1. Created by Mr. Lafferty Maths Dept.
Direct Proportion
Direct Proportion
Inverse Proportion
Direct Proportion (Variation) Graph
Inverse Proportion (Variation) Graph
Direct Variation
Inverse Variation
Joint Variation
19-Apr-17
Created by Mr. Lafferty Maths Dept.

2. Created by Mr. Lafferty Maths Dept.
Starter Questions 9 8 6
19-Apr-17
Created by Mr. Lafferty Maths Dept.

3. Created by Mr. Lafferty Maths Dept.
Direct Proportion
Direct Proportion
Learning Intention
Success Criteria
1. Understand the idea of Direct Proportion.
To explain the term Direct
Proportion.
2. Solve simple Direct Proportional problems.
19-Apr-17
Created by Mr. Lafferty Maths Dept.

4. Direct Proportion www.mathsrevision.com Direct Proportion
Write down two quantities that are in direct proportion.
Direct Proportion
Direct Proportion
Two quantities, (for example, number of cakes and total
cost) are said to be in DIRECT Proportion, if :
Are we expecting more or less
“ .. When you double the number of cakes
you double the cost.”
Easier method
Cakes Pence
6  420 5
Example : The cost of 6 cakes is £4.20. find the cost
of 5 cakes.
Cakes Cost
6  4.20
(less)
1  4.20 ÷ 6 = 0.70
5  0.70 x 5 = £3.50
19-Apr-17
Created by Mr. Lafferty Maths Dept.

5. Direct Proportion Same ratio means in proportion
Example : Which of these pairs are in proportion.
(a) 3 driving lessons for £60 : 5 for £90
(b) 5 cakes for £3 : 1 cake for 60p
(c) 7 golf balls for £4.20 : 10 for £6
19-Apr-17
Created by Mr. Lafferty Maths Dept.

6. Created by Mr. Lafferty Maths Dept.
Direct Proportion
Direct Proportion
Which graph is a direct proportion graph ? x y x y x y
19-Apr-17
Created by Mr. Lafferty Maths Dept.

7. Created by Mr. Lafferty Maths Dept.
Direct Proportion
Direct Proportion
Now try Ex 1.1
Ch7 (page 125)
19-Apr-17
Created by Mr. Lafferty Maths Dept.

8. Created by Mr. Lafferty Maths Dept.
Starter Questions
19-Apr-17
Created by Mr. Lafferty Maths Dept.

9. Created by Mr. Lafferty Maths Dept.
Inverse Proportion
Inverse Proportion
Learning Intention
Success Criteria
1. Understand the idea of Inverse Proportion.
1. To explain the term Inverse Proportion.
2. Solve simple inverse Proportion problems.
19-Apr-17
Created by Mr. Lafferty Maths Dept.

10. Inverse Proportion www.mathsrevision.com Notice xxy = 80
Inverse Proportion is when one quantity increases
and the other decreases. The two quantities are said
to be INVERSELY Proportional
or (INDIRECTLY Proportional) to each other.
Example : Fill in the following table given x and y
are inversely proportional.
Notice xxy = 80
Hence inverse proportion X 1 2 4 8 y 80 40 20 10
19-Apr-17
Created by Mr. Lafferty Maths Dept.

11. Inverse Proportion www.mathsrevision.com Inverse Proportion
Inverse Proportion is the when one quantity increases
and the other decreases. The two quantities are said
to be INVERSELY Proportional
or (INDIRECTLY Proportional) to each other.
Are we expecting more or less
Easier method
Workers Hours
3  8 4
Example : If it takes 3 men 8 hours to build a wall.
How long will it take 4 men. (Less time !!) y x
Men Hours
3  8
(less)
1  3 x 8 = 24 hours
4  24 ÷ 4 = 6 hours
19-Apr-17
Created by Mr. Lafferty Maths Dept.

12. Inverse Proportion www.mathsrevision.com Inverse Proportion
Are we expecting more or less
Example : It takes 10 men 12 months to build a house.
How long should it take 8 men.
Men Months
Easier method
Workers months
10  12 8
10  12
1  12 x 10 = 120 y x
8  120 ÷ 8 = 15 months
(more)
19-Apr-17
Created by Mr. Lafferty Maths Dept.

