Direct and inverse
proportion
CHAPTER 34
34
Direct and inverse proportion
CHAPTER
34.1 Direct proportion
When one quantity increases in the same proportion as another quantity, the quantities are said to
be directly proportional to each other (see Section 20.4).
For example, the cost of a bag of potatoes is directly proportional to the weight of the potatoes.
The symbol is used to denote direct proportion.
If the cost of the bag of potatoes is C pence and the weight of the potatoes is W kg then C W.
If the potatoes cost k pence per kilogram then C kW.
In general if y is directly proportional to x
y x and y kx
where k is a number known as the constant of proportionality.
Since y kx the graph of y against x is a straight line passing through the origin.
y
y kx
O
x
The constant of proportionality, k, is the gradient of this straight line.
Example 1
y is directly proportional to x.
When x 30, y 45
Find y when x 40
Solution 1
y x so y kx
45 k 30
45
k 1.5
30
y 1.5x
Substitute x 30 and y 45
y 1.5 40
Substitute x 40 into y 1.5x
y 60
550
Find k
This is the formula for y in terms of x
34.1 Direct proportion
CHAPTER 34
Example 2
The voltage, V volts, across an electrical circuit is directly proportional to the current, I amps, flowing
through the circuit.
When I 1.2, V 78
a Express V in terms of I.
b Find V when I 2
c Find I when V 162.5
Solution 2
a V I so V kI
78 k 1.2
78
k 65
1.2
V 65I
b V 65 2 130
c 162.5 65 I
162.5
I 2.5
65
Substitute V 78 and I 1.2
Find k
Substitute I 2 into V 65I
Substitute V 162.5 into V 65I
Exercise 34A
1 y is directly proportional to x.
a y 10 when x 2 Find y when x 3
b y 5 when x 3 Find y when x 4.5
c y 6 when x 2 Find y when x 3.3
d y 3 when x 8 Find y when x 6
3 The height, T mm, of a pile of paper is
directly proportional to the number of
sheets, N, in the pile.
When N 250, T 28
a Find a formula for T in terms of N.
b Find the value of T when N 300
c Find the value of N when T 98
2 y is directly proportional to x.
a y 8 when x 2 Find x when y 10
b y 6 when x 4 Find x when y 7.5
c y 7 when x 2 Find x when y 3
d y 8 when x 5 Find x when y 13
4 The distance, D km, travelled by a car is
directly proportional to the amount,
A litres, of petrol used.
When A 5, D 270
a Express D in terms of A.
b Find the value of D when A 4.5
c How many litres of petrol are needed
for a journey of 324 km?
5 The time, T seconds, taken for a pan of water to boil is directly proportional to the amount,
A litres, of water in the pan.
When A 2.4, T 150
a Find a formula for T in terms of A.
b Find the value of T when A 1.8
c If the pan takes 3 minutes to boil, how much water is in it?
6 The perimeter, P cm, of a regular hexagon is proportional to the length, l cm, of its longest
diagonal. When l 3.6, P 10.8
a Find a formula for P in terms of l
b The value of l increases from 4.8 to 5.4 Find the increase in the value of P.
c The value of l increases by 20%. Find the percentage increase in the value of P.
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Direct and inverse proportion
CHAPTER 34
34.2 Further direct proportion
Sometimes one quantity is directly proportional to the square or the cube of another quantity. For
example, the area, A cm2, of a circle is proportional to the square of its radius, r cm.
That is, A r2 or A kr2
In general if y is proportional to the square of x
y x 2 and y kx 2
where k is the constant of proportionality.
y
Since y kx2, the graph is a quadratic curve passing through the origin.
y kx2
k is generally positive.
Similarly if y is proportional to the cube of x
y x 3 and y kx3
where k is the constant of proportionality.
O
Example 3
The area, A cm2, of a square is proportional to the square of its perimeter, P cm.
When P 8, A 4 Find a formula for A in terms of P.
