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© Boardworks Ltd 2005 2 of 67
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N6.1 Ratio
N6 Ratio and proportion
Contents
N6.3 Direct proportion
N6.2 Dividing in a given ratio
N6.6 Graphs of proportional relationships
N6.4 Inverse proportion
N6.5 Proportionality to powers
© Boardworks Ltd 2005 4 of 67
Ratio
A ratio compares the sizes of parts or quantities to each other.
For example,
What is the ratio of red counters to blue counters?
red : blue
= 9 : 3
= 3 : 1
For every three red counters there is one blue counter.
© Boardworks Ltd 2005 5 of 67
Ratio
A ratio compares the sizes of parts or quantities to each other.
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.
For example,
What is the ratio of blue counters to red counters?
© Boardworks Ltd 2005 6 of 67
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.
© Boardworks Ltd 2005 7 of 67
Simplifying ratios
Ratios can be simplified like fractions by dividing each part by the highest common factor.
For example,21 : 35
= 3 : 5÷ 7 ÷ 7
For a three-part ratio all three parts must be divided by the same number.
For example,6 : 12 : 9
= 2 : 4 : 3÷ 3 ÷ 3
© Boardworks Ltd 2005 9 of 67
When a ratio is expressed in different units, we must write the ratio in the same units before simplifying.
Simplify the ratio 90p : £3
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÷ 30 ÷ 30
= 3 : 10
Simplifying ratios with units
© Boardworks Ltd 2005 10 of 67
Simplify the ratio 0.6 m : 30 cm : 450 mm
First, write the ratio using the same units.
60 cm : 30 cm : 45 cm
60 : 30 : 45
÷ 15 ÷ 15
= 4 : 2 : 3
Simplifying ratios with units
© Boardworks Ltd 2005 11 of 67
When a ratio is expressed using fractions or decimals we can simplify it by writing it in whole-number form.
Simplify the ratio 0.8 : 2
We can write this ratio in whole-number form by multiplying both parts by 10.
0.8 : 2
= 8 : 20
× 10 × 10
÷ 4 ÷ 4
= 2 : 5
Simplifying ratios containing decimals
© Boardworks Ltd 2005 12 of 67
Simplifying ratios containing fractions
Simplify the ratio : 4 23
We can write this ratio in whole-number form by multiplying both parts by 3.
23 : 4
× 3 × 3
= 2 : 12
÷ 2 ÷ 2
= 1 : 6
© Boardworks Ltd 2005 13 of 67
Comparing ratios
We can compare ratios by writing them in the form 1 : m or m : 1, where m is 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.
5 : 8÷ 5 ÷ 5
= 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÷ 8 ÷ 8
= 0.625 : 1
© Boardworks Ltd 2005 14 of 67
Comparing ratios
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 ÷ 4
= 1 : 1.25
The ratio of boys to girls in 9G is 5 : 7÷ 5 ÷ 5
= 1 : 1.4
9G has a higher proportion of girls.
© Boardworks Ltd 2005 15 of 67
Writing ratios as fractions
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:
θ
This is the side opposite the angle θ.
This is the side adjacent to the angle θ.
We could write the ratio of the length of the opposite side to the length of the adjacent side as
opposite : adjacent
However in this context we write the ratio as .oppositeadjacent
This ratio is called the tangent of the angle θ.
© Boardworks Ltd 2005 16 of 67
Writing ratios as fractions
What is the ratio of the height to the width of the photographa) using ratio notationb) as a fraction?
7.5 cm
12.5 cm
a) height : width
7.5 : 12.5÷ 2.5 ÷ 2.5
3 : 5
b)heightwidth
=7.512.5
=35
We could say that the height is of the width. 35
© Boardworks Ltd 2005 17 of 67
Finding the missing number in a ratio
Suppose the picture is reduced in size so that its width is 7.5 cm. What is the height of the reduced picture?
?
7.5 cm
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.
We must therefore find a ratio equivalent to 3 : 5 but with the second part equal to 7.5.
