worth 13 euros.

How much is £2 worth?

13

or × 1.625

×

8

8

×

or × 0.615 (to 3 dp)

13

Direct proportion problems

We can convert between any number of pounds or euros using

pounds

euros

Fruit cocktail recipes

y

x

By rearranging the equation we can see that k = .

When two quantities y and x are directly proportional to each other we can link them with the symbol .

We write

y x

We can also link these variables with the equation

y = kx

where k is called the constant of proportionality.

2

6

18

32.4

15

50

65

a

b

6

2

=

k =

15

5

5

2

b= a

a= b

We can write

and

5

2

or,

a = 0.4b

and

b = 2.5a

Two quantities a and b are in direct proportion. By writing an equation in a and b, or otherwise, complete this table:

20

26

5

45

81

a and b are directly proportional so, a = kb

When a = 6, b = 15, so

6 = 15k

x

or k =

F

k = 2 ÷ 10

When F = 10, x = 2 so,

A spring stretches when a weight is attached to the end of it.

The amount that the spring stretches by, x, is directly proportional to the weight attached to it, F.

If a weight of 10 N is attached to a certain spring it stretches 2 cm.

Write a formula in terms of x and F.

x F so x= kF

k= 0.2

x= 0.2F

Substituting:

12 = 0.2F

We can use the formula x = 0.2F to solve problems involving these variables for this spring.

How much would the spring stretch by if a weight of 35 N is attached to it?

Using the formula x= 0.2F and substituting the given value we have

x= 0.2× 35

x= 7 cm

What weight would stretch the spring by 12 cm?

F = 12 ÷ 0.2

F = 60 N

N6 Ratio and proportion

N6.1 Ratio

- A

Contents

N6.2 Dividing in a given ratio

- A

N6.3 Direct proportion

- A

N6.4 Inverse proportion

- A

N6.5 Proportionality to powers

- A

N6.6 Graphs of proportional relationships

- A

How long would it take 5 people, working at the same rate, to put 150 letters into envelopes?

One person takes 1 hour so 5 people take of an hour.

1

1

of 60 minutes =

12 minutes

5

5

It takes one person 1 hour to put 150 letters into envelopes.

Inverse proportion

The more people there are, the less time it will take.

5 people will take a fifth of the time to put the same number of letters in the envelopes.

The number of people and the time they take are said to be inversely proportional.

Two quantities are said to be inversely proportional if, as one quantity increases, the other quantity decreases at the same rate.

Inverse proportion

For example, the speed that a car travels is inversely proportional to the time it takes to cover the same distance.

If the car doubles its speed it will take half the time to cover the same distance.

If the car trebles its speed it will take a third of the time to cover the same distance.

If the car halves its speed it will take double the time to cover the same distance.

What happens when the speed of the car is 0, in other words, when it is at rest?

If the car is at rest then it will never cover the given distance.

Inverse proportion

Even an infinite amount of time isn’t enough. So the answer to how long the car will take at 0 speed is undefined.

We know that the faster the car goes, the less time it takes to cover a given distance.

Is it possible for the distance to be covered in 0 time?

No matter how fast the car goes the journey will always take some time. It can never take no time.

If two variables are inversely proportional, then when one of the variables is 0 the other variable is undefined.

1

y

x

k

y=

x

When two quantities x and y are inversely proportional to each other we can link them with the symbol by writing,

We can also link these variables with the equation,

where k is called the constant of proportionality.

By rearranging the equation we can see that k = xy.

2

4

5

16

25

10

8

a

a and b are inversely proportional, so a =

b

k

k

b

4=

25

100

100

ab = 100

b =

a =

or

We can write

a

b

Two quantities a and b are inversely proportional. By writing an equation in a and b, or otherwise, complete this table:

10

12.5

50

20

6.25

.

When a = 4, b = 25, so

k = 100

=

If then

1

k

f

f

When = 0.4, f = 825, so

k= 0.4 × 825

330

=

f

The wavelength of a sound wave is inversely proportional to its frequency f.

When the wavelength of a sound wave traveling through air is 0.4 m its frequency is 825 Hz.

