Direct and Inverse Proortion

Example 1: Directly Proportional

W is directly proportional to H. When W = 56, H = 8. Find W when H = 4.

W ∞ H

Therefore: W = kH (k is the constant of proportionality introduced)

56 = k (8)

k = ⁵⁶⁄₈

k = 7

Therefore: when H = 4,

W = 7H

W = 7 x 4

W = 28

Example 2: Inversely Proportional

B is inversely proportional to T. When B = 80, T = 40. Find B when T = 10

B ∞ ¹⁄_T

Therefore: B = ^k⁄_T

80 = ^k⁄₄₀

k = 80 x 40

k = 3200

Therefore: When T = 10

B = ³²⁰⁰⁄_T

B = ³²⁰⁰⁄₁₀

B = 320

Try the questions below from Edexcel GCSE Exam Papers, UK:

The weight of a piece of wire is directly proportional to its length.

A piece of wire is 25 cm long and has a weight of 6 grams.
Another piece of the same wire is 30 cm long.

Calculate the weight of the 30 cm piece of wire.
1. 8.1 grams
2. 9 grams
3. 7.2 grams
4. 6 grams
A ball falls vertically after being dropped.
The ball falls a distance d metres in a time of t seconds. d is directly proportional to the square of t.

The ball falls 20 metres in a time of 2 seconds.

(a) Find a formula for d in terms of t.

(b) Calculate the distance the ball falls in 3 seconds.
1. (a) d = 5t² (b) 45 metres
2. (a) d = 5t (b) 15 metres
3. (a) d = 10t (b) 30 metres
4. (a) d = 5t² (b) 20 metres
The time, T seconds, it takes a water heater to boil some water is directly proportional to the mass of water, m kg, in the water heater.

When m = 250, T = 600. Find T when m = 400
1. 400 seconds
2. 1260 seconds
3. 800 seconds
4. 960 seconds
The time, T seconds, it takes a water heater to boil a constant mass of water is inversely proportional to the power, P watts, of the water heater.

When P = 1400, T = 360. Find the value of T when P = 900
1. 400 seconds
2. 560 seconds
3. 1560 seconds
4. 800 seconds
D is proportional to S². D = 900 when S = 20. Calculate the value of D when S = 25.
1. 1406.25
2. 1125
3. 2000
4. 905