MEP Y8 Practice Book A 7 Ratio and Proportion 7.1 Equivalent Ratios Orange squash is to be mixed with water in a ratio of 1 : 6; this means that for every unit of orange squash, 6 units of water will be used. The table gives some examples: Amount of Orange Squash Amount of Water (cm ) (cm ) 3 3 1 6 20 120 5 30 The ratios 1 : 6 and 20 : 120 and 5 : 30 are all equivalent ratios, but 1 : 6 is the simplest form. Ratios can be simplified by dividing both sides by the same number: note the similarity to fractions. An alternative method for some purposes, is to reduce to the form 1 : n or n : 1 by dividing both numbers by either the left-hand-side (LHS) or the right-hand-side (RHS). For example: 4 10 the ratio 4 :10 may be simplified to β 1 : 2.5 : 4 4 the ratio 8 : 5 may be simplified to 8 5 : 5 5 β 1.6 : 1 Example 1 Write each of these ratios in its simplest form: (a) 7 : 14 (b) 15 : 25 Solution (a) Divide both sides by 7, giving 7 14 : 7 : 14 = 7 7 = 1:2 (b) Divide both sides by 5, giving 15 25 15 : 25 = : 5 5 = 3 :5 (c) Divide both sides by 2, giving 10 4 10 : 4 = : 2 2 = 5 :2 114 (c) 10 : 4 MEP Y8 Practice Book A Example 2 Write these ratios in the form 1 : n. (a) 3 : 12 (b) 5:6 (c) 10 : 42 Solution (a) Divide both sides by 3, giving 3 : 12 = 1 : 4 (b) Divide both sides by 5, giving 6 5 = 1 : 1.2 5:6 = 1: (c) Divide both sides by 10, giving 42 10 = 1 : 4.2 10 : 42 = 1 : Example 3 The scale on a map is 1 : 20 000. What actual distance does a length of 8 cm on the map represent? Solution Actual distance = 8 × 20 000 = 160 000 cm = 1600 m = 1.6 km Exercises 1. 2. Write each of these ratios in its simplest form: (a) 2:6 (b) 4 : 20 (c) 3 : 15 (d) 6:2 (e) 24 : 4 (f) 30 : 25 (g) 14 : 21 (h) 15 : 60 (i) 20 : 100 (j) 80 : 100 (k) 18 : 24 (l) 22 : 77 Write in the form 1 : n, each of the following ratios: (a) 2:5 (b) 5:3 (c) 10 : 35 (d) 2 : 17 (e) 4 : 10 (f) 8 : 20 (g) 6:9 (h) 15 : 12 (i) 5 : 12 115 MEP Y8 Practice Book A 7.1 3. Write in the form n : 1, each of the following ratios: (a) 24 : 3 (b) 4:5 (c) 7 : 10 (d) 15 : 2 (e) 18 : 5 (f) 6:5 4. Jennifer mixes 600 ml of orange juice with 900 ml of apple juice to make a fruit drink. Write the ratio of orange juice to apple juice in its simplest form. 5. A builder mixes 10 shovels of cement with 25 shovels of sand. Write the ratio of cement to sand: 6. (a) in its simplest form, (b) in the form 1 : n, (c) in the form n : 1, In a cake recipe, 300 grams of butter are mixed with 800 grams of flour. Write the ratio of butter to flour: (a) in its simplest form, (b) in the form 1 : n, (c) in the form n : 1. 7. In a school there are 850 pupils and 40 teachers. Write the ratio of teachers to pupils: (a) in its simplest form, (b) in the form 1 : n. 8. A map is drawn with a scale of 1 : 50 000. Calculate the actual distances, in km, that the following lengths on the map represent: (a) 2 cm (b) 9 cm (c) 30 cm. 9. A map has a scale of 1 : 200 000. The distance between two towns is 60 km. How far apart are the towns on the map? 10. On a map, a distance of 5 cm represents an actual distance of 15 km. Write the scale of the map in the form 1 : n. 7.2 Direct Proportion Direct proportion can be used to carry out calculations like the one below: If 10 calculators cost £120, then 1 calculator costs £12, and 8 calculators cost £96. 