National 5 Mathematics Course Materials Topic 11: Changing the subject of a formula

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1 SCHOLAR Study Guide National 5 Mathematics Course Materials Topic 11: Changing the subject of a formula Authored by: Margaret Ferguson Reviewed by: Jillian Hornby Previously authored by: Eddie Mullan Heriot-Watt University Edinburgh EH14 4AS, United Kingdom.

2 First published 2014 by Heriot-Watt University. This edition published in 2016 by Heriot-Watt University SCHOLAR. Copyright 2016 SCHOLAR Forum. Members of the SCHOLAR Forum may reproduce this publication in whole or in part for educational purposes within their establishment providing that no profit accrues at any stage, Any other use of the materials is governed by the general copyright statement that follows. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, without written permission from the publisher. Heriot-Watt University accepts no responsibility or liability whatsoever with regard to the information contained in this study guide. Distributed by the SCHOLAR Forum. SCHOLAR Study Guide Course Materials Topic 11: National 5 Mathematics 1. National 5 Mathematics Course Code: C747 75

3 Acknowledgements Thanks are due to the members of Heriot-Watt University's SCHOLAR team who planned and created these materials, and to the many colleagues who reviewed the content. We would like to acknowledge the assistance of the education authorities, colleges, teachers and students who contributed to the SCHOLAR programme and who evaluated these materials. Grateful acknowledgement is made for permission to use the following material in the SCHOLAR programme: The Scottish Qualifications Authority for permission to use Past Papers assessments. The Scottish Government for financial support. The content of this Study Guide is aligned to the Scottish Qualifications Authority (SQA) curriculum. All brand names, product names, logos and related devices are used for identification purposes only and are trademarks, registered trademarks or service marks of their respective holders.

5 1 Topic 1 Changing the subject of a formula Contents 11.1 Changing the subject of a formula Learning points End of topic test

6 2 TOPIC 1. CHANGING THE SUBJECT OF A FORMULA Learning objectives By the end of this topic, you should be able to: change the subject of a formula.

7 TOPIC 1. CHANGING THE SUBJECT OF A FORMULA Changing the subject of a formula Rearranging formulae Look at this formula: A = 3B A is called the subject of the formula. The formula tells you how to calculate A when you know B. Go online Reverse the process and B becomes the subject of the formula. Look at this formula: A = B + 4 A is called the subject of the formula. The formula tells you how to calculate A when you know B. Reverse the process and B becomes the subject of the formula. Look at this formula: C = ah + t C is called the subject of the formula. The formula tells you how to calculate C when you know a, H and t.

8 4 TOPIC 1. CHANGING THE SUBJECT OF A FORMULA Reverse the process and B becomes the subject of the formula. Examples 1. Problem: Make A the subject of the formula B = 3A +2. Solution: Think what the formula does to A: A times 3 then add 2 gives B. Reverse this: B minus 2 then divide by 3 gives A. Write it as a formula: A = (B 2) 3 or as A = B Problem: The formula for the volume of a cylinder is V = πr 2 h. Change the subject of the formula to r. Solution: Think what the formula does to r: r squared times πh gives V. Reverse this: V divided by πh then square root gives r. V Write it as a formula:r = πh 3. Problem: The formula to convert from Fahrenheit to Celsius is: C = 5 9 (F 32). Change the subject of the formula to F. Solution: Think what the formula does to F : F minus 32 multiplied by 5 / 9 gives C. Reverse this: C divided by 5 / 9 then add 32 gives F.

9 TOPIC 1. CHANGING THE SUBJECT OF A FORMULA 5 Remember when we divide by a fraction we change it to multiply by the reciprocal, flipping the fraction over to become 9 / 5. Therefore, 5 9 becomes 9 5. Write it as a formula: F = 9 5 C Problem: a b = 1 g 1+g, change the subject of the formula to g. Solution: Sometimes it is easier to use the rules for rearranging an equation. This is especially true when the subject we want appears more than once in the formula. a = 1 g b 1 + g multiply by (1 + g) a(1 + g) b = 1 g multiply by b a(1 + g) = b(1 g) multiply out the brackets a + ag = b bg add bg a + ag + bg = b subtract a ag + bg = b a factorise the left hand side g(a + b) = b a divide by (a + b) g = b a a + b Key point Remember to re-arrange a step at a time and show all your working. Changing the subject of a formula practice Try the following questions on paper then check your answers. Q1: Make F the subject of the formula E = 4F 5 Go online Q2: Make h the subject of the formula V = 1 2 bhl Q3: Make t the subject of the formula M = 1 2 kt2 Q4: The formula to calculate your BMI (B) isb = w h where w is your weight in kg 2 and h is your height in metres. To find the height of someone when you know their weight and BMI, change the subject of the formula to h.

