Worksheets for GCSE Mathematics. Ratio & Proportion. Mr Black's Maths Resources for Teachers GCSE 1-9. Number

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1 Worksheets for GCSE Mathematics Ratio & Proportion Mr Black's Maths Resources for Teachers GCSE 1-9 Number

2 Ratio & Proportion Worksheets Contents Differentiated Independent Learning Worksheets Simplifying Ratios Proportional Reasoning Sharing an Amount to a Ratio Modelling Direct Proportions Modelling Inverse Variations Modelling Non-Linear Direct Variations Page 3 Page 4 Page 5 Page 6 Page 7 Page 8 Solutions Simplifying Ratios Proportional Reasoning Sharing an Amount to a Ratio Modelling Direct Proportions Modelling Inverse Variations Modelling Non-Linear Direct Variations Page 9 Page 10 Page 11 Page 12 Page 13 Page 14

3 Simplifying Ratios Q1. Express the following ratios in their simplest form. 6 : 18 b) 20 : 15 c) 32 : 12 d) 75 : 120 e) 27 : 18 f) 32 : 48 g) 18 : 108 : 27 h) 270 : 150 : 120 i) 28 : 42 : 63 j) 54 : 62 : 36 k) 1 : 5 : 12 l) 108 : 72 : 45 Simplify the following ratios by considering their units. 20 cm : 1 m b) 70p : 1 c) 600 g : 1 kg d) 14 : 210p e) 40 cm : 0.8 m f) 90p : 1.10 g) 20 mins : 1 hour h) 340p : 2.50 i) 450 m : 1.5 km j) 45 mins : 2.5 hours k) 2 days : 2 weeks l) 4 months : 1 year Q3. Match the ratios to their simplified counterparts. 25 : 40 : 15 5 : 26 : 2 70 mins : 2.5 hours : 1 hour 2 : 9 : : 270 : : 15 : 6 20 weeks : 2 years : 2 months 5 : 8 : 3 Simon is 20 years old and Jane is 15 years old. What is the ratio of Simon s age to Jane s age? b) Elliot has 9 pens and 3 pencils on his desk. What is the ratio of pens to pencils? c) In a class of 24, there are 15 girls. What is the ratio of boys to girls? d) Driving 150 miles from Manchester to Oxford Clare covers 90 miles on the motorway. What is the ratio of the number of miles driven on motorways to the number of miles on other roads? e) While doing his weekly shop Lee spends on on food and 14 on general supplies. What is the ratio of the amount he spent on food to the amount he spent on general supplies? Q5. Calculate the simplified ratio of the shaded regions to the whole shape. 3

4 Proportional Reasoning Q1. Use equivalent ratios to calculate each unknown 3 5 = 6 n b) 2 7 = 8 n c) 4 3 = n 12 d) = 5 30 x e) = 12 f y f) = 10 j k g) 5 12 = y 6 h) 9 2 = t 1 i) = u 0.25 p The ratio of male to female members at a gym is 5 : 3. There are 450 males at the gym. Calculate the number of female members. b) The ages of a father and son are in the ratio 9 : 2. The father is 45 years old. How old is the son? c) Green paint is mixed from blue and yellow paint in the ratio 2 : 5. Ben has lots of blue paint but only 55 litres of yellow paint. How much green paint can he make? d) The distances travelled by Donna and Kate to a party were in the ratio 5 : 2. Kate travelled 8.4 miles. How far did Donna travel? e) The cost of 5 oranges is Calculate the cost of 9 oranges. f) Five books have a total weight of 750 grams. How much would 12 books weigh? g) 42 litres of fuel can cover 230 miles. How far can I travel on 63 litres? h) At a steady rate, a woman drives 72 miles in 2 hours. How far does she drive in 5 hours? i) A clock gains 2 minutes in 6 days. How many days does it take to gain 7 minutes? Q3. To make 12 cakes Sally uses 5 ounces of flour. How many cakes can Sally make with 8 ounces of flour? $8 can be exchanged for 6 Euros. How many dollars can be exchanged for 25 Euros b) How many Euros can be exchanged for $52 Q5. 14 can be exchanged for $18. How many dollars can be exchanged for 20 b) How many pounds can be exchanged for $50 Q6. Parsnips cost 7.50 for an 8.5 kg bag at a farm shop. The same parsnips cost 1.54 for a 2.5 kg bag at a supermarket. Where are the parsnips the better value, at the farm shop or at the supermarket? Q7. Sachets of coffee are sold in packets of 16 for or packets of 5 for Which packet of coffee is better value? 4

