1. Directly and
Indirectly
proportional

2. Related Quantities a) The cost of a Lemonade is €1. 10
Related Quantities a) The cost of a Lemonade is € Complete the following table. (L is the number of Lemonade cans and C is the cost)
What is the gradient of the graph? Gradient= change in y change in x Gradient= =1.1 What is the equation connecting C and L? 𝐶=1.1𝐿 Use the equation to find C when L = 150. C=150×€1.10=€165 L 1 2 3 5 C
€1.10
€5.50
€0.00
€2.20
€3.30
Draw the graph of C against L

3. Related Quantities b) Emily works part time at a local store
Related Quantities b) Emily works part time at a local store. Her wage is €5.50 per hour. (H is the number of hours and W is her weekly wage)
What is the gradient of the graph? Gradient= change in y change in x Gradient= =5.5 What is the equation connecting W and H? W=5.5H Use the equation to find H when W= €46.75 €46.75=€5.5H 𝐻= 𝐻=8.5 ℎ𝑜𝑢𝑟𝑠 H 1 4 8 W
€5.50
€44.00
€55.00 10
€0.00
€22.00
Draw the graph of H against W

4. W directly proportional to H → 𝑊=𝑘𝐻
The wage (W) is DIRECTLY PROPORTIONAL to the number of hours (H) worked.
W directly proportional to H → 𝑊=𝑘𝐻
W is connected to H by the formula W = kH
k is called the constant of proportion.
The graph of W against H is a straight line through the origin.
The gradient represents the value per unit (e.g. wage per hour)
𝑊∝𝐻 is used to mean ‘W is directly proportional to H’
Thus 𝑊∝𝐻 is another way of writing 𝑊=𝑘𝐻
The ratio 𝑊 𝐻 is constant (the same) for every pair of values of W and H

5. CHECKING FOR DIRECT PROPORTION
Which of the following graphs involve direct proportion? a) Revenue of a Quantity of Items Produced Revenue (is, is not) directly proportional to quantity produced. REASON:
Graph is a straight line through the origin.

6. CHECKING FOR DIRECT PROPORTION
b) Cost of a journey by Taxi. Cost (is, is not) directly proportional to distance travelled. REASON:
Graph is a straight line but NOT through the origin.

7. CHECKING FOR DIRECT PROPORTION
c) Distance travelled by a car at constant speed Distance (is, is not) directly proportional to time. REASON:
CHECKING FOR DIRECT PROPORTION
Graph is a straight line through the origin.

8. CHECKING FOR DIRECT PROPORTION
d) Temperature of a cup of tea Temperature (is, is not) directly proportional to time. REASON:
Graph is NOT a straight line

9. CHECKING FOR DIRECT PROPORTION
a) Is B directly proportional to A in the following table? If so give the equation connecting B and A. 𝐵 𝐴 4 1 = = = =4 The ratio 𝐵 𝐴 is constant ∴ B is directly proportional to A Equation: 𝐵=4𝐴 A 1 3 4 10 B 12 16 40
Check the ratio 𝐵 𝐴
for each pair of values

10. CHECKING FOR DIRECT PROPORTION
b) Is B directly proportional to A in the following table? If so give the equation connecting B and A. 𝐵 𝐴 ? = = =2.5 The ratio 𝐵 𝐴 is constant ∴ B is directly proportional to A Equation: 𝐵=2.5𝐴 A 1 2 6 B
2.5 5 15
Check the ratio 𝐵 𝐴
for each pair of values

11. CHECKING FOR DIRECT PROPORTION
c) Is B directly proportional to A in the following table? If so give the equation connecting B and A. 𝐵 𝐴 2 1 =2 3 2 = = =2.5 The ratio 𝐵 𝐴 is not constant ∴ B is NOT directly proportional to A A 1 2 3 4 B 6 10
Check the ratio 𝐵 𝐴
for each pair of values

