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Chapter 8 Formulae This chapter will show you how to ✔ write a formula from a problem ✔ substitute numbers into expressions and formulae ✔ change the subject of a formula 8.1 Formulae Formulae is the plural of formula. You will need to know l the correct order of operations Multiplication Indices Brackets Division " " Addition " Subtraction A formula is a general rule that shows how quantities (or variables) are related to each other. For example, v 5 u 1 at This is a formula that shows the relationship between an object’s final velocity, v, its initial velocity, u, its acceleration, a, and the time it has been moving, t. Deriving formulae When solving a problem, it often helps to write a formula to express the problem. Start by deciding on a letter to represent an unknown value. EXAMPLE 1 Alex buys x melons. Each melon costs 45 cents. Alex pays with a $5 note. Write a formula for the change, C, in cents, Alex should receive. x 5 number of melons C 5 500 2 45x $5 5 500c The melons cost 45c each so the cost, in cents, for x melons is 45x. Algebra 117 M08_CME_SB_IGCSE_6867_U08.indd 117 28/8/09 11:36:53 EXERCISE 8A 1 Nilesh buys y mangoes. Each mango costs 48 cents. Nilesh pays with a $5 note. Write a formula for the change, C, in cents, Nilesh should receive. 2 Apples cost r cents each and bananas cost s cents each. Sam buys 7 apples and 5 bananas. Write a formula for the total cost, t, in cents, of these fruit. 3 To cook a chicken you allow 45 minutes per kg and then a further 20 minutes. Write a formula for the time, t, in minutes, to cook a chicken that weighs w kg. 4 To cook lamb you allow 30 minutes plus a further 65 minutes per kg. Write a formula for the time, t, in minutes, to cook a joint of lamb that weighs w kg. 5 A rectangle has a length of 3x 1 1 and a width of x 1 2. Write down a formula for the perimeter, p, of this rectangle. 3x + 1 x+2 Substitution This section shows you how to use substitution to find the values of different algebraic expressions. Use mathematical operations in the correct order when substituting values into an algebraic expression. EXAMPLE 2 If a 5 5, b 5 4 and c 5 3 work out the value of these expressions. a 1 3 (b) 3b2 2 1 (c) 5c 1 1 (a) 2 2 The dividing line acts like a bracket. You must work out the numerator first. a 1 3 5 5 1 3 2 2 5 8 4 2 5 4 (a) 8 Remember 2 5 8 4 2. (b) 3b 2 1 5 3 × 4 2 1 5 3 × 16 2 1 5 48 2 1 5 47 2 2 continued . You must work out the indices first (42 5 16), then the multiplication (3 × 16), then the subtraction (48 2 1). 118 Algebra M08_CME_SB_IGCSE_6867_U08.indd 118 28/8/09 11:36:56 Formulae (c) 5c 1 1 5 5 × 3 1 1 b 4 5 15 1 1 4 16 5 4 5 4 EXERCISE 8B 1 If r 5 5, s 5 4 and t 5 3, work out the value of these expressions. r13 s15 t17 (a) (b) (c) 2 3 2 (e) 4t2 2 6 (f) 2s2 1 r (d) 3r2 1 1 (g) 4(5s 1 1) (h)t(r 1 s) (i) 5(2s 2 3t) 5t 1 1 4r 2 2 3s 1 t (j) (k) (l) s t r 3r2 5 3 × r2 5 3 × r × r 2 Copy and complete this table. x 1 2 x2 1 2x 3 4 5 3.7 3.8 32 1 2 × 3 5 9 1 6 5 15 15 3 Copy and complete this table. x 3 x 2 x 3 4 3.5 Remember x3 5 x × x × x 43 2 4 5 64 2 4 5 60 60 4 If A 5 6, B 5 24, C 5 3 and D 5 30, work out the value of these expressions. (a) D(B 1 7) (b) A(B 1 1) (c) A2 1 2B 1 C 2A 1 3 4B 1 D A2 1 3B (d) (e) (f) C 2 C M08_CME_SB_IGCSE_6867_U08.indd 119 Algebra 119 28/8/09 11:36:57 Substituting into formulae EXAMPLE 3 A formula for working out acceleration is v2u a 5 t where v is the final velocity, u is the initial velocity and t is the time taken. Work out the value of a when v 5 50, u 5 10, t 5 8. v2u t 50 2 10 5 8 5 40 4 8 5 5 a 5 Remember the line for division acts like a bracket. You must work out the numerator first. EXAMPLE 4 A formula for working out distance travelled is s 5 ut 1 21 at2 where u is the initial velocity, a is the acceleration and t is the time taken. Work out the value of s when u 5 3, a 5 8, t 5 5. s 5 ut 1 21 at2 5 3 × 5 1 21 × 8 × 52 5 15 1 4 × 25 5 15 1 100 Do 52 5 25 first; then 3 × 5 5 15 and 12 × 8 5 4; then 4 × 25 5 100; then 15 1 100 5 115. 