13. Inverse Proportion www.mathsrevision.com Inverse Proportion
Are we expecting more or less
Example : At 9 m/s a journey takes 32 minutes.
How long should it take at 12 m/s.
Speed Time
Easier method
Speed minutes
9  32 12
9  32 mins
1  32 x 9 = 288 mins y x
12  288 ÷ 12 = 24 mins
(less)
19-Apr-17
Created by Mr. Lafferty Maths Dept.

14. Created by Mr. Lafferty Maths Dept.
Inverse Proportion
Inverse Proportion
Exercise 2.1
Ch7 (page 127)
19-Apr-17
Created by Mr. Lafferty Maths Dept.

15. Created by Mr. Lafferty Maths Dept.
Starter Questions
19-Apr-17
Created by Mr. Lafferty Maths Dept.

16. Created by Mr. Lafferty Maths Dept.
Direct Proportion
Direct Proportion Graphs
Learning Intention
Success Criteria
1. Understand that Direct Proportion Graph is a straight line.
1. To explain how Direct Direct Proportion Graph is always a straight line.
2. Construct Direct Proportion Graphs.
19-Apr-17
Created by Mr. Lafferty Maths Dept.

17. Direct Proportion Notice C ÷ P = 20 Hence direct proportion
Direct Proportion Graphs
The table below shows the cost of packets of “Biscuits”.
We can construct a graph to represent this data.
What type of graph do we expect ?
19-Apr-17
Created by Mr. Lafferty Maths Dept.

18. Created by Mr. Lafferty Maths Dept.
Notice that the points lie on a straight line passing through the origin
So direct proportion
Direct Proportion Graphs
C α P
C = k P
k = 40 ÷ 2 = 20
C = 20 P
19-Apr-17
Created by Mr. Lafferty Maths Dept.

19. Direct Proportion KeyPoint Two quantities which are in
Direct Proportion Graphs
KeyPoint
Two quantities which are in
Direct Proportion
always lie on a straight line
passing through the origin.
19-Apr-17
Created by Mr. Lafferty Maths Dept.

20. Created by Mr. Lafferty Maths Dept.
Direct Proportion
Direct Proportion Graphs
Ex: Plot the points in the table below.
Show that they are in Direct Proportion.
Find the formula connecting D and W ? W 1 2 3 4 D 6 9 12
We plot the points (1,3) , (2,6) , (3,9) , (4,12)
19-Apr-17
Created by Mr. Lafferty Maths Dept.

21. Direct Proportion D www.mathsrevision.com D and W are in W
Direct Proportion Graphs D 12
Plotting the points
(1,3) , (2,6) , (3,9) , (4,12) 11 10 9 8 7
Since we have a straight line
passing through the origin
D and W are in
Direct Proportion. 6 5 4 3 2 1
19-Apr-17
Created by Mr. Lafferty Maths Dept. W 1 2 3 4

22. Direct Proportion D D α W D = kW Constant k = 6 ÷ 2 = 3
Direct Proportion Graphs D 12
Finding the formula
connecting D and W we have. 11 10 9
D α W 8 7
D = 6
W = 2
D = kW 6 5
Constant k = 6 ÷ 2 = 3 4 3
Formula is : D= 3W 2 1
19-Apr-17
Created by Mr. Lafferty Maths Dept. W 1 2 3 4

23. Created by Mr. Lafferty Maths Dept.
Direct Proportion
Direct Proportion Graphs
1. Fill in table and construct graph
2. Find the constant of proportion (the k value)
Write down formula
19-Apr-17
Created by Mr. Lafferty Maths Dept.

24. Created by Mr. Lafferty Maths Dept.
Direct Proportion
Direct Proportion
Now try Ex 3.1
Ch7 (page 129)
19-Apr-17
Created by Mr. Lafferty Maths Dept.

25. Direct Proportion S 10 20 30 40 D 5 45 80 Does the distance D
vary directly as speed S ?
Explain your answer
Direct Proportion
Direct Proportion Graphs
Q The distance it takes a car to brake depends on how fast it is going.
The table shows the braking distance
for various speeds. S 10 20 30 40 D 5 45 80
19-Apr-17
Created by Mr. Lafferty Maths Dept.

