This document discusses direct and inverse proportion, including: - Expressing proportional relationships using notation and general equations - Finding equations for variables linked by direct or inverse proportion - Examples of setting up equations for direct proportion when variables are proportional to squares, cubes, or roots

1. Proportion

2. Learning Objectives:
• To write and use formulae to solve problems involving direct
proportion
• To write and use formulae to solve problems involving inverse
proportion

3. Proportion
Wednesday,
January 11, 2023
Express
relationships using
proportional
notation and the
general equation
for those variables
Find a final
equation for
variables linked by
direct proportion
Find equations for
variables directly
proportional to
squares, cubes or
roots
Key
Words
directly
proportional
square cube root equation substitute variable diagonal
Starter Task
Which exchange rate would be best for my trip to France?
£1 = €1.28
£5 = €6.35
£25=€31.25
£20 = €26
How many Euros can I get for £400 at the best available price?
MoneyCo.
XchangeIt
FXX
TraveX

4. Proportion
Wednesday,
January 11, 2023
Express
relationships using
proportional
notation and the
general equation
for those variables
Find a final
equation for
variables linked by
direct proportion
Find equations for
variables directly
proportional to
squares, cubes or
roots
Key
Words
directly
proportional
square cube root equation substitute variable diagonal
Unitary Method
Which exchange rate would be best for my trip to France?
£1 = €1.28
£5 = €6.35
£25=€31.25
£20 = €26
How many Euros can I get for £400 at the best available price?
MoneyCo.
XchangeIt
FXX
TravEX
No change
£1
€1.28
€1.27
€1.25
€1.30
÷ 5
÷ 25
÷ 20

5. Direct Proportion
Wednesday,
January 11, 2023
Express
relationships using
proportional
notation and the
general equation
for those variables
Find a final
equation for
variables linked by
direct proportion
Find equations for
variables directly
proportional to
squares, cubes or
roots
Key
Words
directly
proportional
square cube root equation substitute variable diagonal
If two variables are directly proportional that
means that if one doubles the other doubles
If one is halved the other is halved
The two variables are linked by a constant (k)
Their ratio stays the same as they increase or
decrease

6. Direct Proportion
This relationship can be expressed in any of these ways:
◼ y is directly proportional to x.
◼ y varies directly with x.
◼ y varies as x.
In symbols, direct proportion relationships can be written as y ∝ x.
The ∝ sign can then be replaced by ‘= k’ to give the formula y = kx,
where k is the constant of proportionality.

7. Key
Words
Directly/
inversely
proportional
square cube root equation substitute variable constant
Proportion
Express
relationships using
proportional
notation and the
general equation
for those variables
Find a final
equation for
variables linked by
inverse proportion
Find equations for
variables inversely
proportional to
squares, cubes or
roots
Wednesday,
January 11,
2023
I want to build a wall in my garden.
The builders say it will take 216 man hours to build the wall.
A “man hour” is one hour worked by one man.
It will, therefore, take 1 builder 216 hours to finish the job,
2 builders would do it in half the time (108 hours) and so on….
How long will it take 3 builders?
How many builders would it take to
finish in 12 hours?

8. Time (hours) Number of builders
1 216
2 108
3
4
6
8
9
12
18
24
Key
Words
Directly/
inversely
proportional
square cube root equation substitute variable constant
Inverse Proportion
Express
relationships using
proportional
notation and the
general equation
for those variables
Find a final
equation for
variables linked by
inverse proportion
Find equations for
variables inversely
proportional to
squares, cubes or
roots
Wednesday,
January 11,
2023
Time (hours) Number of builders
1 216
2 108
3 72
4 54
6 36
8 27
9 24
12 18
18 12
24 9
Time (hours) Number of builders 𝑻 × 𝒏
1 216
2 108
3 72
4 54
6 36
8 27
9 24
12 18
18 12
24 9
Time (hours) Number of builders 𝑻 × 𝒏
1 216 1 × 216
2 108 2 × 108
3 72 3 × 72
4 54 4 × 54
6 36 6 × 36
8 27 8 × 27
9 24 9 × 24
12 18 12 × 18
18 12 18 × 12
24 9 24 × 9
Number of builders
Time (hours)
𝑻 × 𝒏
1 216 216
2 108 216
3 72 216
4 54 216
6 36 216
8 27 216
9 24 216
12 18 216
18 12 216
24 9 216
Here is a table of values for some integers that will work:
For the man hours to remain at 216 we get these values:
The relationship can be seen by adding an extra column:
The product of the time and the number of builders…..
……remains CONSTANT.