116 MEP Y8 Practice Book A Example 1 If 6 copies of a book cost £9, calculate the cost of 8 books. Solution If 6 copies cost £9, then 1 copy costs £ 9 6 = £1.50 £1.50 × 8 and 8 copies cost = £12 Example 2 If 25 floppy discs cost £5.50, calculate the cost of 11 floppy discs. Solution If 25 discs cost £5.50 = 550p then 1 disc costs 550 25 so 11 discs cost = 22p 11 × 22 p = 242p = £2.42 Exercises 1. If 5 tickets for a play cost £40, calculate the cost of: (a) 6 tickets (b) 9 tickets (c) 20 tickets. 2. To make 3 glasses of orange squash you need 600 ml of water. How much water do you need to make: (a) 5 glasses of orange squash, (b) 7 glasses of orange squash? 3. If 10 litres of petrol cost £8.20, calculate the cost of: (a) 4. 4 litres (b) 12 litres (c) 30 litres. A baker uses 1800 grams of flour to make 3 loaves of bread. How much flour will he need to make: (a) 2 loaves (b) 7 loaves 117 (c) 24 loaves? 7.2 MEP Y8 Practice Book A 5. Ben buys 21 football stickers for 84p. Calculate the cost of: (a) 6. 7. 7 stickers (b) 12 stickers (c) 50 stickers. A 20 m length of rope costs £14.40. (a) Calculate the cost of 12 m of rope. (b) What is the cost of the rope, per metre? A window cleaner charges n pence to clean each window, and for a house with 9 windows he charges £4.95. (a) What is n ? (b) Calculate the window cleaner's charge for a house with 13 windows. 8. 16 teams, each with the same number of people, enter a quiz. At the semifinal stage there are 12 people left in the competition. How many people entered the quiz? 9. Three identical coaches can carry a total of 162 passengers. How many passengers in total can be carried on seven of these coaches? 10. The total mass of 200 concrete blocks is 1460 kg. Calculate the mass of 900 concrete blocks. 7.3 Proportional Division Sometimes we need to divide something in a given ratio. Malcolm and Alison share the profits from their business in the ratio 2 : 3. This means that, out of every £5 profit, Malcolm gets £2 and Alison gets £3. Example 1 Julie and Jack run a stall at a car boot sale and take a total of £90. They share the money in the ratio 4 : 5. How much money does each receive? Solution As the ratio is 4 : 5, first add these numbers together to see by how many parts the £90 is to be divided. 4 + 5 = 9 , so 9 parts are needed. Now divide the total by 9. 90 = 10 , so each part is £10. 9 118 MEP Y8 Practice Book A Julie gets 4 parts at £10, giving 4 × £10 = £40, Jack gets 5 parts at £10, giving 5 × £10 = £50. Example 2 Rachel, Ben and Emma are given £52. They decide to divide the money in the ratio of their ages, 10 : 9 : 7. How much does each receive? Solution 10 + 9 + 7 = 26 so 26 parts are needed. Now divide the total by 26. 52 = 2, so each part is £2. 26 Rachel gets 10 parts at £2, giving 10 × £2 = £20 gets 9 parts at £2, giving 9 × £2 = £18 Emma gets 7 parts at £2, giving 7 × £2 = £14 Ben Exercises 1. 2. (a) Divide £50 in the ratio 2 : 3. (b) Divide £100 in the ratio 1 : 4. (c) Divide £60 in the ratio 11 : 4. (d) Divide 80 kg in the ratio 1 : 3. (a) Divide £60 in the ratio 6 : 5 : 1. (b) Divide £108 in the ratio 3 : 4 : 5. (c) Divide 30 kg in the ratio 1 : 2 : 3. (d) Divide 75 litres in the ratio 12 : 8 : 5. 3. Heidi and Briony get £80 by selling their old toys at a car boot sale. They divide the money in the ratio 2 : 3. How much money do they each receive? 