26. Does D vary directly as speed S2 ?
Explain your answer D S2
Direct Proportion
Direct Proportion Graphs
The table shows S2 and D
Fill in the missing S2 values. S2 S 10 20 30 40 D 5 45 80
100
400
900
1600
19-Apr-17
Created by Mr. Lafferty Maths Dept.

27. Direct Proportion Find a formula connecting D and S2. D α S2 D = kS2
Direct Proportion Graphs
Find a formula connecting D and S2.
D α S2
D = kS2
D = 5
S2 = 100
Constant k = 5 ÷ 100 = 0.05
Formula is : D= 0.05S2
19-Apr-17
Created by Mr. Lafferty Maths Dept.

28. Created by Mr. Lafferty Maths Dept.
Direct Proportion
Direct Proportion
Now try Ex 3.2
Ch7 (page 131)
19-Apr-17
Created by Mr. Lafferty Maths Dept.

29. Created by Mr. Lafferty Maths Dept.
Starter Questions 9 8 6
19-Apr-17
Created by Mr. Lafferty Maths Dept.

30. Created by Mr. Lafferty Maths Dept.
Inverse Proportion
Inverse Proportion Graphs
Learning Intention
Success Criteria
1. Understand the shape of a Inverse Proportion Graph .
1. To explain how the shape and construction of a Inverse Proportion Graph.
2. Construct Inverse Proportion Graph and find its formula.
19-Apr-17
Created by Mr. Lafferty Maths Dept.

31. Inverse Proportion Notice W x P = £1800 Hence inverse proportion
Inverse Proportion Graphs
The table below shows how the total prize money of
£1800 is to be shared depending on how many winners.
Winners W 1 2 3 4 5
Prize P
£1800
£900
£600
£450
£360
We can construct a graph to represent this data.
What type of graph do we expect ?
19-Apr-17
Created by Mr. Lafferty Maths Dept.

32. Notice that the points lie on a decreasing curve
Inverse Proportion
Notice that the points lie on a decreasing curve
so inverse proportion
Direct Proportion Graphs

33. Inverse Proportion KeyPoint Two quantities which are in
Inverse Proportion Graphs
KeyPoint
Two quantities which are in
Inverse Proportion
always lie on a decrease curve
19-Apr-17
Created by Mr. Lafferty Maths Dept.

34. Created by Mr. Lafferty Maths Dept.
Inverse Proportion
Inverse Proportion Graphs
Ex: Plot the points in the table below.
Show that they are in Inverse Proportion.
Find the formula connecting V and N ? N 1 2 3 4 5 V
1200
600
400
300
240
We plot the points (1,1200) , (2,600) etc...
19-Apr-17
Created by Mr. Lafferty Maths Dept.

35. Inverse Proportion V Note that if we plotted V against
then we would get a straight line.
because v directly proportional to
These graphs
tell us the same thing V N
Inverse Proportion
Inverse Proportion Graphs V
1200
Plotting the points
(1,1200) , (2,600) , (3,400)
(4,300) , (5, 240)
1000
800
Since the points lie on a
decreasing curve
V and N are in
Inverse Proportion.
600
400
200 N 1 2 3 4 5

36. connecting V and N we have.
Inverse Proportion
Inverse Proportion Graphs V
1200
Finding the formula
connecting V and N we have.
1000
800
600
V = 1200
N = 1
400
k = VN = 1200 x 1 = 1200
200 1 2 3 4 5 N

37. Created by Mr. Lafferty Maths Dept.
Direct Proportion
Direct Proportion Graphs
1. Fill in table and construct graph
2. Find the constant of proportion (the k value)
Write down formula
19-Apr-17
Created by Mr. Lafferty Maths Dept.

38. Created by Mr. Lafferty Maths Dept.
Inverse Proportion
Inverse Proportion
Now try Ex 4.1
Ch7 (page 129)
19-Apr-17
Created by Mr. Lafferty Maths Dept.