9. Key
Words
Directly/
inversely
proportional
square cube root equation substitute variable constant
Inverse Proportion
Express
relationships using
proportional
notation and the
general equation
for those variables
Find a final
equation for
variables linked by
inverse proportion
Find equations for
variables inversely
proportional to
squares, cubes or
roots
Wednesday,
January 11,
2023
In this problem, the number of builders and time taken for the job are
INVERSELY PROPORTIONAL, meaning that as one variable increases
the other variable decreases.
i.e. More builders = less time, More time= less builders
This can be expressed as: ‘t is inversely proportional to n’.
In symbols, this is written as t ∝
1
𝑛
.
The ∝ sign can then be replaced by ‘= k’ to give the formula T =
𝑘
𝑛
where k is the
constant of proportionality.

10. Inverse proportion
• The temperature of a cup of coffee decreases as time increases.
A graph of temperature (T) against time (t) shows an
inverse relationship.

11. Direct Proportion
Wednesday,
January 11, 2023
Express
relationships using
proportional
notation and the
general equation
for those variables
Find a final
equation for
variables linked by
direct proportion
Find equations for
variables directly
proportional to
squares, cubes or
roots
Key
Words
directly
proportional
square cube root equation substitute variable diagonal
Example1:
𝑆 is directly proportional to 𝑇. When 𝑆 = 30 𝑇 = 5.
Write an equation linking 𝑆 and 𝑇
𝑆 ∝ 𝑇
𝑆 = 𝑘𝑇
30 = 𝑘 × 5
30 ÷ 5 = 𝑘
6 = 𝑘
The final equation is 𝑆 = 6𝑇

12. Direct Proportion
Wednesday,
January 11, 2023
Express
relationships using
proportional
notation and the
general equation
for those variables
Find a final
equation for
variables linked by
direct proportion
Find equations for
variables directly
proportional to
squares, cubes or
roots
Key
Words
directly
proportional
square cube root equation substitute variable diagonal
Example2:
𝑆 is directly proportional to the square of 𝑇. When 𝑆 = 50 𝑇 = 5.
Write an equation linking 𝑆 and 𝑇
𝑆 ∝ 𝑇2
𝑆 = 𝑘𝑇2
50 = 𝑘 × 52
50 ÷ 25 = 𝑘
2 = 𝑘
The final equation is 𝑆 = 2𝑇2

13. Direct Proportion
Wednesday,
January 11, 2023
Express
relationships using
proportional
notation and the
general equation
for those variables
Find a final
equation for
variables linked by
direct proportion
Find equations for
variables directly
proportional to
squares, cubes or
roots
Key
Words
directly
proportional
square cube root equation substitute variable diagonal
Example3:
𝐶 is directly proportional to the square root of 𝐷. When C = 18 𝐷 = 36.
Write an equation linking 𝐶 and 𝐷
C ∝ 𝐷
C = 𝑘 𝐷
18 = 𝑘 × 36
18 ÷ 6 = 𝑘
3 = 𝑘
The final equation is C = 3 𝐷

15. Answers
Wednesday,
January 11, 2023
Express
relationships using
proportional
notation and the
general equation
for those variables
Find a final
equation for
variables linked by
direct proportion
Find equations for
variables directly
proportional to
squares, cubes or
roots
Key
Words
directly
proportional
square cube root equation substitute variable diagonal