4. In a chemistry lab, acid and water are mixed in the ratio 1 : 5. A bottle contains 216 ml of the mixture. How much acid and how much water were needed to make this amount of the mixture? 5. Blue and yellow paints are mixed in the ratio 3 : 5 to produce green. How much of each of the two colours are needed to produce 40 ml of green paint? 119 MEP Y8 Practice Book A 7.3 6. Simon, Sarah and Matthew are given a total of £300. They share it in the ratio 10 : 11 : 9. How much does each receive? 7. In a fruit cocktail drink, pineapple juice, orange juice and apple juice are mixed in the ratio 7 : 5 : 4. How much of each type of juice is needed to make: (a) 80 ml of the cocktail, (b) 1 litre of the cocktail? 8. Blue, red and yellow paints are mixed to produce 200 ml of another colour. How much of each colour is needed if they are mixed in the ratio: (a) 1 : 1 : 2, (b) 3 : 3 : 2, (c) 9:4:3? 9. To start up a small business, it is necessary to spend £800. Paul, Margaret and Denise agree to contribute in the ratio 8 : 1 : 7. How much does each need to spend? 10. Hannah, Grace and Jordan share out 10 biscuits so that Hannah has 2, Grace has 6 and Jordan has the remainder. Later they share out 25 biscuits in the same ratio. How many does each have this time? 7.4 Linear Conversion The ideas used in this unit can be used for converting masses, lengths and currencies. Example 1 If £1 is worth 9 French francs, convert: (a) £22 to Ff, (b) 45 Ff to £, Solution (a) £22 = 22 × 9 = 198 Ff (b) 1 Ff = £ so 1 9 45 Ff = 45 × = 1 9 45 9 = £5 120 (c) 100 Ff to £. MEP Y8 Practice Book A (c) 100 Ff = 100 × = 1 9 100 9 = £11 1 9 = £11.11 to the nearest pence Example 2 Use the fact that 1 foot is approximately 30 cm to convert: (a) 8 feet to cm, (b) 50 cm to feet, (c) 195 cm to feet. Solution (a) 8 feet = 8 × 30 = 240 cm (b) 1 cm = 1 feet 30 so 50 cm = 50 × = 5 3 = 1 (c) 1 30 2 feet 3 195 cm = 195 × 1 30 195 30 13 = 2 1 = 6 feet 2 = Example 3 If £1 is worth $1.60, convert: (a) £15 to dollars (b) 121 $8 to pounds. MEP Y8 Practice Book A 7.4 Solution (a) £15 = 15 × 1.60 = $24 (b) $1 = £ = £ 1 1.60 10 16 $8 = 8 × 10 16 80 16 = £5 = Exercises 1. 2. 3. 4. 5. If £1 is worth 9 Ff, convert: (a) £6 to Fr, (b) £100 to Ff, (c) 54 Ff to £, (d) 28 Ff to £. Use the fact that 1 inch is approximately 25 mm to convert: (a) 6 inches to mm, (b) 80 inches to mm, (c) 50 mm to inches, (d) 1000 mm to inches. Before 1971, Britain used a system of money where there were 12 pennies in a shilling and 20 shillings in a pound. Use this information to convert: (a) 100 shillings into pounds, (b) 8 shillings into pennies, (c) 132 pennies into shillings, (d) 180 pennies into shillings. Given that a weight of 1 lb is approximately equivalent to 450 grams, convert: (a) 5 lbs to grams, (b) 9 lb into grams, (c) 1800 grams to lb, (d) 3150 grams to lb. Use the fact that 1 mile is approximately the same distance as 1.6 km to convert: (a) 30 miles to km, (b) 21 miles to km, (c) 80 km to miles, (d) 200 km to miles 122 MEP Y8 Practice Book A 6. On a certain day, the exchange rate was such that £1 was worth $1.63. Use a calculator to convert the following amounts to £, giving each answer correct to the nearest pence. (a) 7. 9. $250 (c) $75. 