PROPORTION - ANSWERS
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1. The cost of Chanel T'Unnel perfume is directly proportional to the amount of perfume in the bottle. A bottle containing 50ml of perfume costs £20.00 a) Find a formula which connects the price and the volume of perfume. b) Work out the cost of a 100ml bottle of perfume.
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a) Let the cost = C and the volume = V: C a V Therefore C = a constant x V, we'll call this constant k, Hence C = kV From the question, 50ml of perfume costs £20, so we can subtitute this into the formula to find k: C = kV 20 = 50k hence k = 20/50 = 2/5 Hence the formula is: C = 2/5V b) To find the cost of 100ml, we simply use the equation again (now that we know what k is) C = 2/5 x 100 = 200/5 = £40.00
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2. The extension E of a spring is directly proportional to the force F pulling the spring. If a force of 15N is applied to the spring the extension is measured as 4cm. What is the extension when the force is 12N ? |
E a F hence E = kF When F = 15, E = 4 hence 4 = 15k k = 4/15 E = 12 x 4/15 = 48/15 = 3.2cm |
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3. The distance d travelled by a car is directly proportional to the square of the time t taken for the journey. a) Deduce a formula for d in terms of t. When t = 10 seconds d = 200 metres. b) Calculate: i) The distance travelled in 1 minute. ii) The time taken when d = 3,000 metres.
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a) d a t2 hence d = kt2 If d =200 when t = 10 200 = k(10)2 200 = 100k k = 200/100 = 2 Substituting k = 2 we get the formula d = 2t2 i) If t = 1 minute this is 60 seconds (DON'T get caught out with sneaky changes to quantities like this) Hence d = 2(60)2 = 2 x 3,600 = 7,200 metres ii) 3,000 = 2t2 1,500 = t2 t = Ö 1,500 = 38.73 seconds (to 2 d.p.) |
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4. The time taken to build a house is inversely proportional to the number of builders working on it. If 2 builders are working on the house it will take two months to complete. How long will it take to complete the house if 5 builders are working on it? |
Let number of builders = N and time taken = T, T a 1/N T = k/N 2 = k/2 Hence k = 4 With 5 builders T = 4/5, assuming a month = 30 days, the 5 builders will take just 24 days. |
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5. The variables m and n are related in such a way that n is directly proportional to the cube of m. Complete the table of values below:
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n a m3 Hence n = km3 When m = 3, n = 54, hence we can use these to find k: k = 54/3 = 18 Hence n = 18m3, you can work the rest out from this formula.
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6. q is directly proportional to the square root of p.
a) Work out a formula involving p and q. b) What is the value of q when p = 36?
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a) q a Ö p q = kÖp 36 = kÖ81 i.e. 36 = 9k hence k = 36/9 = 4 The formula is: q = 4Öp b) q = 4Ö36 q = 4 x 6 = 24 |
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7. A football kicked into the air to height h is directly proportional to square of the speed v at which it was kicked. If the ball is kicked at 20 metres per second, it will reach a height of 20 metres. a) How high does the ball reach if it is kicked at 18 m/s ? b) How high does the ball reach if it is kicked at 25 m/s ?
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h a v2 hence h = kv2 20 = 400k k = 1/20 Giving the formula h = 1/20v2 a) h = 1/20 (18)2 = 16.2m b) h = 1/20 (25)2 = 31.25m |
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8. A lorry travels 350km in 2 hours 40 minutes. The distance it travels is directly proportional to the time taken. How far will the lorry travel in 4 hours?
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It is often less confusing to reduce questions like this which give times in hours and minutes to minutes only, then you have a single quantity to deal with which is much easier. 2 hours 40 minutes = 60 + 60 + 40 minutes d a t hence d = kt 350 = 160k k = 350/160 = 2.1875 The formula is: d = 2.1875t 4 hours = 4 x 60 minutes = 240 minutes hence in 4 hours d = 2.1875 x 240 = 525km |