Ratio and proportion 43
Here are some worked examples to help us understand
more about direct proportion.
Problem 9. 3 energy saving light bulbs cost
£7.80. Determine the cost of 7 such light bulbs
(i) 3 light bulbs cost £7.80
(ii) Therefore, 1 light bulb costs
7.80
3
= £2.60
Hence, 7 light bulbs cost 7 × £2.60 = £18.20
Problem 10. If 56 litres of petrol costs £59.92,
calculate the cost of 32 litres
(i) 56 litres of petrol costs £59.92
(ii) Therefore, 1 litre of petrol costs
59.92
56
= £1.07
Hence, 32 litres cost 32 × 1.07 = £34.24
Problem 11. Hooke’s law states that stress, σ , is
directly proportional to strain, ε, within the elastic
limit of a material. When, for mild steel, the stress
is 63 MPa, the strain is 0.0003. Determine (a) the
value of strain when the stress is 42 MPa, (b) the
value of stress when the strain is 0.00072
(a) Stress is directly proportional to strain.
(i) When the stress is 63 MPa, the strain is
0.0003
(ii) Hence, a stress of 1 MPa corresponds to a
strain of
0.0003
63
(iii) Thus, the value of strain when the stress is
42 MPa =
0.0003
63
× 42 = 0.0002
(b) Strain is proportional to stress.
(i) When the strain is 0.0003, the stress is
63 MPa.
(ii) Hence, a strain of 0.0001 corresponds to
63
3
MPa.
(iii) Thus, the value of stress when the strain is
0.00072 =
63
3
× 7.2 = 151.2 MPa.
Problem 12. Charles’s law states that for a given
mass of gas at constant pressure, the volume is
directly proportional to its thermodynamic
temperature. A gas occupies a volume of 2.4 litres
at 600 K. Determine (a) the temperature when the
volume is 3.2 litres, (b) the volume at 540 K
(a) Volume is directly proportional to temperature.
(i) When the volume is 2.4 litres, the temperature is 600 K.
(ii) Hence, a volume of 1 litre corresponds to a
temperature of
600
2.4
K.
(iii) Thus, the temperature when the volume is
3.2 litres =
600
2.4
× 3.2 = 800 K.
(b) Temperature is proportional to volume.
(i) When the temperature is 600 K, the volume
is 2.4 litres.
(ii) Hence, a temperature of 1 K corresponds to
a volume of
2.4
600
litres.
(iii) Thus, the volume at a temperature of
540 K =
2.4
600
× 540 = 2.16 litres.
Now try the following Practice Exercise
Practice Exercise 26 Direct proportion
(answers on page 342)
1. 3 engine parts cost £208.50. Calculate the cost
of 8 such parts.
2. If 9 litres of gloss white paint costs £24.75,
calculate the cost of 24 litres of the same paint.
3. The total mass of 120 household bricks is
57.6 kg. Determine the mass of 550 such
bricks.
4. A simple machine has an effort : load ratio of
3 :37. Determine the effort, in grams, to lift a
load of 5.55 kN.
5. If 16 cans of lager weighs 8.32 kg, what will
28 cans weigh?
6. Hooke’s law states that stress is directly proportional to strain within the elastic limit of
a material. When, for copper, the stress is
60 MPa, the strain is 0.000625. Determine
(a) the strain when the stress is 24 MPa and
(b) the stress when the strain is 0.0005
Here are some worked examples to help us understand
more about direct proportion.
Problem 9. 3 energy saving light bulbs cost
£7.80. Determine the cost of 7 such light bulbs
(i) 3 light bulbs cost £7.80
(ii) Therefore, 1 light bulb costs
7.80
3
= £2.60
Hence, 7 light bulbs cost 7 × £2.60 = £18.20
Problem 10. If 56 litres of petrol costs £59.92,
calculate the cost of 32 litres
(i) 56 litres of petrol costs £59.92
(ii) Therefore, 1 litre of petrol costs
59.92
56
= £1.07
Hence, 32 litres cost 32 × 1.07 = £34.24
Problem 11. Hooke’s law states that stress, σ , is
directly proportional to strain, ε, within the elastic
limit of a material. When, for mild steel, the stress
is 63 MPa, the strain is 0.0003. Determine (a) the
value of strain when the stress is 42 MPa, (b) the
value of stress when the strain is 0.00072
(a) Stress is directly proportional to strain.
(i) When the stress is 63 MPa, the strain is
0.0003
(ii) Hence, a stress of 1 MPa corresponds to a
strain of
0.0003
63
(iii) Thus, the value of strain when the stress is
42 MPa =
0.0003
63
× 42 = 0.0002
(b) Strain is proportional to stress.
(i) When the strain is 0.0003, the stress is
63 MPa.
(ii) Hence, a strain of 0.0001 corresponds to
63
3
MPa.
(iii) Thus, the value of stress when the strain is
0.00072 =
63
3
× 7.2 = 151.2 MPa.
Problem 12. Charles’s law states that for a given
mass of gas at constant pressure, the volume is
directly proportional to its thermodynamic
temperature. A gas occupies a volume of 2.4 litres
at 600 K. Determine (a) the temperature when the
volume is 3.2 litres, (b) the volume at 540 K
(a) Volume is directly proportional to temperature.
(i) When the volume is 2.4 litres, the temperature is 600 K.
(ii) Hence, a volume of 1 litre corresponds to a
temperature of
600
2.4
K.
(iii) Thus, the temperature when the volume is
3.2 litres =
600
2.4
× 3.2 = 800 K.
(b) Temperature is proportional to volume.
(i) When the temperature is 600 K, the volume
is 2.4 litres.
(ii) Hence, a temperature of 1 K corresponds to
a volume of
2.4
600
litres.
(iii) Thus, the volume at a temperature of
540 K =
2.4
600
× 540 = 2.16 litres.
Now try the following Practice Exercise
Practice Exercise 26 Direct proportion
(answers on page 342)
1. 3 engine parts cost £208.50. Calculate the cost
of 8 such parts.
2. If 9 litres of gloss white paint costs £24.75,
calculate the cost of 24 litres of the same paint.
3. The total mass of 120 household bricks is
57.6 kg. Determine the mass of 550 such
bricks.
4. A simple machine has an effort : load ratio of
3 :37. Determine the effort, in grams, to lift a
load of 5.55 kN.
5. If 16 cans of lager weighs 8.32 kg, what will
28 cans weigh?
6. Hooke’s law states that stress is directly proportional to strain within the elastic limit of
a material. When, for copper, the stress is
60 MPa, the strain is 0.000625. Determine
(a) the strain when the stress is 24 MPa and
(b) the stress when the strain is 0.0005