1000 Y (b) 200 Y (c) 50 000 Y. A weight of 1 lb is approximately equivalent to 450 grams. There are 16 ounces in 1 lb. Give answers to the following questions correct to 1 decimal place. (a) Convert 14 oz to lb. (b) Convert 200 grams to lb. (c) Convert 300 grams to ounces. If £1 is worth 2.8 German Marks (DM), and 1 DM is worth 2800 Italian Lira (L), use a calculator to convert: (a) 10. (b) The Japanese currency is the Yen (Y). The exchange rate gives 197 Yen for every £1. Using a calculator, convert the following amounts to pounds, giving each answer correct to the nearest pence. (a) 8. $100 800 DM to £, (b) 10 000 L to DM, (c) 50 000 L to £. There are 8 pints in one gallon. One gallon is equivalent to approximately 4.55 litres. Use a calculator to convert: (a) 12 pints to litres, (b) 20 litres to pints. Give your answers correct to 1 decimal place. 7.5 Inverse Proportion Inverse proportion is when an increase in one quantity causes a decrease in another. The relationship between speed and time is an example of inverse proportionality: as the speed increases, the journey time decreases, so the time for a journey can be found by dividing the distance by the speed. Example 1 (a) Ben rides his bike at a speed of 10 mph. How long does it take him to cycle 40 miles? (b) On another day he cycles the same route at a speed of 16 mph. How much time does this journey take? 123 MEP Y8 Practice Book A 7.5 Solution (a) Time = 40 10 (b) = 4 hours 1 40 = 2 2 16 1 = 2 hours 2 Time = Note: Faster speed β shorter time. Example 2 Jai has to travel 280 miles. How long does it take if he travels at: (a) 50 mph, (b) 60 mph ? (c) How much time does he save when he travels at the faster speed? Solution (a) Time = 280 50 = 5.6 hours = 5 hours 36 minutes (b) Time = 280 60 = 4 2 hours 3 = 4 hours 40 minutes (c) Time saved = 5 hours 36 mins β 4 hours 40 mins = 56 minutes Example 3 In a factory, each employee can make 40 chicken pies in one hour. How long will it take: (a) 6 people to make 40 pies, (b) 3 people to make 240 pies, (c) 10 people to make 600 pies? 124 MEP Y8 Practice Book A Solution (a) 1 person makes 40 pies in 1 hour. 1 6 people make 40 pies in hour (or 10 minutes). 6 (b) 1 person makes 40 pies in 1 hour. 240 1 person makes 240 pies in = 6 hours. 40 3 people make 240 pies in (a) 6 3 = 2 hours. 1 person makes 40 pies in 1 hour. 1 person makes 600 pies in 600 = 15 hours. 40 10 people make 600 pies in 15 10 = 1 1 hours. 2 Exercises 1. How long does it take to complete a journey of 300 miles travelling at: (a) 60 mph, (b) 50 mph, (c) 40 mph ? 2. Alec has to travel 420 miles. How much time does he save if he travels at 70 mph rather than 50 mph?. 3. Sarah has to travel 60 miles to see her boyfriend. Her dad drives at 30 mph and her uncle drives at 40 mph. How much time does she save if she travels with her uncle rather than with her dad? 4. Tony usually walks to school at 3 mph. When Jennifer walks with him he walks at 4 mph. He walks 1 mile to school. How much quicker is his journey when he walks with Jennifer? 5. One person can put 200 letters into envelopes in 1 hour. How long would it take for 200 letters to be put into envelopes by: (a) 4 people, (b) 6 people, (c) 10 people? 125 7.5 MEP Y8 Practice Book A 6. A person can make 20 badges in one hour using a machine. How long would it take: (a) 4 people with machines to make 20 badges, (b) 10 people with machines to make 300 badges, (c) 12 people with machines to make 400 badges? 