39. Created by Mr. Lafferty Maths Dept.
Starter Questions
19-Apr-17
Created by Mr. Lafferty Maths Dept.

40. Created by Mr. Lafferty Maths Dept.
Direct Variation
Learning Intention
Success Criteria
1. Understand the process for calculating direct variation formula.
1. To explain how to work out direct variation formula.
2. Calculate the constant k from information given and write down formula.
19-Apr-17
Created by Mr. Lafferty Maths Dept.

41. Direct Variation Given that y is directly proportional to x,
and when y = 20, x = 4.
Find a formula connecting y and x.
Since y is directly proportional to x
the formula is of the form y x
k is a constant
y = kx
20 = k(4)
y = 20
x =4
k = 20 ÷ 4 = 5
y = 5x
19-Apr-17

42. Direct Variation The number of dollars (d) varies directly as the
number of £’s (P). You get 3 dollars for £2.
Find a formula connecting d and P.
Since d is directly proportional to P
the formula is of the form d P
k is a constant
d = kP
3 = k(2)
d = 3
P = 2
k = 3 ÷ 2 = 1.5
d = 1.5P

43. Created by Mr. Lafferty Maths Dept.
Direct Variation
How much will I get for £20 d P
d = 1.5P
d = 1.5 x 20 = 30 dollars
19-Apr-17
Created by Mr. Lafferty Maths Dept.

44. Direct Variation
Harder Direct Variation
Given that y is directly proportional to the square of x, and when y = 40, x = 2.
Find a formula connecting y and x .
Since y is directly proportional to x squared
the formula is of the form y x2
y = kx2
40 = k(2)2
y = 40
x = 2
k = 40 ÷ 4 = 10
y = 10x2
19-Apr-17

45. Created by Mr. Lafferty Maths Dept.
Direct Variation
Harder Direct Variation
Calculate y when x = 5
y = 10x2 y x2
y = 10(5)2 = 10 x 25 = 250
19-Apr-17
Created by Mr. Lafferty Maths Dept.

46. Direct Variation www.mathsrevision.com Harder Direct Variation
The cost (C) of producing a football magazine
varies as the square root of the number of
pages (P). Given 36 pages cost 48p to produce.
Find a formula connecting C and P.
Since C is directly proportional to “square root of” P
the formula is of the form C √P
C = 48
P = 36
k = 48 ÷ 6 = 8
19-Apr-17

47. Created by Mr. Lafferty Maths Dept.
Direct Variation
Harder Direct Variation
How much will 100 pages cost. C √P
19-Apr-17
Created by Mr. Lafferty Maths Dept.

48. Created by Mr. Lafferty Maths Dept.
Direct Variation
Harder Direct Variation
Ex 5.1 & 5.2
Ch7 (page 135)
19-Apr-17
Created by Mr. Lafferty Maths Dept.

49. Created by Mr. Lafferty Maths Dept.
Starter Questions
19-Apr-17
Created by Mr. Lafferty Maths Dept.

50. Created by Mr. Lafferty Maths Dept.
Inverse Variation
Learning Intention
Success Criteria
1. Understand the process for calculating inverse variation formula.
1. To explain how to work out inverse variation formula.
2. Calculate the constant k from information given and write down formula.
19-Apr-17
Created by Mr. Lafferty Maths Dept.

51. Inverse Variation Given that y is inverse proportional to x,
and when y = 40, x = 4.
Find a formula connecting y and x.
Since y is inverse proportional to x
the formula is of the form y x 1
k is a constant
y = 40
x =4
k = 40 x 4 = 160
19-Apr-17

52. Inverse Variation Speed (S) varies inversely as the Time (T)
When the speed is 6 kmph the Time is 2 hours Find a formula connecting S and T.
Since S is inversely proportional to T
the formula is of the form S T 1
k is a constant
S = 6
T = 2
k = 6 x 2 = 12

53. Created by Mr. Lafferty Maths Dept.
Inverse Variation
Find the time when the speed is 24mph. S T 1
S = 24
T = ?
19-Apr-17
Created by Mr. Lafferty Maths Dept.

