Direct or inverse proportion?
Sheet A
1. For



each set of values:
use a graph to show the values are directly proportional
find the constant of proportionality and thus write an equation linking x and y
find the missing value in the table.
a.
x
0
1
2
3
4
10
y
0
3
6
9
12
b.
x
0
2.5
5
7.5
10
y
0
10
20
30
40
100
c.
x
-0.2
-0.1
0
0.4
0.8
10
y
-0.1
-0.05
0
0.2
0.4
d.
x
-4
-2
5
10
12
y
-4.8
-2.4
6
12
14.4
0.6
2. x and y are directly proportional. When x = 0.15, y = 15. Find the constant of
proportionality, k, and write an equation linking x and y.
3. For



each set of values:
show that the values are inversely proportional
find the constant of proportionality and thus write an equation linking x and y
find the missing value in the table.
a.
x
1
2
3
4
y
1
0.5
0.333...
0.25
2
b.
x
1
2
4
10
20
y
50
25
12.5
5
c.
x
0.2
0.4
2
8
-0.1
y
4
2
0.4
0.1
d.
x
-2
-3
-5
-12
90
y
-30
-20
-12
-5
4. x and y are inversely proportional. When x = 3, y = 4.5. Find the constant of
proportionality, k, write an equation linking x and y, and hence find the value of x when y
is 2.
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22321
Page 1 of 5
Direct or inverse proportion?
Sheet B
1. For each set of values, determine whether they are directly proportional, inversely
proportional, or neither. If x and y are proportional, find an equation linking them.
a.
x
0
1
2
3
4
y
5
2
8
6
9
b.
x
0
1
2
3
4
y
0
1.5
3
4.5
6
c.
x
0
1
2
3
y
1
2
3
4
d.
x
1
2
5
10
y
25
12.5
5
2.5
e.
x
7
9
11
13
y
350
450
550
650
f.
x
1
2
3
4
y
1
2
4
8
2. f is directly proportional to v. When v = 2.5, f = 1.5. Find f when v = 10.
3. g is inversely proportional to h. When g = 10, h = 4. Find g when h = 1.
4. r is inversely proportional to s. When r = 8, s = ½. Find s when r = 5. Explain why we
cannot find s when r = 0.
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22321
Page 2 of 5
Direct or inverse proportion?
Extension sheet
5ml of paint is required to cover a square tile with side 10cm.
10cm
10cm
10cm
20cm
10cm
20cm
How much paint would be required for a square tile with side 20cm?
Complete the table to show how much paint is required for each size of a square tile:
Length of tile (cm)
Amount of paint (ml)
10
20
30
40
50
Explain how this shows the amount of paint required is not proportional to the length of the tile.
Suggest what is proportional to the amount of paint needed, and form an equation to link the
two.
Think up scenarios giving the following relationships:
1
yx
y
x
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22321
y  x2
y
1
x2
Page 3 of 5
Direct or inverse proportion?
Teaching notes
An accompanying PowerPoint, with worked examples, is available from Teachit Maths
(www.teachitmaths.co.uk, Quick search ‘22321’).
Sheet A addresses direct and inverse proportion separately (suitable after PowerPoint slides 7
and 12), Sheet B asks mixed questions, and the Extension sheet could provide homework.
Answers – Sheet A
1. Each set of values produces a graph with a straight line going through the origin.
a.
k 3
c.
k 
2.
1
2
k
y  3x
x  10
y  30
b.
k 4
y  4x
y  100
x  25
1
x
2
x  10
y5
d.
k  1 .2
y  1.2 x
y  0 .6
x  0 .5
y
15
y

 100
x 0.15
y  100 x
3. Students should show xy for each pair of numbers gives the same value, k.
a.
k 1
c.
k  0 .8
4.
y
1
x
y2
x  0 .5
b.
k  50
y
4
5x
x  0.1
y  8
d.
k  60
k  xy  3  4.5  13.5
y
13.5 27

2x
x
y
50
x
x  20
y  2 .5
y
60
x
x  90
2
y
3
y  2, x 
27
3
6
4
4
Answers – Sheet B
1. (X means the values are not in proportion)
a.
X
b.
direct
k  1. 5
c.
X
d.
inverse
k  25
e.
direct
k  50
f.
X
y  50 x
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22321
y  1. 5 x
y
25
x
Page 4 of 5
Direct or inverse proportion?
2.
3.
4.
k 
f
1 .5