3 : 5
? : 7.5
© Boardworks Ltd 2005 18 of 67
Finding the missing number in a ratio
To find the missing number in the ratio we have to work out what we have multiplied 5 by to get 7.5:
3 : 5
? : 7.5
To do this divide 7.5 by 5.
7.5 ÷ 5 = 1.5
The 5 is multiplied by 1.5 …
× 1.5 × 1.5… so the 3 must be multiplied by 1.5. 4.5
So when the width of the rectangle is 7.5 cm this height is 4.5 cm.
© Boardworks Ltd 2005 19 of 67
Finding the missing number in a ratio
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?
6 : 7
72 : ?
To do this divide 72 by 6.
72 ÷ 6 = 12
… so the 7 must be multiplied by 12.
× 12 × 12The 6 is multiplied by 12 … 84
Again we can work this out by finding the missing number in the ratio.
If there are 72 boys there must be 84 girls.
© Boardworks Ltd 2005 20 of 67
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N6.2 Dividing in a given ratio
N6 Ratio and proportion
Contents
N6.3 Direct proportion
N6.1 Ratio
N6.6 Graphs of proportional relationships
N6.4 Inverse proportion
N6.5 Proportionality to powers
© Boardworks Ltd 2005 22 of 67
Dividing in a given ratio
A ratio is made up of parts.
We can write the ratio 2 : 3 as
2 parts : 3 parts
The total number of parts is
2 parts + 3 parts = 5 parts
Divide £40 in the ratio 2 : 3.
£40 ÷ 5 = £8
We need to divide £40 by the total number of parts.
© Boardworks Ltd 2005 23 of 67
Dividing in a given ratio
Divide £40 in the ratio 2 : 3.
Each part is worth £8 so
2 parts = 2 × £8 = £16
and 3 parts = 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
© Boardworks Ltd 2005 24 of 67
Dividing in a given ratio
A citrus twist cocktail contains orange juice, lemon juice and lime juice in the ratio 6 : 3 : 1.
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
How much of each type of juice is contained in 750 ml of the cocktail?
© Boardworks Ltd 2005 25 of 67
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
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?
© Boardworks Ltd 2005 27 of 67
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N6.3 Direct proportion
Contents
N6.1 Ratio
N6 Ratio and proportion
N6.2 Dividing in a given ratio
N6.4 Inverse proportion
N6.6 Graphs of proportional relationships
N6.5 Proportionality to powers
© Boardworks Ltd 2005 28 of 67
Direct proportion
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.
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.
© Boardworks Ltd 2005 30 of 67
Direct proportion problems
3 packets of crisps weigh 84 g.How much do 12 packets weigh?
3 packets weigh 84 g.× 4
12 packets weigh× 4
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.
© Boardworks Ltd 2005 31 of 67
Direct proportion problems
3 packets of crisps weigh 84 g.How much does 1 packet weigh?
3 packets weigh 84 g.÷ 3
1 packet weighs÷ 3
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.
© Boardworks Ltd 2005 32 of 67
3 packets of crisps weigh 84 g.How much do 7 packets weigh?
3 packets weigh 84 g.÷ 3 ÷ 3
1 packet weighs 28 g.× 7 × 7
7 packets weigh 196 g.
This is called using a unitary method.
Direct proportion problems
© Boardworks Ltd 2005 33 of 67
3 packets of crisps weigh 84 g.How much do 7 packets weigh?
3 packets weigh 84 g.
7 packets weigh 196 g.
We could also work this out in a single step as follows,
Direct proportion problems
What do we multiply 3 by to get 7?
To work this out we divide 7 by 3 to get73
×73
×73
.
© Boardworks Ltd 2005 34 of 67
3 packets of crisps weigh 84 g.How much do 7 packets weigh?
3 packets weigh 84 g.
7 packets weigh 196 g.
Alternatively, we could scale from 3 to 84 by multiplying by 28.
Direct proportion problems
× 28
× 28
© Boardworks Ltd 2005 35 of 67
Direct proportion problems
To scale from £8 to £2 we × 14
or × 0.25
£8 is worth 13€
£2 is worth
× 14
or × 0.25
× 14
or × 0.25(13 ÷ 4)€
= 3.25€
£8 is worth 13 euros.How much is £2 worth?