5 Sharing to a Ratio Q1. Divide the following amounts according to the given ratios. 50 in the ratio 2 : 3 b) 36 m in the ratio 2 : 7 c) 120 kg in the ratio 3 : 2 d) $60 in the ratio 7 : 8 e) 200 g in the ratio 1 : 9 f) 400g in the ratio 8 :12 g) 150 cm in the ratio 2 : 3 : 10 h) 8 km in the ratio 1 : 2 : 1 i) 200 cm in the ratio 15 : 9 : 1 j) 5 in the ratio 8 : 7 : 10 k) 140 litres in the ratio 2 : 4 : 1 l) 240 in the ratio 5 : 3 : 12 David has 64 tokens. He shares them between his 3 friends in the ratio 4 : 3 : 1 How many tokens are in the largest share? b) Mrs Rutterford inherits She divides the between her three children Clare, Katie and Sarah in the ratio 7 : 8 : 9, respectively. How much does Laura receive? c) The sizes of the interior angles of a quadrilateral are in the ratio 3 : 4 : 6 : 7 Calculate the size of the largest angle. d) Brian and Pauline win 2500 on the lottery. They share the money in the ratio 1 : 4. How much more money does Pauline receive than Brian? e) Jamie, Craig and Vicki share a lottery win in the ratio 11: 8: 6 How much does each of them receive? f) The town of Ashton has houses. The ratio of detached houses to other types of houses is 2 : 3 How many detached houses are there? Q3. Rachel, Georgina and Ellie share some money. Rachel gets 3 of the money. 10 Rachel and Georgina share the rest of the money in the ratio 3 : 2 What is Georgina s share of the money? b) David and Carol share some money in the ratio 2 : 5 David gets 45 more than Carol. How much money did Carol get? c) There are 6 more boys than girls in Mr Unett s mathematics class of 32 pupils. What is the ratio of girls to boys in his class? 5

6 Direct Proportion Q1. Calculate the values of the unknown in this table. State the equation connecting the two variables. x b) a 4 16 y b c) w n e) v 7 20 c g) m n d) i t f) e d h) r f 6 60 h n. When h = 18, n = 4. i) Find the value of h when n = 20 ii) Find the value of n when h = 63. b) G b. When b = 5, G = 8. i) Find the value of G when b = 15 ii) Find the value of b when h = 20. c) f and j are in direct proportion. When f = 24, j = 30. i) State the equation connecting f and j. ii) Find the value of f when j = 40. d) t varies directly with x. When x = 5, t = 32. i) State the equation connecting x and t. ii) Find the value of t when x = 17.5 e) e is directly proportional to s. When e = 10 when s = 2.5 i) State the formula connecting e and s. ii) Find the value of s when e = 6. Q3. b) c) b) c) Q5. b) c) A rock is dropped from a cliff. Its speed, v, varies directly with the time, t seconds, after release. Its speed after 0.5 seconds is 4.8 m/s. State the formula connecting the time and speed. What is the speed after 1.5 seconds? How long will it take to reach 6 m/s. An elastic band s extension is in direct proportion to the mass attached to it. With a 60 g weight attached the elastic band is stretched 3.5 cm. State the formula connecting the band s extension to the mass attached. Calculate the extension when a mass of 100 g is attached. Calculate the mass when the elastic band is stretch to 4.2 cm. The width of a tree trunk varies directly with its age. A 10 years old tree has a trunk diameter of 1.5 m. State the formula connecting the width of a tree s diameter with its age. What is the width of a tree that is 9 months old? What is the age of a tree with trunk width of 1.6 m. 6

7 Q1. b) c) d) e) Inverse Proportion Use the proportion symbol ( ) to describe the variation in each of these scenarios. The time, t, of a train journey and the speed, s, of the train. The number of pages, p, in a book and the number of words, w. The number of people, p, working on a project and the time, t, the project takes. The distance, d, of a flight and the time, t, taken to arrive. The empty space, s, in a cup and the length of time, t, it has been filling. Each of the following sets of variables are in inverse proportion. State the equation connecting the two variables and calculate the unknowns. x 2 4 b) x 4 8 y y c) x 5 20 y e) x y 20 4 d) x 2 8 y f) x y Q3. a 1. When a = 2 b = 4. b i) Calculate the value of a when b = 16 ii) Calculate the value of b when a = 5 b) m 1. When m = 6 b = 1.5. n i) Calculate the value of m when n = 4 ii) Calculate the value of n when m = 20 c) r varies inversely with d. When r = 2 d = 5. i) Calculate the value of r when d = 8 ii) Calculate the value of d when r = 12.5 d) V is indirectly proportion to n. When v = 4.2 n = 3. i) Calculate the value of v when n = 3.5 ii) Calculate the value of n when v = 0.8 Q is inversely proportional to W. When Q = 100, W = 32 State the formula connecting Q and W. b) Sketch a graph of the relationship between Q and W on this diagram. c) Calculate the value of W when Q is twice as big as W. Q5. The number of days, N, to complete a project is inversely proportional to the number of people, P, who work on the project. The project takes 9 days to complete if 75 people work on it. State the equation connecting N and P. b) How many people are needed to complete the project in 7 days? c) Sketch a graph of the relationship between N and P on this diagram. 7