1 hours. How much 2 faster would it have to travel to complete the journey in 4 hours? 7. A train normally complete a 270-mile journey in 4 8. On Monday Tom takes 15 minutes to walk one mile to school. On Tuesday he takes 20 minutes to walk the same distance. Calculate his speed in mph for each day's walk. 9. Joshua shares a 2 kg tin of sweets between himself and three friends. 10. (a) How many kg of sweets do they each receive? (b) How much less would they each have received if there were four friends instead of three? Nadina and her friends can each make 15 Christmas cards in one hour. How long would it take Nadina and four friends to make: (a) 300 cards, (b) 1000 cards? 126 MEP: Demonstration Project UNIT 7 1. 2. 3. 4. Teacher Support Y8A Ratio and Proportion Extra Exercises 7.1 Write each of these ratios in its simplest form: (a) 3 : 12 (b) 4 : 30 (c) 7 : 21 (d) 9 : 21 (e) 8 : 64 (f) 12 : 15 (g) 6 : 22 (h) 11 : 99 (i) 15 : 27 Write in the form 1 : n, each of the following ratios: (a) 2:5 (b) 8 : 40 (c) 5:7 (d) 5:2 (e) 10 : 37 (f) 20 : 50 In a class there are 18 girls and 12 boys. Write the ratio of boys to girls: (a) in its simplest form, (b) in the form 1 : n. In a drink, 80 cm 3 of pineapple juice is mixed with 20 cm 3 of orange juice. Write the ratio of pineapple juice to orange juice: (a) in its simplest form, (b) in the form 1 : n. © The Gatsby Charitable Foundation MEP: Demonstration Project UNIT 7 1. Ratio and Proportion Teacher Support Y8A Extra Exercises 7.2 If 10 m of chain costs £6, calculate the cost of: (a) 1 m, (b) 7 m, (c) 22 m. 2. If the mass of 20 wooden blocks is 3000 grams, calculate the mass of 16 wooden blocks. 3. To make 8 glasses of squash, you need 1280 cm 3 of water. How much water would you need to make: 4. 5. (a) 4 glasses of squash, (b) 1 glass of squash, (c) 7 glasses of squash? A car uses 22 litres of petrol to travel 176 miles. How far would the car travel using: (a) 10 litres of petrol, (b) 35 litres of petrol? Four identical minibuses carry a total of 64 passengers. How many passengers can be carried in 6 of these minibuses? © The Gatsby Charitable Foundation MEP: Demonstration Project UNIT 7 Ratio and Proportion Teacher Support Y8A Extra Exercises 7.3 1. Divide £60 in the ratio 2 : 3. 2. Divide £170 in the ratio 9 : 8. 3. Divide 240 kg in the ratio 1 : 2 : 5. 4. Divide 320 kg in the ratio 1 : 5 : 10. 5. John and Steve have earned £150 by doing extra work. They divide the money in the ratio 2 : 3. How much do they each receive? 6. Red, blue and yellow paint are mixed in the ratio 4 : 7 : 3 to produce 308 cm 3 of another colour. How much of each colour paint is used? © The Gatsby Charitable Foundation MEP: Demonstration Project UNIT 7 1. 2. 3. Ratio and Proportion Teacher Support Y8A Extra Exercises 7.4 If 1 pound sterling (£) is worth 300 Portuguese Escudos (Esc), convert: (a) £5 to Esc, (b) £7 to Esc, (c) 3600 Esc to £, (d) 1740 Esc to £. If one pound sterling (£) is worth 60 Belgian Francs (BFr), convert: (a) £45 to BFr, (b) £8.60 to BFr, (c) 1140 BFr to £, (d) 495 BFr to £. If there are 4.55 litres in one gallon, convert: (a) 20 gallons to litres, (b) 42 gallons to litres, (c) 37 gallons to litres, (d) 45.5 litres to gallons, (e) 182 litres to gallons, (f) 236.6 litres to gallons. © The Gatsby Charitable Foundation MEP: Demonstration Project UNIT 7 1. 