© Boardworks Ltd 2005 36 of 67
Direct proportion problems
£8 is worth 13€
× 138
or × 1.625
£2 is worth (2 × 1.625)€ = 3.25€
× 138
or × 1.625 Alternatively, to scale from 8 to 13 we
× 138
or × 1.625
£8 is worth 13 euros.How much is £2 worth?
© Boardworks Ltd 2005 37 of 67
Direct proportion problems
× 138
or × 1.625
We can convert between any number of pounds or euros using
× 813
or × 0.615 (to 3 dp)
pounds euros
£8 is worth 13 euros.How much is £2 worth?
© Boardworks Ltd 2005 39 of 67
Equations and direct proportion
When two quantities y and x are directly proportional to each other we can link them with the symbol .
We writey x
We can also link these variables with the equation
y = kx
where k is called the constant of proportionality.
By rearranging the equation we can see that k = . y
x
© Boardworks Ltd 2005 40 of 67
Equations and direct proportion
Two quantities a and b are in direct proportion. By writing an equation in a and b, or otherwise, complete this table:
a
b
2 6
15
18
50 65
32.4
a and b are directly proportional so, a = kb
When a = 6, b = 15, so 6 = 15k
5 45
20 26
81
k =615
= 25
We can write anda = b25 b = a5
2
or, a = 0.4b and b = 2.5a
© Boardworks Ltd 2005 42 of 67
Using proportionality to write formulae
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
When F = 10, x = 2 so, k = 2 ÷ 10
k = 0.2
x = 0.2Fx = 0.2F
or k =x
F
© Boardworks Ltd 2005 43 of 67
Using proportionality to write formulae
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?
Substituting: 12 = 0.2F
F = 12 ÷ 0.2
F = 60 N
© Boardworks Ltd 2005 44 of 67
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N6.4 Inverse proportion
Contents
N6.3 Direct proportion
N6.1 Ratio
N6 Ratio and proportion
N6.2 Dividing in a given ratio
N6.6 Graphs of proportional relationships
N6.5 Proportionality to powers
© Boardworks Ltd 2005 45 of 67
Inverse proportion
It takes one person 1 hour to put 150 letters into envelopes.
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.
One person takes 1 hour so 5 people take of an hour.15
of 60 minutes =15
12 minutes
The number of people and the time they take are said to be inversely proportional.
How long would it take 5 people, working at the same rate, to put 150 letters into envelopes?
© Boardworks Ltd 2005 46 of 67
Inverse proportion
Two quantities are said to be inversely proportional if, as one quantity increases, the other quantity decreases at the same rate.
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.
© Boardworks Ltd 2005 47 of 67
Inverse proportion
What happens when the speed of the car is 0, in other words, when it is at rest?
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 the car is at rest then it will never cover the given distance.
We know that the faster the car goes, the less time it takes to cover a given distance.
Even an infinite amount of time isn’t enough. So the answer to how long the car will take at 0 speed is undefined.
If two variables are inversely proportional, then when one of the variables is 0 the other variable is undefined.If two variables are inversely proportional, then when
one of the variables is 0 the other variable is undefined.
© Boardworks Ltd 2005 48 of 67
Equations and inverse proportion
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.
y 1x
y =kx
© Boardworks Ltd 2005 49 of 67
Equations and inverse proportion
Two quantities a and b are inversely proportional. By writing an equation in a and b, or otherwise, complete this table:
a
b
2 4
25
5
10 8
16
When a = 4, b = 25, so
50 20
10 12.5
6.25
4 = k25
k = 100
We can write ab = 100a = 100b b = 100
a or
a and b are inversely proportional, so a = kb
.
© Boardworks Ltd 2005 51 of 67
Using proportionality to write formulae
The wavelength of a sound wave is inversely proportional to its frequency f.
Write a formula in terms of and f.
When = 0.4, f = 825, so k = 0.4 × 825
k = 330
When the wavelength of a sound wave traveling through air is 0.4 m its frequency is 825 Hz.