16. Key
Words
Directly/
inversely
proportional
square cube root equation substitute variable constant
Inverse Proportion
Express
relationships using
proportional
notation and the
general equation
for those variables
Find a final
equation for
variables linked by
inverse proportion
Find equations for
variables inversely
proportional to
squares, cubes or
roots
𝑡 is inversely proportional to 𝑛.
When 𝑛 = 2, 𝑡 = 108.
(a) Find an equation linking 𝑡 and 𝑛.
(b) When 𝑛 = 6 find the value of 𝑡
(c) When 𝑡 = 24 find the value of 𝑛
Example1:
Wednesday,
January 11,
2023
(a) 𝑡 ∝
1
𝑛
𝑡 =
𝑘
𝑛
108 =
𝑘
2
108 × 2 = 𝑘
216 = 𝑘
𝑡 =
216
𝑛
Inversely proportional
General equation
substitution
constant
Final equation
(b) When 𝑛 = 6
𝑡 =
216
𝑛
𝑡 =
216
6
𝑡 = 36
(c) When 𝑡 = 24
𝑡 =
216
𝑛
24 =
216
𝑛
24𝑛 = 216
𝑛 =
216
24
𝑛 = 9

17. Key
Words
Directly/
inversely
proportional
square cube root equation substitute variable constant
Inverse Proportion
Express
relationships using
proportional
notation and the
general equation
for those variables
Find a final
equation for
variables linked by
inverse proportion
Find equations for
variables inversely
proportional to
squares, cubes or
roots
𝑥 is inversely proportional to 𝑦2
.
When 𝑥 = 5, 𝑦 = 4.
(a) Find an equation linking 𝑥 and y.
(b) When 𝑦 = 2 find the value of 𝑥
(c) When 𝑥 = 5 find the value of y
Example2:
Wednesday,
January 11,
2023
(a) 𝑥 ∝
1
𝑦2
𝑥 =
𝑘
𝑦2
5 =
𝑘
42
16 × 5 = 𝑘
80 = 𝑘
𝑥 =
80
𝑦2
Inversely proportional
General equation
substitution
constant
Final
equation
(b) When 𝑦 = 2
𝑥 =
80
𝑦2
𝑥 =
80
22
𝑥 = 20
(c) When 𝑥 = 5
𝑥 =
80
𝑦2
5 =
80
𝑦2
𝑦2
=
80
5
𝑦2
= 16
𝑦 = 16
𝑦 = 4

18. Your turn…
1 ) y is inversely proportional to x. If y = 4 when x = 3, find
a) the formula for y in terms of x
b) y when x = 2
c) x when y = 3
d) Sketch the graph of y against x.
2) V varies inversely with the cube of w. If V = 12.5 when w = 2, find
a) the formula for V in terms of w
b) V when w = 1
c) w when V = 0.8

19. I dropped a stone from the top of the Shard in London.
The distance the stone falls is directly proportional to
the square of the time it has been falling.
After 5 seconds the stone has fallen 122.50 metres.
How far will it have fallen after 8 seconds?
Wednesday,
January 11, 2023
Express
relationships using
proportional
notation and the
general equation
for those variables
Find a final
equation for
variables linked by
direct proportion
Find equations for
variables directly
proportional to
squares, cubes or
roots
Key
Words
directly
proportional
square cube root equation substitute variable diagonal
Home Problem Solving for next time
Before next lesson consider this problem:
You should use direct proportion to find a solution. Look up the building online…..

20. Direct Proportion
Wednesday,
January 11, 2023
Express
relationships using
proportional
notation and the
general equation
for those variables
Find a final
equation for
variables linked by
direct proportion
Find equations for
variables directly
proportional to
squares, cubes or
roots
Key
Words
directly
proportional
square cube root equation substitute variable diagonal
Review:
4 people can paint a fence in 3 hours.
How long will it take 6 people to paint it?
(Assume everyone works at the same rate)
Is this a Direct Proportion question? How do you know? How might we describe the link?