5 115 EXERCISE 8C v2u to work out the value of a when t (a) v 5 15, u 5 3, t 5 2 (b) v 5 29, u 5 5, t 5 6 (c) v 5 25, u 5 7, t 5 3 (d) v 5 60, u 5 10, t 5 4. 1 Use the formula a 5 120 Algebra M08_CME_SB_IGCSE_6867_U08.indd 120 28/8/09 11:36:58 Formulae 2 A person’s Body Mass Index, b, is calculated using the formula m b 5 2 h where m is their mass in kilograms and h is their height in metres. Work out the value of b when (a) m 5 70, h 5 1.8 (b) m 5 38, h 5 1.4 (c) m 5 85, h 5 1.9 (d) m 5 59, h 5 1.7. a 3 The formula for the area of a trapezium is A 5 12 (a 1 b)h Work out the value of A when (a) a 5 10, b 5 6, h 5 4 (b) a 5 13, b 5 9, h 5 8 (c) a 5 9, b 5 6, h 5 4 (d) a 5 15, b 5 10, h 5 6. h b 4 Use the formula s 5 ut 1 12 at2 to work out the value of s when (a) u 5 3, a 5 10, t 5 2 (b) u 5 7, a 5 6, t 5 5 (c) u 5 2.5, a 5 5, t 5 4 (d) u 5 24, a 5 8, t 5 3. 5 A formula for working out the velocity of a car is v 5 √u2 1 2as where u is the initial velocity, a is the acceleration and s is the distance travelled. Work out the value of v when (a) u 5 3, a 5 4, s 5 5 (b) u 5 6, a 5 8, s 5 4 (c) u 5 9, a 5 10, s 5 2 (d) u 5 7, a 5 4, s 5 15. 6 A formula for the surface area of a cone, including the base, is where r is the radius and l is the slant height. Work out the surface area of a cone with these dimensions. (a) r 5 2 cm, l 5 13 cm (b) r 5 4 cm, l 5 10 cm. Give your answers to 3 s.f. surface area of cone 5 pr(r 1 l) l r 7 A formula for the total surface area of a cylinder is where r is the radius and h is the height. Work out the surface area of a cylinder with these dimensions. (a) r 5 3.7 cm, h 5 10.8 cm (b) r 5 4.2 cm, h 5 4.1 cm. Give your answers to 3 s.f. surface area of cylinder 5 2pr(r 1 h) r h M08_CME_SB_IGCSE_6867_U08.indd 121 Algebra 121 28/8/09 11:36:59 Using formulae EXAMPLE 5 The perimeter of a rectangle is given by P 5 2l 1 2w, where l is the length and w is the width. Work out the value of l when P 5 24 and w 5 5. P 5 2l 1 2w 24 5 2l 1 2 × 5 24 5 2l 1 10 Substitute the values you know, of P and w, into the formula, then solve the equation to find l. 24 2 10 5 2l 14 5 2l 7 5 l EXERCISE 8D 1 The formula for the area of a rectangle is A 5 lw, where l is the length and w is the width. Work out the value of w when (a) A 5 12 and l 5 4 (b) A 5 36 and l 5 9 (c) A 5 42 and l 5 7 (d) A 5 60 and l 5 15. l w 2 The formula for the voltage, V, in an electrical circuit is V 5 IR, where I is the current and R is the resistance. Work these out. (a) The value of R when V 5 18 and I 5 2. (b) The value of R when V 5 35 and I 5 5. (c) The value of I when V 5 240 and R 5 30. (d) The value of I when V 5 240 and R 5 40. 3 The perimeter of a rectangle is given by l P 5 2l 1 2w where l is the length and w is the width. Use the formula to (a) find l when P 5 18 and w 5 4. (b) find l when P 5 32 and w 5 7. (c) find w when P 5 60 and l 5 17. (d) find w when P 5 50 and l 5 13.5. w 122 Algebra M08_CME_SB_IGCSE_6867_U08.indd 122 28/8/09 11:37:00 Formulae 4 Use the formula v 5 u 1 at (a) to find u when v 5 30, a 5 8, t 5 3 (b) to find a when v 5 54, u 5 19, t 5 7 (c) to find t when v 5 60, u 5 15, a 5 5 (d) to find u when v 5 20, a 5 7, t 5 4. 