2. Ratio and Proportion Teacher Support Y8A Extra Exercises 7.5 If Ernie drives at 60 mph, how long will it take him to travel: (a) 240 miles, (b) 75 miles, (c) 165 miles, (d) 140 miles? Jane has to drive 60 miles to visit her parents. How long will the journey take if she drives at: (a) 40 mph, (b) 50 mph, (c) 60 mph? 3. Amy has to drive 350 miles. How much time does she save if she drives at 70 mph rather than at 50 mph? 4. One person can address 50 envelopes in 1 hour. How long will it take: (a) 3 people to address 300 envelopes, (b) 2 people to address 500 envelopes, (c) 5 people to address 1000 envelopes? © The Gatsby Charitable Foundation MEP: Demonstration Project Teacher Support Y8A Extra Exercises 7.1 Answers 1. (a) 1:4 (b) 2 : 15 (c) 1:3 (d) 3:7 (e) 1:8 (f) 4:5 (g) 3 : 11 (h) 1:9 (i) 5:9 (a) 1 : 2:5 (b) 1:5 (c) 1 : 1.4 (2 s.f.) (d) 1 : 0.4 (e) 1 : 3.7 (f) 1 : 2.5 3. (a) 2:3 (b) 1 : 1.5 4. (a) 4:1 (b) 1 : 0.25 2. Extra Exercises 7.2 Answers 1. (a) 60p (b) £4.20 (c) £13.20 2. 2400 grams 3. (a) 640 cm 3 (b) 160 cm 3 (c) 1120 cm 3 4. (a) 80 miles (b) 280 miles 5. 96 passengers Extra Exercises 7.3 Answers 1. £24, £36 2. £90, £80 3. 30 kg, 60 kg, 150 kg 4. 20 kg, 100 kg, 200 kg 5. £60, £90 6. 88 cm 3 , 154 cm 3 , 66 cm 3 © The Gatsby Charitable Foundation MEP: Demonstration Project Teacher Support Y8A Extra Exercises 7.4 Answers 1. (a) 1500 Esc (b) 2100 Esc (c) £12 (d) £5.80 2. (a) 2700 BFr (b) 516 BFr (c) £19 (d) £8.25 3. (a) 91 litres (b) 191.1 litres (c) 168.35 litres (d) 10 gallons (e) 40 gallons (f) 52 gallons Extra Exercises 7.5 Answers 1. (a) (c) 1 hours (1 hour 15 mins) 4 3 1 (d) 2 hours (2 hours 45 mins) 2 hours (2 hours 20 mins) 4 3 4 hours 1 2. (a) 3. 2 hours 4. (a) (b) 1 1 hours (1 hour 30 mins) 2 2 hours (b) © The Gatsby Charitable Foundation 5 hours 1 hours (1 hour 12 mins) 5 (b) 1 (c) 4 hours (c) 1 hour MEP: Demonstration Project UNIT 7 Y8ATeacher Support M7 Ratio and Proportion M 7.1 Standard Route Mental Tests (no calculator) 1. Simplify the ratio 3 : 6. (1 : 2) 2. Simplify the ratio 4 : 20. (1 : 5) 3. Simplify the ratio 5 : 40. (1 : 8) 4. Simplify the ratio 6 : 8. (3 : 4) 5. If 10 bars of chocolate cost £3, how much does 1 bar cost? (30p) 6. If 2 tickets for a swimming session cost £5, how much do 6 tickets cost? (£15) 7. If 5 packets of sweets cost £1, how much do 2 packets cost? (40p) 8. If £1 is worth 3 German Marks, convert: (a) £20 to German Marks, (60 DM) (b) £2.50 to German Marks, (7.5 DM) (c) 45 German Marks to pounds. M 7.2 Academic Route (£15) (no calculator) 1. Simplify the ratio 4 : 20. (1 : 5) 2. Simplify the ratio 6 : 15. (2 : 5) 3. Simplify the ratio 14 : 24. (7 : 12) 4. If 8 bars of chocolate cost £2.40, how much do 3 bars cost? 5. If 5 tickets for a swimming session cost £6, how much do 3 tickets cost? 6. Share £12 in the ratio 1 : 2. 7. Share £18 in the ratio 1 : 2 : 3. 