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CHAPTER 6 RATIO AND PROPORTION EXERCISE 26 Page 48 1. In a box of 333 paper clips, 9 are defective. Express the non-defective paper clips as a ratio of the defective paper clips, in its simplest form. Non-defective paper clips = 333 – 9 = 324 The ratio of non-defective clips to defective clips is: 324:9 Dividing both by 9 gives the ratio as: 36:1 2. A gear wheel having 84 teeth is in mesh with a 24-tooth gear. Determine the gear ratio in its simplest form. Gear ratio = 84:24 Dividing both by 4 gives: 21:6 Dividing both by 3 gives: 3. 7:2 or 3.5:1 In a box of 2000 nails, 120 are defective. Express the non-defective nails as a ratio of the defective ones, in its simplest form. Non-defective nails = 2000 – 120 = 1880 The ratio of non-defective clips to defective clips is: 1880:120 Dividing both by 10 gives the ratio as: 188:12 Dividing both by 4 gives the ratio as: 47:3 4. A metal pipe 3.36 m long is to be cut into two in the ratio 6 to 15. Calculate the length of each piece. Since the ratio is 6:15, the total number of parts is 6 + 15 = 21 parts 88 © 2014, John Bird 21 parts corresponds to 3.36 m = 336 cm, hence, 1 part corresponds to 336 = 16 21 Thus, 6 parts corresponds to 6 × 16 = 96 cm and 15 parts corresponds to 15 × 16 = 240 cm Hence, 3.36 m divides in the ratio of 6:15 as 96 cm to 240 cm 5. On the instructions for cooking a turkey it says that it needs to be cooked 45 minutes for every kilogram. How long will it take to cook a 7 kg turkey? It takes 45 min for 1 kg of turkey Hence, for 7 kg, time required = 7 × 45 min = 315 min = 5 h 15 min 6. In a will, £6440 is to be divided between three beneficiaries in the ratio 4:2:1. Calculate the amount each receives. Since the ratio is 4:2:1 the total number of parts is 4 + 2 + 1 = 7 parts 7 parts corresponds to £6440 1 part corresponds to 6440 = £920, 7 2 parts corresponds to 2 × £920 = £1840 and 4 parts corresponds to 4 × £920 = £3680 Hence, £6440 divides in the ratio of 4:2:1 as £3680 to £1840 to £920 7. A local map has a scale of 1:22 500. The distance between two motorways is 2.7 km. How far are they apart on the map? 2.7 km 2700 m 270 000 cm Distance between motorways = = = = 12 cm 22 500 2250 22 500 8. Prize money in a lottery totals £3801 and is shared among three winners in the ratio 4:2:1. How much does the first-prize winner receive? 89 © 2014, John Bird Since the ratio is 4:2:1 the total number of parts is 4 + 2 + 1 = 7 parts 7 parts corresponds to £3801 1 part corresponds to 3801 = £543 and 4 parts corresponds to 4 × £543 = £2172 7 Hence, the first-prize winner receives £2172 90 © 2014, John Bird EXERCISE 27 Page 48 1. Express 130 g as a ratio of 1.95 kg. Changing both quantities to the same units, i.e. to grams, gives a ratio of: 130:1950 130:1950 ≡ 13:195 Dividing both quantities by 10 gives: 13:195 ≡ 1:15 Dividing both quantities by 13 gives: Thus, 130 g as a ratio of 1.95 kg is: 2. 1:15 In a laboratory, acid and water are mixed in the ratio 2:5. How much acid is needed to make 266 ml of the mixture? In a ratio of 2:5 there are 2 + 5 = 7 parts 1 part = 266 ml ÷ 7 = 38 ml Amount of acid in the mixture = 2 parts = 2 × 38 = 76 ml 3. A glass contains 30 ml of gin, which is 40% alcohol. If 18 ml of water is added and the mixture stirred, determine the new percentage alcoholic content. The 30 ml of gin contains 40% alcohol = 40 × 30 = 12 ml 100 After 18 ml of water is added we have 30 + 18 = 48 ml of fluid, of which alcohol is 12 ml Fraction of alcohol present = 12 48 Percentage of alcohol present = 4. 12 ×100% = 25% 48 A wooden beam 4 m long weighs 84 kg. Determine the mass of a similar beam that is 60 cm long. 4 m of beam weighs 84 kg, hence, 1 m of beam = 84 kg ÷ 4 = 21 kg 91 © 2014, John Bird 60 cm, i.e. 0.6 m, will weigh 0.6 × 21 = 12.6 kg 5. An alloy is made up of metals P and Q in the ratio 3.25:1 by mass. How much of P has to be added to 4.4 kg of Q to make the alloy. For every 1 part of Q there is 3.25 parts of P For 4.4 kg of Q, P = 3.25 × 4.4 = 14.3 kg 6. 