Proportionality Proportionality. Direct proportion. Stretch objectives. Check-in questions. y = k x

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1 lesson: Proportionalit 13Stretch Stretch objectives Before ou start this chapter, mark how confident ou feel about each of the statements below: I can set up and use equations to solve problems involving direct and inverse proportion. Check-in questions Complete these questions to assess how much ou remember about each topic. Then mark our work using the answers at the end of the lesson. If ou score well on all sections, ou can go straight to the Revision Checklist and Eam-stle uestions at the end of the lesson. If ou don t score well, go to the lesson section indicated and work through the eamples and practice questions there. 1 Match each statement with the correct equation. Go to 13.1 a is proportional to b is inversel proportional to = k = k 2 a is directl proportional to. When = 4, a = 8. a What is the constant of proportionalit? b What is the value of when a = 64? Go to is inversel proportional to. When = 3, = 1. a Calculate the value of when = 2. b Calculate the value of when = 6. Go to Proportionalit Direct proportion Two quantites are in direct proportion to each other when the increase at the same rate. The smbol is used to denote direct proportion. For eample, the cost of a bag of apples (C pence) is directl proportional to the mass (M kg) of the apples. The smbol means is directl proportional to, so we can write this as C M. If the apples cost k pence per kilogram, then C = km. GCSE (9-1) Maths for Post-16 HarperCollinsPublishers 217

2 In general, if is directl proportional to : and = k, where k is known as the constant of proportionalit. Since = k, the graph of against is a straight line passing through the origin. (See Chapter 1.) The constant of proportionalit, k, is the gradient of this straight line. You can use given values of the variables to work out the constant of proportionalit. k Eample 1 a is proportional to b and a = 5 when b = 4. a Work out the value of k (the constant of proportionalit). b Work out the value of a when b = 8. a a b a = kb 5 = k 4 k = 5 4 b a = 5 4 b a = a = 1 Write out a proportion statement using the smbol. Write an equation using k as the constant of proportionalit. Substitute the values given to find the value of k. Substitute k = 5 into the equation. 4 When b = 8 Eample 2 The voltage, V volts, across an electrical circuit is directl proportional to the current, I amps, flowing through the circuit. When I = 2.4, V = 156. a Work out the formula connecting V and I. b Work out the value of V when I = 4. c Work out the value of I when V = volts. a V I b V = 65 4 c V = 65I V = ki 156 = k = k 24. k = 65, so V = 65I V = 26 volts I = V 65 I = I = 5.5 amps Inverse proportion Two quantities are in inverse proportion when as one variable increases, the other decreases at the same rate and vice versa. You write is inversel proportional to as 1 or = k When k is positive, the graph = k has a similar shape to = 1. 1 GCSE (9-1) Maths for Post-16 HarperCollinsPublishers 217

3 Eample 3 p is inversel proportional to w. When p = 5, w = 2. What is the value of w when p = 2? p 1 w p = k w 5 = k = k k = 1 p = 1 w 2 = 1 w w = 1 2 Write the information with the proportionalit sign. Replace with an equation using the constant of proportionalit. Substitute known values to find the value of k. Rewrite the formula with k = 1. Find the value of w when p = 2. w = 1 2 or.5 Eam tips Check carefull whether ou are dealing with direct or inverse proportion. Find the value of k then write down the equation connecting the variables. Practice questions 1 T is directl proportional to P. When P = 4, T = 24. a Calculate the value of T when P = 3. b Calculate the value of P when T = 9. 2 R is inversel proportional to S. When R = 1, S = 2. a Calculate the value of R when S = 4. b Calculate the value of S when R = 6. 3 is directl proportional to the square of D. When D = 3, = 27. a Calculate the value of when D = 5. b Calculate the value of D when = Here are four graphs. Graph Graph B Graph C Graph D GCSE (9-1) Maths for Post-16 HarperCollinsPublishers 217

4 Write down the letter of the graph that represents these relationships. a is proportional to 2 b is proportional to c is proportional to 1 5 R is inversel proportional to the square of S. When S = 2, R = 2. a Calculate the value of R when S = 6. b Calculate the value of S when R =.5. REVISION CHECKLIST The notation means is directl proportional to. This is often abbreviated to is proportional to or varies as. Eam-stle questions 1 T is directl proportional to P. When T = 1.5, P = 15. a Calculate the value of T when P = 3. b Calculate the value of P when T = R is inversel proportional to S. When R =.5, S = 16. a Calculate the value of R when S = 4. b Calculate the value of S when R = is directl proportional to the square of D. When D = 4, = 8. a Calculate the value of when D = 3. b Calculate the value of D when = R is proportional to the square of S. When S = 2, R = a Calculate the value of R when S = 3. b Calculate the value of S when R = GCSE (9-1) Maths for Post-16 HarperCollinsPublishers 217

5 Chapter 13 Stretch lesson: nswers Check-in questions 1 a = k b = k 2 a 2 (a, a = k, 8 = 4k, k = 2, a = 2) b 64 = 2, = 32 ( ) 3 ( = ) k k 3 3 a 15 =, 1 =, k = 3, = 3 b Proportionalit 1 a 18 b a.5 b a 75 b 6 4 a D b c B 5 a 2 9 b 4 Eam-stle questions 1 a 3 b.95 2 a 2 b a 45 b 5 4 a 56.7 b 5 GCSE (9-1) Maths for Post-16 HarperCollinsPublishers 217