Solution 3
A P2 so A kP2
4 k 82
4
1
k 64 16
1
A P2
16
Substitute A 4, P 8 into A kP2.
This is the formula for A in terms of P.
Example 4
y is proportional to the square of x. y 60 when x 6
b Find y when x 4.5
a Find a formula for y in terms of x.
c Find a value of x for which y 135
Solution 4
a y x 2 so y kx 2
60 k 62
60 5
k 36 3
5
y x2
3
5
5
b y 4.52 20.25
3
3
y 33.75
5
c 135 x 2
3
135 3
x 2
5
2
x 81
x 9 (or x 9)
552
Substitute y 60, x 6 into y kx2
5
Substitute x 4.5 into y x2
3
5
Substitute y 135 into y x 2
3
x
34.2 Further direct proportion
CHAPTER 34
Example 5
The mass, M kg, of a solid cube made from lead is proportional to the cube of the length, L cm, of an
edge.
When L 0.2, M 90
a Find a formula for M in terms of L.
b Find the value of M when L 0.3
c Find the value of L when M 2000 Give your answer correct to 3 significant figures.
Solution 5
a M L3 so M kL3
90 k 0.23
90
k 11 250
0.008
M 11 250L3
Substitute M 90, L 0.2 into M kL3
b M 11 250 0.33
M 303.75
Substitute L 0.3 into M 11 250L3
c 2000 11 250 L3
2000
L3 11 250
3 2000
L 0.5622…
11 250
L 0.562 (to 3 s.f.)
Substitute M 2000 into M 11 250L3
Exercise 34B
1 y is proportional to the square of x.
a When x 2, y 8 Find y when x 3
b When x 2, y 10 Find y when x 8
c When x 2, y 7 Find y when x 6
d When x 3, y 12 Find y when x 15
2 y is proportional to the square of x.
a When x 2, y 12 Find x when y 108
b When x 3, y 18 Find x when y 162
c When x 4, y 40 Find x when y 160
d When x 4, y 200 Find x when y 32
3 The area, A cm2, of a regular hexagon is
proportional to the square of the length,
l cm, of the longest diagonal.
When A 65, l 10
a Find a formula for A in terms of l.
b Find the value of A when l 4
c Find the value of l when A 200
Give your answer correct to
3 significant figures.
4 The rate of heat loss, H calories per second,
from a sphere is proportional to the square
of the radius, r cm, of the sphere.
When H 2.5, r 5
a Find a formula for H in terms of r.
b Find the value of H when r 4
c Find the value of r when H 90
5 The quantity of light, Q, given out
by a lamp is proportional to the square
of the current, I, passing through the lamp.
When Q 1000, I 2
a Find a formula for Q in terms of I.
b Find the value of Q when I 3
c Find the value of I when Q 2000
Give your answer correct to
3 significant figures.
6 The power, P watts, of an engine is
proportional to the square of
the speed, s m/s, of the engine.
When s 30, P 1260
a Find a formula for P in terms of s.
b Find the value of P when s 25
c Find the value of s when P 1715
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Direct and inverse proportion
CHAPTER 34
7 y is proportional to the cube of x.
a When x 2, y 16 Find y when x 3
b When x 2, y 10 Find y when x 3
c When x 4, y 20 Find y when x 6
d When x 5, y 800 Find y when x 8
8 y is proportional to the cube of x.
When x 8, y 1000
a Find a formula for y in terms of x.
b Find the value of x when y 1728
34.3 Inverse proportion
When one quantity increases at the same rate as another quantity decreases, the quantities are said
to be inversely proportional to each other (see Section 20.5).
In general if y is inversely proportional to x
k
1
y and y x
x
where k is the constant of proportionality.
y
k
The graph of y when k is positive has a similar
x
1
shape to the graph of y (see Section 25.1).
x
Similarly if y is inversely proportional to the square of x
k
yx
O
x
k
1
y 2 and y 2
x
x
where k is the constant of proportionality.
k
Here is the graph of y 2 when k is positive.
x
y
y
k
x2
O
x
Example 6
When a fixed volume of water is poured into a cylindrical jar, the depth, D cm, of the water is
inversely proportional to the cross-sectional area, A cm2, of the cylindrical jar.