8 Non-Linear Direct Proportion Q1. a is directly proportional to the square of b. When b = 5, a = 4. Sketch the graph to show the relationship between a and b. b) Find the value of a when a = 8. The surface area, A, of a solid is directly proportional to the square of the depth, d. When d = 4 cm A = 48 cm 2. Sketch the graph to show the relationship between A and d. b) Calculate the surface area for a solid with depth 9 cm. c) A solid has a surface area of cm 2. Calculate its depth. Q3 Complete the following grids by modelling the variations using formulae. y x 2 b) b a 2 State the formula connecting x and y. State the formula connecting a and b. c) h f d) t u 3 State the formula connecting h and f. State the formula connecting t and u. In an experiment measures g and u were taken and recorded as shown. Which of these rules fits the results? (i) u g (ii) u g 2 (iii) u g 3 Q5. b) A compound is sold in different sized blocks. The weight of each block, W kilograms, is directly proportional to the cube of its height, h metres. A block of weight 56 kg has height 2 m. Find an equation connecting h and W. Another block has a weight of 1512 kg. Find its height. 8

9 Simplifying Ratios Solutions Q1. Express the following ratios in their simplest form. 1 : 3 b) 4 : 3 c) 8 : 3 d) 5 : 8 e) 3 : 2 f) 2 : 3 g) 2 : 12 : 3 h) 9 : 5 : 4 i) 4 : 6 : 9 j) 27 : 31 : 18 k) 1 : 5 : 12 l) 12 : 8 : 5 Simplify the following ratios by considering their units. 1 : 5 b) 7 : 10 c) 3 : 5 d) 20 : 3 e) 1 : 20 f) 9 : 11 g) 1 : 3 h) 34 : 25 i) 3 : 10 j) 3 : 10 k) 1 : 7 l) 1 : 3 Q3. Match the ratios to their simplified counterparts. 25 : 40 : 15 5 : 26 : 2 70 mins : 2.5 hours : 1 hour 2 : 9 : : 270 : : 15 : 6 20 weeks : 2 years : 2 months 5 : 8 : 3 4 : 3 b) 3 : 1 c) 5 : 3 d) 5 : 3 e) 45 : 28 Q5. 2 : 1 b) 20 : 9 c) 3 :1 9

10 Proportional Reasoning Solutions Q1. n = 10 b) n = 28 c) n = 16 d) x = 20 e) f = 30, y = 54 f) j = 30, k = 25 g) y = 2.5 h) t = 4.5 i) u = 0.75, p = b) 10 years old c) 77 litres d) 21 miles e) 2.52 f) 1.8 kg g) 345 miles h) 180 miles i) 21 minutes Q3. 19 cakes $33.33 b) 39 Euros Q5. $15.56 b) Q6. Farm Shop = 1.13 / kg Supermarket = 1.62 / kg Q7. Packet of 16 = / sachet Packet of 5 = / sachet 10

11 Sharing to a Ratio Solutions Q1. 20 : 30 b) 8 m :28 m c) 72 kg : 48 kg d) $ $28 : $32 e) 20 g : 180 g f) 160 g :240 g g) 20 cm : 30 cm : 100 cm h) 2 km : 4 km : 2 km i) 120 cm : 72 cm : 8 cm j) 1.60 : 1.40 : 2.00 k) 40 l : 80 l : 20 l l) 60 : 36 : b) 8000 c) 126 d) 1500 e) 6600 : 4800 : 3600 f) Q b) 30 c) 13 : 19 11

12 Solutions Direct Proportion Q1. x y y 7x c) w n n 6w e) v c c 1.5v g) m n y 7x b) a b b 9a d) i t t 2.5i f) e d e 0.2e h) r f r 1.2f i) h = 90, ii) n = 14 b) i) G = 24, ii) b = 12.5 c) i) F = 4j, ii) F = 32 5 d) i) t = 6.4x, ii) t = 112 e) i) e = 4s, s = 1.5 Q3. V = 9.6t b) V = 14.4 m/s c) t = seconds e = 7m, 20 b) e = 5.83 cm c) Mass = 72 g Q5. w = 3a 20 b) width = 1.35 m c) years old. 12

13 Inverse Proportion Solutions Q1. t 1 s b) w p c) t 1 p d) t 1 d e) s 1 t x y b) x y c) x y d) x y e) x y f) x y Q3. i) a = 0.5 ii) b = 1.6 b) i) m = 2.25 ii) n = 0.45 c) i) r = 1.25 ii) d = 0.8 d) i) v = 3.6 ii) n = b) Q = 3200 W c) W = 40 Q5. N = 675 P b) 97 people c) 13

14 Non-Linear Direct Proportion Solutions Q1. b) a = b) A = 243 cm 2. c) d = 4.5 cm Q3 b) y = 2.5x 2 b = 2.5a 2 c) d) h = 7.4 f t = 1.4u 3 Q5. b) u g 2 w = 7w 3 6 m 14