15 000 kg of a mixture of sand and gravel is 20% sand. Determine the amount of sand that must be added to produce a mixture with 30% gravel. Amount of sand in 15 000 kg = 20% of 15 000 kg = 20 ×15 000 = 3000 kg 100 If the mixture has 3000 kg of sand, then amount of gravel = 15 000 – 3000 = 12 000 kg We want this 12 000 kg of gravel to be 30% of the new mixture 1% would be 12 000 12 000 t and 100% of mixture would be ×100 kg = 40 000 kg 30 30 If there is 12 000 kg of gravel then amount of sand = 40 000 – 12 000 = 28 000 kg We already have 3000 kg of sand, so amount of sand to be added to produce a mixture with 30% gravel = 28 000 – 3000 = 25 000 kg 92 © 2014, John Bird EXERCISE 28 Page 50 1. Three engine parts cost £208.50. Calculate the cost of eight such parts. If three engine parts cost £208.50, then one engine part costs £208.50 ÷ 3 = £69.50 Hence, the cost of eight engine parts = 8 × £69.50 = £556 2. If nine litres of gloss white paint costs £24.75, calculate the cost of 24 litres of the same paint. If nine litres of paint cost £24.75, then one litre costs £24.75 ÷ 9 = £2.75 Hence, the cost of 24 litres = 24 × £2.75 = £66 3. The total mass of 120 household bricks is 57.6 kg. Determine the mass of 550 such bricks. If 120 bricks have a mass of 57.6 kg, then 1 brick has a mass of 57.6 kg ÷ 120 = 0.48 kg Hence, the mass of 550 bricks = 550 × 0.48 kg = 264 kg 4. A simple machine has an effort:load ratio of 3:37. Determine the effort, in newtons, to lift a load of 5.55 kN. If 37 parts is equivalent to 5.55 kN = 5550 N Hence, 1 part is equivalent to 5550 ÷ 37 = 150 N and 3 parts is equivalent to 3 × 150 N = 450 N, i.e. the required effort is 450 N 5. If 16 cans of lager weighs 8.32 kg, what will 28 cans weigh? If 16 cans of lager weighs 8.32 kg, then 1 can weighs 8.32 kg ÷ 16 = 0.52 kg Hence, the weight of 28 cans = 28 × 0.52 kg = 14.56 kg 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 93 © 2014, John Bird (a) Stress is directly proportional to strain. When the stress is 60 MPa, the strain is 0.000625, hence a stress of 1 MPa corresponds to a strain of 0.000625 60 and the value of strain when the stress is 24 MPa = 0.000625 × 24 = 0.00025 60 (b) If when the strain is 0.000625 the stress is 60 MPa, then a strain of 0.0001 corresponds to 60 MPa 6.25 and the value of stress when the strain is 0.0005 = 7. 60 × 5 = 48 MPa 6.25 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 volume is 6.4 litres, and (b) the volume when the temperature is 396 K. (a) Volume is directly proportional to temperature When the volume is 4.8 litres, the temperature is 330 K hence a volume of 1 litre corresponds to a temperature of and the temperature when the volume is 6.4 litres = 330 K 4.8 330 × 6.4 = 440 K 4.8 (b) Temperature is proportional to volume When the temperature is 330 K, the volume is 4.8 litres hence a temperature of 1 K corresponds to a volume of and the volume at a temperature of 396 K = 4.8 litres 330 4.8 × 396 = 5.76 litres 330 8. A machine produces 320 bolts in a day. Calculate the number of bolts produced by four machines in seven days. 94 © 2014, John Bird The machine produces 320 bolts in one day If there were four machines, then 4 × 320 bolts would be produced daily, i.e. 1280 bolts. In seven days, number of bolts produced = 7 × 1280 = 8960 bolts 95 © 2014, John Bird EXERCISE 29 Page 51 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 24 µA flows. Determine: (a) the current flowing when the p.d. is 5 V, and (b) the p.d. when the current is 10 mA. (a) Current is directly proportional to the voltage When p.d. is 60 mV, the current is 24 µA, hence a p.d. of 1 mV corresponds to a current of 24 µA 60 and a p.d. of 1 volt corresponds to a current of 1000 × and when the p.d. is 5 V, the current = 5 × 1000 × 24 µA 60 24 = 2000 µA = 2 mA 60 (b) Voltage is directly proportional to the current When current is 24 µA, the voltage is 60 mV, hence a current of 1 A corresponds to a voltage of 60 ×103 = 2.5 kV 24 ×10−6 and when the current is 10 mA, the voltage = 10 ×10−3 × 2.5 kV = 25 V 2. The tourist rate for the Swiss franc is quoted in a newspaper as £1 = 1.90 fr. How many francs can be purchased for £310? Number of Swiss francs that can be purchased for £310 = 310 × 1.90 = 589 fr 3. If 1 inch = 2.54 cm, find the number of millimetres in 27 inches. In 27 inches, number of centimetres = 27 × 2.54 = 68.58 cm Number of millimetres in 27 inches = 10 × 68.58 = 685.8 mm 96 © 2014, John Bird 4. If 2.2 lb = 1 kg, and 1 lb = 16 oz, determine the number of pounds and ounces in 38 kg (correct to the nearest ounce). 