12. CALCULATING WITH DIRECT PROPORTION
a) In the following table B is directly proportional to A. Find the equation connecting B and A. Hence complete the table. ∴𝐵=𝑘𝐴 10.5=𝑘×3 k= k=3.5 A 3 5 B
10.5 28 8
17.5
B is directly proportional to A
Equation connecting B and A
𝐵=3.5𝐴
Substitute values of A and B
Rearrange
When A = 5
When B = 28
Use the equation 𝐵=3.5𝐴 to find the missing values
𝐵=3.5𝐴
𝐵=3.5𝐴
𝐵=3.5×5=17.5
28=3.5𝐴
𝐴= =8

13. M is directly proportional to V Equation connecting M and V is 𝑀=3.6𝑉
c) The mass, M kg, of oil is directly proportional to its volume, V litres. 6.5 litres of oil have a mass of 23.4kg. What is the equation connecting M and V? Hence calculate: i) the mass of 10 litres of oil ii) the volume of 171kg of oil ∴𝑀=𝑘𝑉 23.4=𝑘×6.5 k= k=3.6
M is directly proportional to V
Equation connecting M and V is
𝑀=3.6𝑉
Substitute values of M and V
Rearrange
When V = 10
When M = 171
Use the equation M=3.6V to find the missing values
𝑀=3.6𝑉
𝑀=3.6𝑉
𝑀=3.6×10=36kg
171=3.6𝑉
V= =47.5 𝑙𝑖𝑡𝑟𝑒𝑠

14. CALCULATING WITH DIRECT PROPORTION 𝐵∝ 𝐴 2
a) In the following table B is directly proportional to the square of A. Find the equation connecting B and A. Hence complete the table. ∴𝐵=𝑘 𝐴 2 14=𝑘× =4𝑘 k= 14 4 k=3.5 A 2 5 B 14
61.74
B is directly proportional to 𝐴 2
Substitute values of A and B
Rearrange
Equation connecting B and A is 𝐵=3.5 𝐴 2

15. CALCULATING WITH DIRECT PROPORTION 𝐵∝ 𝐴 2
a) In the following table B is directly proportional to the square of A. Find the equation connecting B and A. Hence complete the table. A 2 5 B 14
61.74
4.2
87.5
When B = 61.74
Use the equation 𝐵=3.5 𝐴 2 to find the missing values
𝐵=3.5 𝐴 2
61.74=3.5 𝐴 2
𝐴 2 =
When A = 5
𝐴 2 =17.64
𝐵=3.5 𝐴 2
𝐴= =4.2
𝐵=3.5× 5 2 =87.5

16. CALCULATING WITH DIRECT PROPORTION
b) 𝑃∝ 𝑄 2 and when P is 8 Q is 4. Find the equation connecting P and Q. Hence find i) P when Q = 7 ii) Q when P = 84.5 ∴𝑃=𝑘 𝑄 2 8=𝑘× 4 2 8=16𝑘 k= 8 16 k=0.5
𝑃∝ 𝑄 2
Substitute values of P and Q
Rearrange
Equation connecting P and Q is 𝑃=0.5 𝑄 2

17. CALCULATING WITH DIRECT PROPORTION
b) 𝑃∝ 𝑄 2 and when P is 8 Q is 4. Find the equation connecting P and Q. Hence find i) P when Q = 7 ii) Q when P = 84.5
Use the equation P=0.5 Q 2 to find the missing values
When P = 84.5
𝑃=0.5 𝑄 2
When Q = 7
84.5=0.5 𝑄 2
𝑃=0.5 𝑄 2
𝑄 2 =
𝑃=0.5× 7 2 =24.5
𝑄 2 =169
𝑄= 169 =13

18. CALCULATING WITH DIRECT PROPORTION 𝐵∝ 𝐴 3
c) A factory produces spheres used as garden ornaments. The weight W kg is directly proportional to the cube of its diameter D cm. i) Write down a formula connecting W and D ii) Find the value of k given that an ornament of diameter 30cm weighs 9kg. (Give k as a fraction in its lowest terms). W=𝑘 𝐷 3 9=𝑘× =27000𝑘 k= k=
W=k D 3
Substitute values of W and D
Rearrange
∴𝑊= 𝐷 3