 0 .6
v
2. 5
k  gh  10  4  40
k  rs  8 
1
4
2
f  0 .6 v
g
40
h
s
4
r
v  10 , f  6
h  1 , g  40
r  5, s 
4
5
We are unable to divide by 0. If r  0 , then s   .
Extension sheet
The amount of paint required is directly proportional to the area of the tile, which is the length
of the tile squared: p  l 2 .
k 
l2
1
.
therefore p 
20
20
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22321
Page 5 of 5
What do these all have in common?
Apples cost 12p each.
I buy x apples.
They cost y pence in
total.
y
y
x
x
Learning objectives
x
y
-3
-9
-2
-6
-1
-3
0
0
1
3
2
6
3
9
• Understand the terms direct proportion and indirect
proportion.
• Work out whether two quantities are directly or
inversely proportional.
• Find an equation linking two quantities which are
directly or inversely proportional.
y = 7.2x
When x increases, y increases.
y and x are in proportion.
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22321
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22321
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Direct proportion
If two values are in direct proportion, as one value
increases, the other increases at the same rate.
For example:
Two pencils cost 10p
Direct proportion
×2
×2
Four pencils cost 20p
×3
×3
Six pencils cost 30p
×½
×½
One pencil costs 5p
×0
×0
Zero pencils cost 0p
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When y is directly proportional to x ...
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Examples of direct proportion
we write y  x
if one value is zero, the other is zero,
if one value doubles, the other doubles,
s
1
2
3
4
6
12
t
5
10
15
20
30
60
t
s
5
5
5
5
5
5
Are s and t directly
proportional?
if one value trebles, the other trebles ...
y
t
is constant, therefore t  s
s
their graph is a straight line going
through the origin
the ratio between x and y is constant:
y
k
x
therefore
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y  kx
22321
(in fact, t  5s )
x
(k is called the ‘constant
of proportionality’)
5
We could also show that the
graph is a straight line
going through the origin:
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22321
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1
Examples of direct proportion
p and q are directly proportional. When q is 7, p is 28.
a) Find the constant of proportionality, k.
b) Write an equation linking p and q.
c) When q is 3, find p.
d) When p is 1.6, find q.
p  q therefore p  kq
p
q
a) k 

28
4
7
writing this
equation helps
us remember if
we need to
calculate p/q or
q/
p
d) p  4 q
1. 6  4 q
q
b) p  4 q
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Inverse proportion
c) p  4 q
 4  3  12
1. 6
 0. 4
4
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22321
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Inverse proportion
When y is inversely proportional to x ...
A taxi costs £24. Complete this table to show the cost for
each person as the number of passengers changes:
pc
Passengers (p)
Cost per passenger (c)
the product of x and y is constant: xy  k therefore y 
1
£24
1 × 24 = 24
2
£12
2 × 12 = 24
3
£8
3 × 8 = 24
4
£6
4 × 6 = 24
6
£4
6 × 4 = 24
12
£2
12 × 2 = 24
1
2
3
4
6
12
t
36
18
12
8
6
3
ts
36
36
36
32
36
36
y
9
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When s  4 , t should be 9:
a) Find the constant of proportionality, k
b) Write an equation linking p and t.
c) How long would 3 painters take to paint the fence?
1
?
s
t
1
k
therefore t 
p
p
c) t 
a) k  pt
 2  8  16
t  s  4  9  36
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The time to paint a fence is inversely proportional to the
number of painters. Two painters (p), take 8 hours (t).
b) t 
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x
22321
Examples of inverse proportion
Are s and t inversely
proportional?
t  s is not constant, so s and t are not inversely proportional.
What would need to change so t 
1
x
their graph is a hyperbola: a
special curve which never quite
reaches either axis.
Examples of inverse proportion
s
y
if one value trebles, the other is divided by three ...
but neither value can ever be zero, or the other would be
infinite!
What value is constant?
22321
1
:
x
if one value doubles, the other halves,
c
These values are not in direct proportion, as
is not
p
constant.
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this is the same as y being proportional to
k
x
11
16
p
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22321
16
16

p
3
1
 5 hours
3
 5 hrs 20 mins
12
2
Summary
directly
y is __________
proportional to x.
1
x
inversely
y is __________
proportional to x.
If one value doubles,
doubles
the other __________.
If one value doubles,
halves
the other __________.
y  kx
y
Their graph is ...
a straight line through
the origin.
Their graph is ...
a hyperbola (a curve
which doesn’t touch
either axis).
y
yx
k
x
k is called ... the constant of proportionality.
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