38 kg = 38 × 2.2 = 83.6 lb 0.6 lb = 0.6 × 16 oz = 9.6 oz = 10 oz to the nearest ounce Hence, 38 kg = 83 lb 10 oz 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. (a) 35 gallons = 35 × 8 pints = 280 pints Number of litres in 35 gallons = 280 ÷ 1.76 = 159.1 litres (b) 75 litres = 75 × 1.76 pints = 132 pints 132 pints = 132 ÷ 8 gallons = 16.5 gallons 6. Hooke’s law states that stress is directly proportional 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 Stress is directly proportional to strain. If when the strain is 0.00025, the stress is 21 MPa, then a strain of 0.0001 corresponds to 21 MPa 2.5 and the value of stress when the strain is 0.00035 = 21 × 3.5 = 29.4 MPa 2.5 7. If 12 inches = 30.48 cm, find the number of millimetres in 23 inches. If 12 inches = 30.48 cm, then 1 inch = 30.48 ÷ 12 = 2.54 cm and 23 inches = 23 × 2.54 cm = 58.42 cm = 10 × 58.42 mm = 584.2 mm 97 © 2014, John Bird 8. The tourist rate for the Canadian dollar is quoted in a newspaper as £1 = $1.84. How many Canadian dollars can be purchased for £550? Number of Canadian dollars that can be purchased for £550 = 550 × $1.84 = $1012 98 © 2014, John Bird EXERCISE 30 Page 53 1. A 10 kg bag of potatoes lasts for a week with a family of seven people. Assuming all eat the same amount, how long will the potatoes last if there were only two in the family? If a 10 kg bag of potatoes lasts for a week with a family of seven people, then it would last for seven weeks for one person. For two persons, it would last for 7/2 = 3.5 weeks 2. If eight men take five days to build a wall, how long would it take two men? If eight men take five days to build a wall, then one man would take 8 × 5 = 40 days. Hence, two men would take 40/2 = 20 days to build the wall. 3. If y is inversely proportional to x and y = 15.3 when x = 0.6, determine (a) the coefficient of proportionality, (b) the value of y when x is 1.5 and (c) the value of x when y is 27.2 yα 1 x i.e. y = k x where k is the constant of proportionality (a) When y = 15.3, x = 0.6, hence 15.3 = k 0.6 from which, constant of proportionality, k = (15.3)(0.6) = 9.18 (b) When x = 1.5, y = (c) When y = 27.2, x = k 9.18 = 6.12 = x 1.5 k 9.18 = = 0.3375 y 27.2 4. A car travelling at 50 km/h makes a journey in 70 minutes. How long will the journey take at 70 km/h? If the car takes 70 minutes at 50 km/h then distance travelled = 50 km/h × 99 70 h = 175/3 km 60 © 2014, John Bird If the speed is 70 km/h then time for journey = 175 / 3km 5 5 = h= × 60 min = 50 minutes 70 km/h 6 6 5. Boyle’s law states that for a gas at constant temperature, the volume of a fixed mass of gas is inversely proportional to its absolute pressure. If a gas occupies a volume of 1.5 m3 at a pressure of 200 × 103 Pascal’s, determine (a) the constant of proportionality, (b) the volume when the pressure is 800 × 103 Pascals and (c) the pressure when the volume is 1.25 m3. Volume α 1 absolute pressure i.e. Vα 1 p or V = k where k is the constant of proportionality p (a) When V = 1.5 m3 , p = 200 ×103 Pa, hence, 1.5 = k 200 ×103 from which, constant of proportionality, k = (1.5)( 200 ×103 ) = 300 ×103 (b) When p = 800 ×103 , V = (c) When V = 1.25 m3 , p = k 300 ×103 = = 0.375 m3 p 800 ×103 k 300 ×103 = 240 ×103 Pa = 1.25 V 6. The energy received by a surface from a source of heat is inversely proportional to the square of the distance between the heat source and the surface. A surface 1 m from the heat source receives 200 J of energy. Calculate (a) the energy received when the distance is changed to 2.5 m and (b) the distance required if the surface is to receive 800 J of energy. (a) Energy, Wα 1 d2 or W = k where k is the coefficient of proportionality d2 When d = 1 m, W = 200 J, i.e. 200 = When d = 2.5 m, energy received, W = (b) When energy, W = 800 J, then 800 = k =k 12 k 200 = 32 J = d 2 2.52 k 200 = 2 d d2 100 © 2014, John Bird from which, = d2 200 1 = 800 4 and distance to surface, d = 1 1 = m = 0.5 m 4 2 101 © 2014, John Bird