8.2 Changing the subject of a formula You will need to know that addition and subtraction are inverse operations multiplication and division are inverse operations squaring and finding the square root are inverse operations l l l The subject of a formula appears only once, on its own, on one side of the formula. In the formula v 5 u 1 at the variable v is called the subject of the formula. P is the subject of the formula P 5 2l 1 2w. P is on its own on one side of the formula. You can rearrange a formula to make a different variable the subject. EXAMPLE 6 Rearrange d 5 a 1 8 to make a the subject. You need to have a on its own on one side. Subtract 8 from both sides as you would when solving an equation. d 5 a 1 8 d 2 8 5 a EXAMPLE 7 Make x the subject of the formula y 5 5x 2 2. Add 2 to both sides to leave 5x on its own. y 5 5x 2 2 y 1 2 5 5x y 1 2 5 x 5 Then divide both sides by 5 to leave x on its own. M08_CME_SB_IGCSE_6867_U08.indd 123 Algebra 123 28/8/09 11:37:01 EXAMPLE 8 Rearrange P 5 4g 1 2h to make g the subject. P 5 4g 1 2h P 2 2h 5 4g P 2 2h 5 g 4 Subtract 2h from both sides. Divide both sides by 4. EXAMPLE 9 Rearrange V 5 √(w 1 y) to make w the subject. V 5 √(w 1 y) Square both sides first. V 2 5 w 1 y V 2 2 y 5 w Then subtract y from both sides. EXAMPLE 10 q2 Rearrange p 5 2 s to make q the subject. r q2 p 5 2 s r q2 p 1 s 5 r r (p 1 s) 5 q2 6 √r(p 1 s) 5 q Add s to both sides to leave the term with q on its own. Multiply both sides by r. Square root both sides. EXERCISE 8e 1Rearrange these formulae to make y the subject. (a) x 5 5y 2 6 (b) x 1 2y 5 8 (c) x 1 3y 2 8 5 0 (d) 2x 1 5y 2 7 5 0 (e) x 2 y 5 4 (f) 2x 2 5y 5 20 2Make x the subject of these formulae. (a) y 5 5x 2 6 (b) y 5 12 x 2 8 (c) y 1 12 x 5 2 (d) y 2 x 5 7 (e) 2y 1 x 5 9 (f) 2y 1 5x 5 9 124 Algebra M08_CME_SB_IGCSE_6867_U08.indd 124 28/8/09 11:37:03 Formulae 3Make r the subject of these formulae. (a) p 5 3(4r 1 5t) (b) v 5 5(7r + h) 3r 2 2s 6p 2 5r (c) w 5 (d) y 5 5 8 4Rearrange these formulae to make a the subject. (b) b 5 12 a 1 7 (a) b 5 12 a 1 6 (c) b 5 13 a 2 1 (d) b 5 14 a 2 3 (e) b 5 2(a 1 1) (f) b 5 3(a 2 5) Multiply 12 a by 2 to get a. 5Rearrange these formulae to make x the subject. (a) 3(x 1 y) 5 5y (b) 2(x 2 y) 5 y 1 5 x 2x (c) z 5 2 5 (d) 3p 5 2 s y q 6Rearrange these formulae to make w the subject. (b) A 5 √w 2 a (a) K 5 √w 1 t (c) h 5 2√w 1 l (d) T 5 √wr 1 5 Use Example 9 to help. 7Rearrange these formulae to make r the subject. r2 r2 (a) t 5 2 m (b) h 5 1 3a g 4 (d) A 5 p(r2 2 s2) (c) V 5 13 pr2h Use Example 10 to help. 8A formula for the total surface area of a cylinder is A 5 2pr(r 1 h) where r is the radius and h is the height. Rearrange the formula to make h the subject. 9A formula for the period of a pendulum is l T 5 2p g Rearrange the formula to make l the subject. The period T is the time for one complete swing; l is the length and g is a constant. 10The formula F 5 1.8C 1 32 can be used to convert degrees Celsius, °C, to degrees Fahrenheit, °F. (a) Convert 15°C to degrees Fahrenheit. (b) Rearrange the formula to make C the subject. 11A formula is given as v2 5 u2 + 2as. (a) Find v when u 5 10, a 5 4 and s 5 12. (b) Rearrange the formula to make a the subject. (c) Rearrange the formula to make u the subject. 12A formula is given by h2 1 g2 T 5 g2 (a) Find T when g = 10 and h = 2. (b) Rearrange the formula to make h the subject. M08_CME_SB_IGCSE_6867_U08.indd 125 Algebra 125 28/8/09 11:37:04 EXAMINATION QUESTIONS 1 Make h the subject of the formula g 5 √h 1 i. [2] (CIE Paper 2, Nov 2000) 2 Make y the subject of the formula x 5 4 1 √y . 3 [3] 3x 1 5. 2 [3] (CIE Paper 2, Jun 2001) 3 Make x the subject of the formula y 5 (CIE Paper 2, Nov 2001) 4 5 Make V the subject of the formula T = V 1 1 . [3] (CIE Paper 2, Jun 2002) 5 The surface area of a person’s body, A square metres, is given by the formula hm A 5 3600 where h is the height in centimetres and m is the mass in kilograms. (a) Dolores is 167 cm high and has a mass of 70 kg. Calculate the surface area of her body. (b) Erik has a mass of 80 kg. Find his height if A 5 1.99. (c) Make h the subject of the formula. [1] [2] [3] (CIE Paper 4, Nov 2003) 6 Make c the subject of the formula √3c 2 5. [3] (CIE Paper 2, Nov 2004) 126 Algebra M08_CME_SB_IGCSE_6867_U08.indd 126 28/8/09 11:37:05