19. CALCULATING WITH DIRECT PROPORTION 𝐵∝ 𝐴 3
c) A factory produces spheres used as garden ornaments. The weight W kg is directly proportional to the cube of its diameter D cm. iii) For safety reasons an ornament cannot weigh more than 30kg. Find the largest diameter of an ornament correct to the nearest cm 𝑊= 𝐷 3 30= 𝐷 3 D 3 =30×3000 D 3 =90000 D= =
𝑊= 𝐷 3
For SAFETY the largest diameter can be
44cm

20. INVERSE PROPORTION A inversely proportional to B → 𝐴= 𝑘 𝐵 →𝐴𝐵=𝑘
A is connected to B by the formula 𝐴= 𝑘 𝐵 thus AB = k
k is called the constant of proportion.
The product 𝐴×𝐵 is constant
𝐴∝ 1 𝐵 is used to mean ‘A is inversely proportional to B’
Thus 𝐴∝ 1 𝐵 is another way of writing 𝐴= 𝑘 𝐵
A inversely proportional to 𝐵 → 𝐴= 𝑘 𝐵 2

21. CALCULATING WITH INVERSE PROPORTION A∝ 1 𝐵
a) In the following table A is inversely proportional to B. Find the equation connecting A and B. Hence complete the table. ∴𝐴= 𝑘 𝐵 5= 𝑘 10 5×10=𝑘 k=50 A 5 8 B 10 4
A is inversely proportional to B
Substitute values of A and B
Rearrange
Equation connecting A and B is A= 50 𝐵

22. CALCULATING WITH INVERSE PROPORTION 𝐴∝ 1 𝐵
a) In the following table A is inversely proportional to B. Find the equation connecting B and A. Hence complete the table. A 5 8 B 10 4
12.5
6.25
When B = 4
Use the equation
𝐴= 50 𝐵
to find the missing values
When A = 8
𝐴= 50 𝐵
𝐴= 50 𝐵
𝐴= 50 4 =12.5
8= 50 𝐵
8𝐵=50
𝐵= 50 8 =6.25

23. CALCULATING WITH INVERSE PROPORTION
b) In an electrical circuit, the resistance, R ohms, is inversely proportional to the square of the current, I amps. When the resistance is 4 ohms, the current flowing is 6 amps.
i) Find the equation connecting R and I
∴R= k I 2
4= 𝑘 6 2
4= 𝑘 36
k=4×36=144
R is inversely proportional to the square of I
Substitute values of R and I
Rearrange
Equation connecting R and I is R= I 2

24. CALCULATING WITH INVERSE PROPORTION
ii) Find the resistance when the current is 7 amps (correct to 3 s.f.)
ii) Find the current when
the resistance is 3 ohms
(correct to 3 s.f.)
R= I 2
R= I 2
3= I 2 R=
3 I 2 =144 R=
I 2 = 144 3
I 2 =48
R=2.94𝑜ℎ𝑚𝑠
I= 48 =6.93𝑎𝑚𝑝𝑠

25. CALCULATING WITH INVERSE PROPORTION
c) If 𝑦 varies inversely with the square of 𝑥 and 𝑦=12 when 𝑥=3, find i) the equation connecting 𝑦 and 𝑥 ii) 𝑥 when 𝑦=75 ∴ 𝑦 𝑥 2 =𝑘 𝑦= 𝑘 𝑥 2 12× 3 2 =𝑘 108=𝑘
𝑦 is inversely proportional to the square of 𝑥
𝑦𝑥 2 =108
75𝑥 2 =108
𝑥 2 =
𝑥 2 =1.44
Equation connecting
𝑦 and 𝑥 is
𝑦𝑥 2 =108
𝑥 2 = 1.44
𝑥 2 =1.2