When A 40, D 120
a Find a formula for A in terms of D.
b Find A when D 150
c Find D when A 60
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34.3 Inverse proportion
Solution 6
k
1
a A so A D
D
k
40 120
k 40 120 4800
4800
A D
4800
b A 32
150
4800
c 60 D
4800
D 80
60
CHAPTER 34
k
Substitute A 40, D 120 into A D
4800
Substitute D 150 into A D
4800
Substitute A 60 into A D
Example 7
The force of attraction, F newtons, between two spheres is inversely proportional to the square of
the distance, d m, between the centres of the spheres.
When d 2, F 0.006
a Express F in terms of d.
b Find F when d 2.5
c Find d when F 0.001 Give your answer correct to 3 significant figures.
Solution 7
k
1
a F 2 so F 2
d
d
k
0.006 4
k 0.006 4 0.024
0.024
F d2
0.024
b F 0.003 84
2.52
0.024
c 0.001 d2
0.024
d 2 24
0.001
d 24
4.898…
d 4.90 (to 3 s.f.)
k
d
Substitute F 0.006, d 2 into F 2
0.024
Substitute d 2.5 into F 2
d
0.024
Substitute F 0.001 into F 2
d
Exercise 34C
1 y is inversely proportional to x.
a y 8 when x 2 Find y when x 4
b y 10 when x 4 Find y when x 16
c y 16 when x 10 Find y when x 8
d y 21 when x 10 Find y when x 15
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CHAPTER 34
2 y is inversely proportional to x.
a y 20 when x 4 Find x when y 5
c y 30 when x 9 Find x when y 20
3
Direct and inverse proportion
b y 25 when x 8 Find x when y 10
d y 45 when x 4 Find x when y 54
The time, T seconds, taken for a pan of water to boil on a gas ring is inversely proportional to the
setting, N, of the gas ring.
When N 4.5, T 108
a Find a formula for T in terms of N.
b Find the value of T when N 6
c If the pan takes 3 minutes to boil, what was the setting?
4 For rectangles with the same area, the length, l metres, of the rectangle is inversely proportional
to the width, w metres, of the rectangle.
When l 2.5, w 2.4
a Express l in terms of w.
b Find the value of w when l 3.2
c Given that the values of l and w are the same, find the value of l.
5 The frequency, f cycles per second, of a sound wave is inversely proportional to the wavelength,
l cm, of the sound wave.
When f 256, l 133
a Find a formula for f in terms of l.
b Find f when l 250 Give your answer correct to 3 significant figures.
c Find l when f 300 Give your answer correct to 3 significant figures.
6 y is inversely proportional to the square of x.
a y 4 when x 2 Find y when x 4
c y 20 when x 3 Find y when x 2
b y 10 when x 2 Find y when x 4
d y 45 when x 4 Find y when x 5
7 y is inversely proportional to the square of x.
a y 12 when x 2 Find x when y 0.75
c y 0.5 when x 6 Find x when y 1800
b y 10 when x 4 Find x when y 6.4
d y 12.5 when x 2 Find x when y 2
8 When a fixed volume of liquid is poured into any cylinder, the depth, D cm, of the liquid is
inversely proportional to the square of the radius, r cm, of the cylinder.
When r 5, D 40
a Find a formula for D in terms of r.
b Find the value of D when r 4
c Find the value of r when D 15 Give your answer correct to 3 significant figures.
d For what value of r is the depth equal to the diameter of the cylinder? Give your answer
correct to 3 significant figures.
9 The intensity, I, of the light at a distance, d, from a lamp is inversely proportional to the square of
the distance.