If then 1
f =
k
for k = f
=330
f
© Boardworks Ltd 2005 52 of 67
Using proportionality to write formulae
A sound wave has a frequency of 500 Hz. What is the wavelength?
= 330 ÷ 500
= 0.66 m
We can use the formula to solve problems involving =330
f
the wavelength and frequency of sound waves. For example,
Substituting the values into the formula gives,
=330
f
© Boardworks Ltd 2005 53 of 67
Using proportionality to write formulae
A sound wave has a wavelength of 1.1 m. What is the frequency?
f = 330 ÷ 1.1
f = 300 Hz
We can rearrange the formula to give =330
ff =
330
Substituting the given values,
We can use the formula to solve problems involving =330
f
the wavelength and frequency of sound waves. For example,
© Boardworks Ltd 2005 54 of 67
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N6.5 Proportionality to powers
Contents
N6.3 Direct proportion
N6.1 Ratio
N6 Ratio and proportion
N6.2 Dividing in a given ratio
N6.4 Inverse proportion
N6.6 Graphs of proportional relationships
© Boardworks Ltd 2005 55 of 67
Proportionality to powers
It many situations, one variable may be directly proportional to a power of the other variable.
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.
When the object is at rest it will have no kinetic energy.
If the speed of the object trebles its kinetic energy will be nine times greater.
© Boardworks Ltd 2005 56 of 67
Equations and square proportion
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
We can also link these variables with the equation,
where k is called the constant of proportionality.
y = kx2
By rearranging the equation we can see that k = . y
x2
This means that the ratio between y and x2 is constant.
© Boardworks Ltd 2005 57 of 67
Equations and square proportion
In this table b is directly proportional to a2. By writing an equation in a and b, or otherwise, complete the table:
a
b
1 2
16
3
64 81
5.5
4 36
4 4.5
121
b is proportional a2 so, b = ka2
When a = 2, b = 16, so 16 = 4k
k =164
= 4
We can write orb = 4a2 a = √b2
© Boardworks Ltd 2005 58 of 67
Using proportionality to write formulae
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.
We can write this as S h2
or S = kh2
© Boardworks Ltd 2005 59 of 67
Using proportionality to write formulae
Suppose the largest doll is 11 cm high and has a surface area of 193.6 cm2.
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 = 1.6h2S = 1.6h2
Write a formula in terms of S and h.
© Boardworks Ltd 2005 60 of 67
Using proportionality to write formulae
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
The smallest doll in the set has a surface area of 3.6 cm2. What is its height?
Substituting into the formula S = 1.6h2,
3.6 = 1.6h2
h2 = 3.6 ÷ 1.6
h2 = 2.25
h = 1.5 cm
© Boardworks Ltd 2005 61 of 67
Inverse proportionality to powers
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,
R 1d2
R =kd2
© Boardworks Ltd 2005 62 of 67
Using proportionality to write formulae
Suppose the electrical resistance of a metre of wire with a diameter of 2 mm is 1.2 Ohms.
Substituting the given values into gives:R =kd2
1.2 = k22
1.2 = k4
Write a formula linking the resistance R to the diameter d.
k = 4 × 1.2
k = 4.8
R =4.8d2
© Boardworks Ltd 2005 63 of 67
Using proportionality to write formulae
What is the electrical resistance when the diameter is 5 mm?
R =4.8d2Substituting the given values into gives,
R = 4.852
= 0.192 ohms
What diameter of wire would have a resistance of 0.3 ohms?
3 =4.8d2
d2 = 4.80.3
d2 = 16
d = 4 mm
© Boardworks Ltd 2005 65 of 67
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AN6.6 Graphs of proportional relationships
Contents
N6.3 Direct proportion
N6.1 Ratio
N6 Ratio and proportion
N6.2 Dividing in a given ratio
N6.4 Inverse proportion
N6.5 Proportionality to powers
© Boardworks Ltd 2005 66 of 67
Graphs of proportional relationships
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.
If y xn, four different shaped graphs are possible:
n = 1n = 1 n > 1n > 1
0 < n < 10 < n < 1 n < 0n < 0