8. If £1 is worth 3 German Marks, convert: (a) £22 to German Marks, (b) £7.50 to German Marks, (c) 57 German Marks to pounds. © The Gatsby Charitable Foundation (90p) (£3.60) (£4, £8) (£3, £6, £9) (66 DM) (22.5 DM) (£19) MEP: Demonstration Project UNIT 7 Y8ATeacher Support M7 Ratio and Proportion M 7.3 Express Route Mental Tests (no calculator) 1. Simplify the ratio 6 : 42. (1 : 7) 2. Simplify the ratio 20 : 45. (4 : 9) 3. Simplify the ratio 63 : 81. (7 : 9) 4. If 7 packets of sweets cost £2.80, how much do 5 packets cost?. 5. If you travel 100 miles at 60 mph, how long does your journey take? 6. Share £49 in the ratio 1 : 2 : 4. (£7, £14, £28) 7. Share £56 in the ratio 5 : 2 : 7. (£20, £8, £28) 8. If £1 is worth 11 Danish Krone, convert: (£2) (1 hr 40 mins) (a) £3.50 to Danish Krone, (b) 275 Danish Krone to pounds, (£25) (c) 209 Danish Krone to pounds. (£19) © The Gatsby Charitable Foundation (38.5 Dkr) ππ»π΄ππΈ π΄πππ πππ πΆππππ’ππππππππ ππ π πΆπππππ ππ΄ππ»π ππ»π΄ππΈ π΄πππ πππ πΆππππ’ππππππππ ππ π πΆπππππ ππ΄ππ»π ππ»π΄ππΈ π΄πππ πππ πΆππππ’ππππππππ ππ π πΆπππππ ππ΄ππ»π ππ»π΄ππΈ π΄πππ πππ πΆππππ’ππππππππ ππ π πΆπππππ ππ΄ππ»π MEP: Demonstration Project UNIT 7 1. Teacher Support Y8A Ratio and Proportion Revision Test 7.1 (Standard) Write each of the following ratios in its simplest form: (a) 5 : 15 (b) 6 : 14 (c) 3 : 12 (6 marks) 2. Write in the form of 1 : n, each of the following ratios: (a) 2 : 10 (b) 2:5 (c) 10 : 22 (6 marks) 3. Given that one pound sterling (£) is worth 200 Japanese Yen (Y), convert: (a) £60 to Y , (b) £2.50 to Y, (c) 1600 Y to £ . (6 marks) 4. Share £99 in the ratio 4 : 7. (4 marks) 5. If 7 books cost £35, calculate the cost of: (a) 1 book, (b) 4 books, (c) 20 books. (6 marks) 6. Sugar and flour are mixed in the ratio 1 : 3 in a recipe. How much sugar should be mixed with 900 grams of flour? (2 marks) © The Gatsby Charitable Foundation MEP: Demonstration Project UNIT 7 1. Teacher Support Y8A Ratio and Proportion Revision Test 7.2 (Academic) Write each of the following ratios in its simplest form: (a) 21 : 35 (b) 42 : 49 (c) 24 : 40 (6 marks) 2. Write in the form of 1 : n, each of the following ratios: (a) 10 : 45 (b) 5:8 (c) 2 : 17 (6 marks) 3. Given that one pound sterling (£) is worth 240 Spanish Pesetas (Pta), convert: (a) £8 to Pta, (b) £3.50 to Pta, (c) 2640 Pta to £ . (6 marks) 4. Share £90 in the ratio 7 : 6 : 5. (4 marks) 5. If 9 hockey sticks cost £126, calculate the cost of: (a) 1 stick, (b) 7 sticks. (4 marks) 6. If Debbie drives 150 miles at 60 mph, how long does her journey take? (2 marks) 7. A map uses a scale of 1 : 20 000. What distance on the map would represent an actual distance of 2 km? (2 marks) © The Gatsby Charitable Foundation MEP: Demonstration Project UNIT 7 1. Teacher Support Y8A Ratio and Proportion Revision Test 7.3 (Express) Write each of the following ratios in its simplest form: (a) 42 : 49 (b) 60 : 150 (c) 45 : 63 (6 marks) 2. Write in the form of 1 : n, each of the following ratios: (a) 10 : 4.7 (b) 5 : 22 (4 marks) 3. Given that one pound sterling (£) is worth 2.4 Swiss Francs (SFr), convert: (a) £40 to SFr, (b) 480 SFr to £, (c) 4.2 SFr to £ . (6 marks) 4. Share £198 in the ratio 3 : 6 : 9. (4 marks) 5. If 12 footballs cost £168, calculate the cost of: (a) 4 footballs, (b) 9 footballs (4 marks) 6. Debbie and Karen each drive 60 miles. Debbie drives at 40 mph and Karen at 50 mph. How long does each journey take? (4 marks) 7. A map uses a scale of 1 : 25 000. What distance on the map would represent an actual distance of 4.5 km? (2 marks) © The Gatsby Charitable Foundation MEP: Demonstration Project Teacher Support Y8A Revision Test 7.1 (Standard) 1. 2. 3. 4. 5. 