When I 4.5, d 2.4
a Find a formula for I in terms of d.
b Find I when d 1.8
c Find d when I 6 Give your answer correct to 3 significant figures.
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34.4 Proportion and square roots
CHAPTER 34
10 The pressure, P pascals, that a constant force exerts on a square with an edge of length, x m, is
inversely proportional to x.
When x 0.4, P 50
a Find a formula for P in terms of x.
b Find P when x 0.5
c Find x when P 600 Give your answer correct to 3 significant figures.
34.4 Proportion and square roots
Sometimes one quantity is directly proportional to the square root of another quantity.
In general if y is proportional to the square root of x
y
y x and y kx
yk x
where k is the constant of proportionality.
Here is the graph of y kx when k is positive.
x
O
Example 8
The speed, s, of a particle is directly proportional to the square root of its kinetic energy, E.
When E 225, s 40
a Find a formula for s in terms of E.
b Find s when E 900
c Rearrange the formula to find E in terms of s.
Solution 8
a s E
so s kE
40 k225
k 15
40 8
k 15 3
8
s E
3
8
8
b s 900
30
3
3
s 80
8
s E
c
3
3s
E
8
3s 2
9s2
E or E 8
64
Substitute s 40, E 225 into s kE
8
Substitute E 900 into s E
3
Multiply both sides by 3 and then divide both sides by 8
Square both sides.
Either formula is acceptable.
Sometimes one quantity is inversely proportional to the square root of another quantity.
In general if y is inversely proportional to the square root of x
y
k
y
1
k
x
where k is the constant of proportionality.
y and y x
x
k
Here is the graph of y when k is positive.
x
O
x
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Direct and inverse proportion
CHAPTER 34
Example 9
y is inversely proportional to the square root of x.
When x 64, y 20
a Find a formula for y in terms of x.
b Find y when x 100
c Find x when y 5
Solution 9
1
k
a y so y x
x
k
k
20 64
8
k
x
Substitute y 20, x 64 into y k 20 8 160
160
y x
160
b y 16
100
c
160
5 x
5x 160
160
x , x 32
5
x 322
x 1024
558
160
Substitute x 100 into y x
160
Substitute y 5 into y x
Multiply both sides by x
Square both sides.
Exercise 34D
1 y is directly proportional to the square root of x.
a When x 4, y 6 Find y when x 25
c When x 9, y 4 Find y when x 81
b When x 16, y 20 Find y when x 49
d When x 100, y 40 Find y when x 14
2 y is directly proportional to the square root of x.
a When x 1, y 4 Find x when y 8
c When x 16, y 10 Find x when y 25
b When x 4, y 10 Find x when y 25
d When x 49, y 21 Find x when y 27
3 y is inversely proportional to the square root of x.
a When x 4, y 2 Find y when x 25
c When x 4, y 4 Find y when x 1
b When x 1, y 5 Find y when x 16
d When x 100, y 0.3 Find y when x 900
4 y is inversely proportional to the square root of x.
a When x 4, y 2.5 Find x when y 2
c When x 9, y 2 Find x when y 6
b When x 4, y 12 Find x when y 2
d When x 25, y 0.8 Find x when y 12
Chapter 34 review questions
CHAPTER 34
5 When a ball is thrown upwards, the time, T seconds, the ball remains in the air is directly
proportional to the square root of the height, h metres, reached.
When h 25, T 4.47
a Find a formula for T in terms of h.
b Find the value of T when h 50 Give your answer correct to 3 significant figures.
The ball is thrown upwards and remains in the air for 5 seconds.
c Find the height reached. Give your answer correct to 3 significant figures.
Chapter summary
You should now know:
how to set up and use equations to solve problems involving direct proportion, for example
● if y is directly proportional to x, y x and y kx
● if y is directly proportional to the square of x, y x 2 and y kx2
how to set up and use equations to solve problems involving inverse proportion, for example
k
1
● if y is inversely proportional to x, y and y x
x
k
1
● if y is inversely proportional to the square of x, y 2 and y 2
x
x
that k is a number known as the constant of proportionality
the shapes of the graphs that represent the different types of proportionality.