54. y x2
Inverse Variation
Harder Inverse variation
Given that y is inversely proportional to the square of x, and when y = 100, x = 2.
Find a formula connecting y and x .
Since y is inversely proportional to x squared
the formula is of the form y x2 1
k is a constant
y = 100
x = 2
k = 100 x 22 = 400
19-Apr-17

55. Created by Mr. Lafferty Maths Dept.
Inverse Variation
Harder Inverse variation
Calculate y when x = 5 y x2 1
y = ?
x = 5
19-Apr-17
Created by Mr. Lafferty Maths Dept.

56. Inverse Variation www.mathsrevision.comy r3
Inverse Variation
Harder Inverse variation
The number (n) of ball bearings that can be made from a
fixed amount of molten metal varies inversely as the
cube of the radius (r). When r = 2mm ; n = 168
Find a formula connecting n and r.
Since n is inversely proportional to the cube of r
the formula is of the form n r3 1
k is a constant
n = 100
r = 2
k = 168 x 23 = 1344
19-Apr-17

57. Created by Mr. Lafferty Maths Dept.
Inverse Variation
Harder Inverse variation
How many ball bearings radius 4mm
can be made from the this amount of metal. n r3 1
r = 4
19-Apr-17
Created by Mr. Lafferty Maths Dept.

58. Created by Mr. Lafferty Maths Dept.
Inverse Variation
Ex 6.1 & 6.2
Ch7 (page 137)
19-Apr-17
Created by Mr. Lafferty Maths Dept.

59. Created by Mr. Lafferty Maths Dept.
Starter Questions
19-Apr-17
Created by Mr. Lafferty Maths Dept.

60. Created by Mr. Lafferty Maths Dept.
Joint Variation
Learning Intention
Success Criteria
1. Understand the process for calculating joint variation formula.
1. To explain how to work out Joint Variation formula.
2. Calculate the constant k from information given and write down formula.
19-Apr-17
Created by Mr. Lafferty Maths Dept.

61. Joint Variation Given that y is inverse proportional to x,
directly to z and y = 10 when x = 2, z = 4.
Find a formula connecting y, x and z.
Since y is inversely proportional to x
and directly to z the formula is of the form
k is a constant
y = 10
x = 2
z = 4
k = 10 x 2 ÷ 4 = 5
19-Apr-17

62. Inverse Variation T varies directly as N and inversely as S
Find a formula connecting T, N and S
given T = 144 when N = 24 S = 50
Since T is directly proportional to N
and inversely to S the formula is of the form
k is a constant
T = 144
N = 24
S = 50
k = 144 x 50 ÷ 24= 300

63. Created by Mr. Lafferty Maths Dept.
Joint Variation
Find T when N = 30 and S = 40.
19-Apr-17
Created by Mr. Lafferty Maths Dept.

64. Joint Variation www.mathsrevision.com
The height of a cone varies directly as its volume (V cm3) and inversely as the square of the radius of the base (r cm). When r = 3cm and V = 47.1 cm , h = 5 cm. Find a formula connecting h , V and r.
Since h is directly proportional to V and inversely proportional to r squared, the formula is of the form
V = 100
r = 2
h = 5
k = 5 x 9 ÷ 47.1
= 0.96
19-Apr-17

65. Created by Mr. Lafferty Maths Dept.
Joint Variation
Calculate h, when r = 4cm and V = 75cm3
h = ?
r = 4
V = 75
19-Apr-17
Created by Mr. Lafferty Maths Dept.

66. Joint Variation y varies directly as √x and inversely as z3.
When y = 40, x = 25 and z = 3.
Find a formula connecting y, x and z.
Since y is directly as √x and inversely as cube of z
the formula is of the form
y = 40
x = 25
z = 3
k = 40 x 33 ÷ √25
= 216
19-Apr-17

67. Created by Mr. Lafferty Maths Dept.
Joint Variation
Harder Inverse variation
Calculate x, when y = 81 and z = 2
x = ?
y = 81
z = 2
19-Apr-17
Created by Mr. Lafferty Maths Dept.

68. Created by Mr. Lafferty Maths Dept.
Joint Variation
Ex 7.1 & 7.2
Ch7 (page 140)
19-Apr-17
Created by Mr. Lafferty Maths Dept.