10 6 TOPIC 1. CHANGING THE SUBJECT OF A FORMULA Changing the subject of a formula exercise Go online Key point Remember to re-arrange a step at a time and show all your working. Q5: a) Make P the subject of the formula Q = 8P +6 b) Make t the subject of the formula s = At + B Q6: a) Make l the subject of the formula A = 1 2 lh b) Make a the subject of the formula C = a m + b c) Make h the subject of the formula V = 1 3 πr2 h Q7: a) Make H the subject of the formula Z = H2 R b) Make r the subject of the formula V = 4 3 πr2 c) Make b the subject of the formula d = ab + c Q8: Make y the subject of the formula jy + ky = q Q9: Make p the subject of the formula a p = b p 1 Q10: Make x the subject of the formula x +5 2 x = y z Q11: Height (h) of a rocket can be calculated using the formula h = 16t 2 + vt, where t is the time taken and v is the speed. To find the speed of the rocket, change the subject of the formula to v. Q12: In Physics, acceleration (a) is calculated using the formula a = v u t, where v is the final velocity (speed), u is the initial velocity (speed) and t is time taken (seconds). To find the initial velocity, change the subject of the formula to u.

11 TOPIC 1. CHANGING THE SUBJECT OF A FORMULA Learning points By reversing the operations performed in the original formula you can change the subject. The rules for solving equations can be applied. Re-arrange a step at a time and show all working.

12 8 TOPIC 1. CHANGING THE SUBJECT OF A FORMULA 1.3 End of topic test End of topic test Go online Key point Remember to re-arrange a step at a time and show all your working. Change the subject of these equations to the variable stated. Q13: a) x + a = b to x. b) x d = e to x. c) g = 1 2 hxto x. Change the subject of these equations to the variable stated. Q14: a) A = πr 2 to r. b) t = r2 s to r. c) h 2 = p 2 + r 2 to r. d) k = m to r. r 3 Change the subject of these equations to the variable stated. Q15: a) G = 2(y z) to y. b) Q = 3 4 (y + v) to y. Q16: Make g the subject of the formula ag bh = cg Q17: In Physics the formula for the magnetic force of a particle (F )is F = mv2 r Change the subject of the formula to v. Q18: Make R the subject of the formula L = kr2 3 j

13 TOPIC 1. CHANGING THE SUBJECT OF A FORMULA 9 Q19: The number of rolls (R) of wallpaper required to paper a room can be calculated ph using the formula R = 0 53 (10 r) + 1, where p is the perimeter of the room in metres, h is the height of the room in metres and r is the length of the pattern repeat in metres. To find the length of the pattern repeat, change the subject of the formula to r.

14 10 ANSWERS: TOPIC 11 Answers to questions and activities 11 Changing the subject of a formula Changing the subject of a formula practice (page 5) Q1: Think what the formula does to F : F times 4 minus 5 gives E. Reverse this: E plus 5 all divided by 4 gives F. Write it as a formula: F = E +5 4 Q2: Think what the formula does to h and reverse this. Write it as a formula: h = 2 V bl remember 1 2 becomes 2 1 Q3: Think what the formula does to t and reverse this. Write it as a formula: t = 2 M k Q4: Think what the formula does to h and reverse this. Write it as a formula: h = w B Changing the subject of a formula exercise (page 6) Q5: a) P = Q 6 8 b) t = s B A Q6: a) l = 2 A h b) a = m (C b) c) h = 3 V πr 2 Q7: a) H = ZR b) r = 3 V 4 π c) b = d2 c a Q8: y = Q9: Steps: q j + k

15 ANSWERS: TOPIC What do you get when you cross-multiply? a(p 1) = bp What do you get when you expand the brackets and collect all terms on the left hand side of the equation? ap a bp = 0 Answer: p = a a b Q10: x = 2y 5z y + z Q11: v = h +16t2 t Q12: u = v at End of topic test (page 8) Q13: a) x = b a b) x = de c) x = 2 g h Q14: a) r = A π b) r = st c) r = h 2 p 2 d) r = 3 m k Q15: a) y = G 2 + z b) y = 4 3 Q v Q16: g = bh (a c) Q17: v = Q18: R = Q19: Hint: Fr m R 1 = 3 jl k (10 r) = r = ph 0 53 (10 r) ph 0 53 (R 1) ph 0 53 (R 1) 10 Answer: r = 10 ph 0 53 (R 1)

First published 2014 by Heriot-Watt University. This edition published in 2017 by Heriot-Watt University SCHOLAR. Copyright 2017 SCHOLAR Forum.

First published 2014 by Heriot-Watt University. This edition published in 2017 by Heriot-Watt University SCHOLAR. Copyright 2017 SCHOLAR Forum. SCHOLAR Study Guide National 5 Mathematics Assessment Practice 3: Expanding brackets, factorising, completing the square, changing the subject of a formula and algebraic fractions (Topics 8-12) Authored