Revision Test 2 : Decimals, calculators and percentages
6.3 Direct proportion
Two quantities are in direct proportion when they increase or decrease in thesame ratio. For example, if 12 cans of lager have a mass of 4 kg, then 24 cans of lager will have a mass of 8 kg; i.e., if the quantity of cans doubles then so does the mass. This is direct proportion.
In the previous section we had an example of mixing 1 shovel of cement to 4 shovels of sand; i.e., the ratio of cement to sand was 1 : 4. So, if we have a mix of 10 shovels of cement and 40 shovels of sand and we wanted to double the amount of the mix then we would need to double both the cement and sand, i.e. 20 shovels of cement and 80 shovels of sand. This is another example of direct proportion.
Here are three laws in engineering which involve direct proportion:
(a) Hooke’s lawstates that, within the elastic limit of a material, the strainεproduced is directly propor-tional to the stressσproducing it, i.e.ε∝σ(note than ‘∝’ means ‘is proportional to’).
(b) Charles’s lawstates that, for a given mass of gas at constant pressure, the volumeV is directly pro-portional to its thermodynamic temperatureT, i.e.
V∝T.
(c) Ohm’s law states that the current I flowing through a fixed resistance is directly proportional to the applied voltageV, i.e.I∝V.
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 costs7.80
3 =£2.60 Hence,7 light bulbs cost7×£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 cost32×1.07=£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 of0.0003
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 tempera-ture is 600 K.
(ii) Hence, a volume of 1 litre corresponds to a temperature of600
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
600litres.
(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 pro-portional 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
44 Basic Engineering Mathematics
7. Charles’s law states that volume is directly proportional to thermodynamic temperature for a given mass of gas at constant pressure.
A gas occupies a volume of 4.8 litres at 330 K.
Determine (a) the temperature when the vol-ume is 6.4 litres and (b) the volvol-ume when the temperature is 396 K.
Here are some further worked examples on direct proportion.
Problem 13. Some guttering on a house has to decline by 3 mm for every 70 cm to allow rainwater to drain. The gutter spans 8.4 m. How much lower should the low end be?
(i) The guttering has to decline in the ratio 3 :700 or 3
700
(ii) Ifdis the vertical drop in 8.4 m or 8400 mm, then the decline must be in the ratiod: 8400 or d
8400 (iii) Now d
8400= 3 700
(iv) Cross-multiplying gives 700×d=8400×3 from
which, d=8400×3
700 i.e. d=36 mm, which is how much the lower end should be to allow rainwater to drain.
Problem 14. Ohm’s law state that the current flowing in a fixed resistance is directly proportional to the applied voltage. When 90 mV is applied across a resistor the current flowing is 3 A.
Determine (a) the current when the voltage is 60 mV and (b) the voltage when the current is 4.2 A (a) Current is directly proportional to the voltage.
(i) When voltage is 90 mV, the current is 3 A.
(ii) Hence, a voltage of 1 mV corresponds to a current of 3
90A.
(iii) Thus, when the voltage is 60 mV, the current =60× 3
90=2A.
(b) Voltage is directly proportional to the current.
(i) When current is 3 A, the voltage is 90 mV.
(ii) Hence, a current of 1 A corresponds to a voltage of90
3 mV=30 mV.
(iii) Thus, when the current is 4.2 A, the voltage=30×4.2=126 mV.
Problem 15. Some approximate imperial to metric conversions are shown in Table 6.1. Use the table to determine
(a) the number of millimetres in 12.5 inches (b) a speed of 50 miles per hour in kilometres
per hour
(c) the number of miles in 300 km
(d) the number of kilograms in 20 pounds weight (e) the number of pounds and ounces in
56 kilograms (correct to the nearest ounce) (f) the number of litres in 24 gallons
(g) the number of gallons in 60 litres Table 6.1
length 1 inch=2.54 cm 1 mile=1.6 km weight 2.2 lb=1 kg
(1 lb=16 oz) capacity 1.76 pints=1 litre
(8 pints=1 gallon)
(a) 12.5 inches=12.5×2.54 cm=31.75 cm 31.73 cm=31.75×10 mm=317.5 mm (b) 50 m.p.h.=50×1.6 km/h=80 km/h (c) 300 km=300
1.6 miles=186.5 miles (d) 20 lb= 20
2.2kg=9.09 kg (e) 56 kg=56×2.2 lb=123.2 lb
0.2 lb=0.2×16 oz=3.2 oz=3 oz, correct to the nearest ounce.
Thus, 56 kg=123 lb 3 oz, correct to the nearest ounce.
(f ) 24 gallons=24×8 pints=192 pints 192 pints= 192
1.76litres=109.1 litres
Ratio and proportion 45
(g) 60 litres=60×1.76 pints=105.6 pints 105.6 pints=105.6
8 gallons=13.2 gallons Problem 16. Currency exchange rates for five countries are shown in Table 6.2. Calculate (a) how many euros £55 will buy
(b) the number of Japanese yen which can be bought for £23
(c) the number of pounds sterling which can be exchanged for 6405 kronor
(d) the number of American dollars which can be purchased for £92.50
(e) the number of pounds sterling which can be exchanged for 2925 Swiss francs
Table 6.2
France £1=1.25 euros
Japan £1=185 yen
Norway £1=10.50 kronor Switzerland £1=1.95 francs USA £1=1.80 dollars (a) £1=1.25 euros, hence £55=55×1.25 euros
=68.75 euros.
(b) £1=185 yen, hence £23=23×185 yen
=4255 yen.
(c) £1=10.50 kronor, hence 6405 lira=£6405 10.50
=£610.
(d) £1=1.80 dollars, hence
£92.50=92.50×1.80 dollars=$166.50 (e) £1=1.95 Swiss francs, hence
2925 pesetas=£2925
1.95 =£1500 Now try the following Practice Exercise
Practice Exercise 27 Further direct proportion (answers on page 342)
1. Ohm’s law states that current is proportional to p.d. in an electrical circuit. When a p.d. of
60 mV is applied across a circuit a current of 24μA flows. Determine (a) the current flowing when the p.d. is 5 V and (b) the p.d. when the current is 10 mA.
2. The tourist rate for the Swiss franc is quoted in a newspaper as £1=1.92 fr. How many francs can be purchased for £326.40?
3. If 1 inch=2.54 cm, find the number of mil-limetres in 27 inches.
4. If 2.2 lb=1 kg and 1lb=16 oz, determine the number of pounds and ounces in 38 kg (correct to the nearest ounce).
5. If 1 litre=1.76 pints and 8 pints=1 gallon, determine (a) the number of litres in 35 gallons and (b) the number of gallons in 75 litres.
6. Hooke’s law states that stress is directly pro-portional to strain within the elastic limit of a material. When for brass the stress is 21 MPa, the strain is 0.00025. Determine the stress when the strain is 0.00035.
7. If 12 inches=30.48 cm, find the number of millimetres in 23 inches.
8. The tourist rate for the Canadian dollar is quoted in a newspaper as £1=1.84 fr. How many Canadian dollars can be purchased for
£550?