Write a formula in terms of and f.

or k = f

k= 330

=

We can use the formula to solve problems involving

=

the wavelength and frequency of sound waves. For example,

Substituting the values into the formula gives,

330

330

f

f

A sound wave has a frequency of 500 Hz. What is the wavelength?

= 330 ÷ 500

= 0.66 m

=

f=

We can use the formula to solve problems involving

=

the wavelength and frequency of sound waves. For example,

We can rearrange the formula to give

330

330

330

f

f

A sound wave has a wavelength of 1.1 m. What is the frequency?

Substituting the given values,

f= 330 ÷ 1.1

f= 300 Hz

N6 Ratio and proportion

N6.1 Ratio

- A

Contents

N6.2 Dividing in a given ratio

- A

N6.3 Direct proportion

- A

N6.5 Proportionality to powers

N6.4 Inverse proportion

- A

- A

N6.6 Graphs of proportional relationships

- A

It many situations, one variable may be directly proportional to a power of the other variable.

Proportionality to powers

For example, the kinetic energy of an object is proportional to the square of its speed.

This means that if the speed of an object doubles its kinetic energy will be four times greater.

If the speed of the object trebles its kinetic energy will be nine times greater.

When the object is at rest it will have no kinetic energy.

We can also link these variables with the equation,

y= kx2

where k is called the constant of proportionality.

y

By rearranging the equation we can see that k = .

x2

If one quantity y is directly proportional to the square of another quantity x we can link them to each other with the symbol by writing,

y x2

This means that the ratio between y and x2 is constant.

1

2

3

5.5

16

64

81

a

b

16

4

k =

b= 4a2

a =

We can write

or

√b

2

In this table b is directly proportional to a2. By writing an equation in a and b, or otherwise, complete the table:

4

4.5

4

36

121

b is proportional a2 so, b = ka2

When a = 2, b = 16, so

16 = 4k

= 4

We can write this as

S h2

or

S= kh2

These Russian dolls fit inside each other. They are all the same shape but have different heights.

How are the surface areas and the heights of the dolls related?

The dolls are mathematically similar. This means that the surface area S of each doll is directly proportional to the square of its height h.

Suppose the largest doll is 11 cm high and has a surface area of 193.6 cm2.

Write a formula in terms of S and h.

We can substitute these values into S= kh2 to find k.

193.6 = 121k

k = 193.6 ÷ 121

k = 1.6

The formula linking the surface area and height is therefore:

S = 1.6h2

The smallest doll in the set has a surface area of 3.6 cm2. What is its height?

One of the dolls is 7 cm tall. What is its surface area?

Substituting into the formula S = 1.6h2,

S = 1.6 × 72

= 78.4 cm2

Substituting into the formula S = 1.6h2,

3.6 = 1.6h2

h2 = 3.6 ÷ 1.6

h2 = 2.25

h = 1.5 cm

1

k

R

R=

d2

d2

It some situations, one variable can be inversely proportional to a power of the other variable.

For example, the electrical resistance R of a metre of wire is inversely proportional to the square of its diameter d.

We can write this relationship as,

Or as an equation,

Substituting the given values into gives:

k

k

1.2 =

R=

d2

22

k

1.2 =

4

4.8

R=

d2

Suppose the electrical resistance of a metre of wire with a diameter of 2 mm is 1.2 Ohms.

Write a formula linking the resistance R to the diameter d.

k = 4 × 1.2

k = 4.8

Substituting the given values into gives,

4.8

R=

52

4.8

d2=

0.3

4.8

4.8

R=

3 =

d2

d2

What is the electrical resistance when the diameter is 5 mm?

= 0.192 ohms

What diameter of wire would have a resistance of 0.3 ohms?

d2= 16

d= 4 mm

Complete the table of values

N6 Ratio and proportion

N6.1 Ratio

- A

Contents

N6.2 Dividing in a given ratio

- A

N6.3 Direct proportion

- A

N6.6 Graphs of proportional relationships

N6.4 Inverse proportion

- A

N6.5 Proportionality to powers

- A

- A

n = 1

n > 1

0 <n< 1

n < 0

When trying to find the relationship between two variables it is often useful to construct a table of values and use these to plot a graph.

Graphs of proportional relationships

If y xn, four different shaped graphs are possible:

Graphs of proportional relationships