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Algebra WORKING OUT FORMULAE A formula is a quick way of writing down a general method for solving a problem, which can be used whatever the numbers are. For example, to find the area of any rectangle we multiply the length by the width. The formula is A = L x W or just A = LW where A = Area L = Length W = Width To find the speed a car drives at we take the distance travelled and divide it by the time taken. S = D T where S = Speed D = Distance and T = Time We can then use the formula for a particular case. For example: to find the area of a room which is 5m long and 3m wide, we substitute the numbers for the letters in the formula. A = L x W L= 5 W= 3 Area = 5 x 3 m = 15m² To find the speed of a car which travels 200 miles in 4 hours, we replace the letters in the formula by the numbers. S = D T D = 200 T = 4 S = 200 mph 4 f lex i p = 50mph ac k s 1 Algebra A) Try working out the solutions to these questions 1) A = LW Find A when L = 20 and W = 9 2) V = lwh Find V when l = 5 w = 3 and h = 4 3) l= Find l when z = 150 d = 25 4) C= L+8 5) Rob’s total wage is found with the formula t = w + 30. What is his total wage (t) when w = 175? 6) Jean’s take home pay is found with the formula w = g – d. Find w when g = 300 and d = 140 7) The distance travelled by a bike can be found with the formula d = st. Find d when s = 12 and t = 3 8) A bakery makes bread using the formula: T = b + 50 to find the total needed. How many loaves are baked when b = 95? 9) The number of bricks in one course of a wall is found with z-d the formula Find C when L = 125 L . 20 How many are used when L = 800? 10) The perimeter of a garden is P = 2L + 2W. What is P when L = 12 and W = 8 11) PRT Interest on money is found by the formula I = 100 How much interest is due when P = 500 R = 2 and T = 2? When formulae are more complicated it helps to remember the order of operations: Brackets come first Of Divide (BODMAS) Multiply Add f lex i p Subtract ac k s 2 Algebra For example the formula: 2h + ab means 2xh + axb We know from the BODMAS rule that multiplying has to be done before adding. So if h = 3, a = 4 and b = 10 2h + ab = 2 x 3 + 4 x 10 = 6 + 40 = 46 If a number is multiplying a bracket, the multiplication sign is usually left out. eg. c (3 + b) Means c x (3 + b) We know that the bracket has to be done first. So if c = 6 and b = 5 c (3 + b) = 6 x (3 + 5) = 6 x 8 = 48 B) Try solving these: f lex i p 1) a=2 b=4 c=5 y = 10 a) ab + c c) bc + ay e) 3( a + b) g) 2ay + 3c b) 2y – a d) ab + cy f) b(2a + c) h) 3ac + 2bc 2) A = lb + 3c Find A when l = 4 b = 7 and c = 5 3) The cost of hiring a motorbike is given in the formula C = 12 + 6d where d is the number of days. Find C when d = 5. ac k s 3 Algebra 4) The formula for the time it takes to cook a recipe is T = 15w + 20 Find T when w = 3. 5) V = lbh + 4lh Find V when l = 5 b = 3 and h = 2. 6) C = y (2 + b) Find C if y = 3 b = 12. 7) T = r (a + 2c) Find T when r = 5 a = 2 and c = 3 When an expression has to be divided in a formula, any complex parts can be worked out first. For example: R= + 2a All of these have to be divided by 3, so 3 it’s simpler to work out + 2a first. So if = 2 and a = 5 R= 2 + 10 3 R= 12 3 R= 4 C) Try working out these formulae: 1) 2) P = y + z y 3) C = t(2 + s) when t = 3 s = 6 = 4 4) Try finding the area of these trapezium shapes using the formula = b + 2a f lex i p A = ac k s when b = 2 and a = 7 when = 3 y = 2 and z = 10 h x (a + b) 2 where h is the height and a and b are the parallel sides. a) h = 6 a = 7 b = 2 b) h = 10 a = 5 b = 3 c) h = 4 a = 6 b = 5 4 Algebra 5) a) Try to work out a formula to find the area of this apartment a b t s b) What is the actual area if a = 4 b = 3 s = 6 t = 2 ? 6) a) You buy cinema tickets for yourself and 2 children. Your ticket will cost twice as much as a child's ticket. You also pay 3 pounds in bus fares. If the child's ticket cost t pounds, what is the total cost of the outing? b) The child's ticket cost £1.50 at this cinema. Substitute this into your formula and find the actual cost. f lex i p Check your answers then discuss with your tutor what you need to work on next. ac k s 5 Algebra Answers Working Out Formulae A) 1) A = 180 2) V = 60 3) l= 4) c = 133 5) t = 205 6) w = 160 7) d = 36 8) T = 145 9) 40 bricks 125 10) P = 40 11) I = 20 f lex i p B) 1) a) b) c) d) e) f) g) h) 2) 43 3) 42 4) 65 5) 70 6) 42 7) 40 ac k s 13 18 40 58 18 36 55 70 C) 6 1) = 16 2) P = 8 3) C = 6 4) a) A = 27 b) A = 40 c) A = 22 5) a) ab + st b) 24 6) a) 4t + 3 b) £9