Chapter 34 review questions
1 Here are three examples of proportionality.
i y is directly proportional to x.
ii V is directly proportional to the cube of r.
iii T is inversely proportional to the square root of s.
a Express each of i to iii as a formula. Include a constant of proportionality.
b Draw a sketch of the graph that represents the type of proportionality described in each of i to iii.
2 V is directly proportional to r.
When r 2, V 8
a Find V when r 6
b Find r when V 2
3 y is inversely proportional to x.
When x 10, y 12
a Find a formula for y in terms of x.
b Find the value of y when x 20
c Find the value of x when y 25
4 The time, T seconds, it takes a pendulum to swing once is proportional to the square root of the
length, l metres, of the pendulum.
When l 0.16, T 0.8
a Find a formula for T in terms of l.
b Find the value of T when l 1.44
c What length of pendulum will give a swing of 1 second? Give your answer correct to
3 significant figures.
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CHAPTER 34
Direct and inverse proportion
5 The drag force, F newtons, on an object moving with a speed, s metres per second, is
proportional to the square of the speed.
When s 20, F 80
a Express F in terms of s.
b Find F when s 30
c Find the speed when the drag force is 300 newtons.
6 d is directly proportional to the square of t.
d 80 when t 4
a Express d in terms of t.
b Work out the value of d when t 7
c Work out the positive value of t when d 45
(1387 June 2005)
7 P is inversely proportional to V.
When V 2, P 7.5
a Find a formula for P in terms of V.
The value of V is increased by 25%.
b Work out the percentage change in the value of P.
8 The temperature, T °, at a distance, d metres, from a heat source is inversely proportional to the
square of the distance.
When d 4, T 275
a Find T when d 6
b Find d when T 1000 Give your answer correct to 3 significant figures.
9 The oscillation frequency, f cycles per second, of a spring is inversely proportional to the square
root of the mass, m kg, of the spring.
When m 2.56, f 2
Find f when m 4
10 The time taken, T seconds, for a particle to slide down a smooth slope of length, l m, is directly
proportional to the square root of the length.
When T 1.5, l 6.25
a Find a formula for T in terms of l.
b Rearrange the formula to find l in terms of T.
11 The rate of melting, M grams per second, of a sphere of ice is proportional to the square of the
radius, r cm.
When r 20, M 0.6
a Show that M 0.0015 r 2
b Find the rate of melting when the radius is 40 cm.
c Find the radius when the rate of melting is 1 gram per second. Give your answer correct to
3 significant figures.
d Hannah claims that the rate of melting is directly proportional to the surface area, A cm2, of
the sphere. Is Hannah correct? You must justify your answer.
560
12 The force, F, between two magnets is inversely proportional to the square of the distance, x,
between them.
When x 3, F 4
a Find an expression for F in terms of x.
b Calculate F when x 2
(1387 June 2003)
c Calculate x when F 64
Chapter 34 review questions
CHAPTER 34
13 In a factory, chemical reactions are carried out in spherical containers.
The time, T minutes, the chemical reaction takes is directly proportional to the square of the
radius, R cm, of the spherical container.
When R 120, T 32
(1387 November 2004)
Find the value of T when R 150
14 The shutter speed, S, of a camera varies inversely as the square of the aperture setting, f.
When f 8, S 125
a Find a formula for S in terms of f.
(1387 June 2004)
b Hence, or otherwise, calculate the value of S when f 4
15
y
y
x
x
Graph A
Graph B
y
y
x
Graph C
x
Graph D
The graphs of y against x represent four different types of proportionality.
Copy the table and write down the letter of the graph which represents the type of
proportionality.
Type of proportionality
y is directly proportional to x
y is inversely proportional to x
y is proportional to the square of x
y is inversely proportional to the square of x
Graph letter
..............................
..............................
..............................
..............................
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