6. Answers (a) 1:3 B2 (b) 3:7 B2 (c) 1:4 B2 (a) 1:5 B2 (b) 1 : 2.5 B2 (c) 1 : 2.2 B2 (a) 60 × 200 = 12 000 Y M1 A1 (b) 2.5 × 200 = 500 Y M1 A1 (c) 1600 ÷ 200 = £8 M1 A1 4 + 7 = 11 M1 99 ÷ 11 = 9 M1 4 × 9 = £36 B1 7 × 9 = £63 B1 (6 marks) (6 marks) (6 marks) (4 marks) (a) 35 ÷ 7 = £5 (b) 4 × 5 = £20 M1 A1 (c) 20 × 5 = £100 M1 A1 (6 marks) 900 ÷ 3 = 300 grams M1 A1 (2 marks) - M1 A1 (TOTAL MARKS 30) © The Gatsby Charitable Foundation MEP: Demonstration Project Teacher Support Y8A Revision Test 7.2 (Academic) 1. 2. 3. 4. (a) 3:5 B2 (b) 6:7 B2 (c) 3:5 B2 (a) 1 : 4.5 B2 (b) 1 : 1.6 B2 (c) 1 : 8.5 B2 (a) 8 × 240 = 1920 Pta M1 A1 (b) 3.5 × 240 = 840 Pta M1 A1 (c) 2640 ÷ 240 = £11 M1 A1 7 + 6 + 5 = 18 M1 90 ÷ 18 = 5 A1 7 × 5 = £35 6 × 5 = £30 5 × 5 = £25 5. Answers ο£Ό ο£΄ ο£΄ ο£½ ο£΄ ο£΄ ο£Ύ M1 A1 (6 marks) (6 marks) (6 marks) (4 marks) (a) 126 ÷ 9 = £14 M1 A1 (b) 7 × 14 = £98 M1 A1 (4 marks) 6. 150 1 = 2 hours 60 2 M1 A1 (2 marks) 7. 2000 × 100 = 10 cm 20 000 M1 A1 (2 marks) (TOTAL MARKS 30) © The Gatsby Charitable Foundation MEP: Demonstration Project Teacher Support Y8A Revision Test 7.3 (Express) 1. 2. 3. 4. 5. 6. 7. Answers (a) 6:7 B2 (b) 2:5 B2 (c) 5:7 B2 (a) 1 : 0.47 B2 (b) 1 : 4.4 B2 (a) 40 × 2.4 = 96 SFr M1 A1 (b) 480 ÷ 2.4 = £200 M1 A1 (c) 4.2 ÷ 2.4 = £1.75 M1 A1 4 + 6 + 9 = 18 M1 198 ÷ 18 = 11 A1 3 × 11 = 33 ο£Ό ο£΄ ο£΄ 6 × 11 = 66 ο£½ ο£΄ 9 × 11 = 99 ο£΄ ο£Ύ M1 A1 (a) 168 × 4 = £56 12 M1 A1 (b) 168 × 9 = £126 12 M1 A1 (6 marks) (4 marks) (6 marks) (4 marks) (4 marks) 60 1 = 1 hours 40 2 M1 A1 60 1 = 1 hours 50 5 M1 A1 (4 marks) 4500 × 100 450 = = 18 cm 25000 25 M1 A1 (2 marks) (TOTAL MARKS 30) © The Gatsby Charitable Foundation Converting between units of area and volume 1cm 1cm 1cm 1) 2) 3) 4) 5) ππππ2 π‘π‘π‘π‘ ππ2 11) 12) 13) 14) 15) 2ππππ2 = 36ππππ2 = 17ππππ2 = 2468ππππ2 = 0.5ππππ2 = = ππππππππ 5ππππ3 = 89ππππ3 = 20ππππ3 = 0.7ππππ3 = 0.9ππππ3= ππ2 π‘π‘π‘π‘ ππππ2 9ππ2 = 87ππ2 = 253ππ2 = 0.6ππ2 = 4.6ππ2 = ππππππππππππππ Extensions 21) 22) 23) 24) 25) ππππ, ππππππππππππ 100cm 6) 7) 8) 9) 10) ππππ2 π‘π‘π‘π‘ ππππ2 700ππππ2 = 2300ππππ2 = 540ππππ2 = 0.2ππππ2 = 0.09ππππ2 = = ππππππ = 1m 100cm ππππ3 π‘π‘π‘π‘ ππππ3 9000ππππ3 = 64000ππππ3 = 500ππππ3 = 30ππππ3 = 28.8ππππ3 = Find the volume of this cylinder in ππππ3 . Then convert your answer to ππππ3 . 16) 17) 18) 19) 20) ππ3 π‘π‘π‘π‘ ππππ3 8ππ3 = 4ππ3 = 0.5ππ3 = 1.2ππ3 = 0.00007ππ3= Find the area of this trapezium in ππππ3 . Then convert your answer to ππ3 . 30cm 7cm 3cm 100cm 1m 10mm ππππ3 π‘π‘π‘π‘ ππππ3 = ππππππ 1m 10mm 1cm 26) 27) 28) 29) 30) 10mm ππππππππππππ 100cm = ππππππππ 60cm 10.1. Distance-Time Graphs (1) Paul was travelling in his car to a meeting. This distance-time graph illustrates his journey. a. How long after he set off did he: i) Stop for his break ii) iii) Set off after this break Get to his meeting place? b. At what average speed was he travelling: i) Over the first hour ii) Over the second hour iii) For the last part of his journey? c. The meeting was scheduled to start at 10.30am. What is the latest time he should have left home? (2) James was travelling to Cornwall on his holiday. This distance-time graph illustrates his journey. a. His greatest speed was on the motorway. i) How far did he travel on the ii) motorway? What was his average speed on the motorway? b. i) When did he travel the most slowly? ii) What was his lowest average speed? 10.1. Distance-Time Graphs (3) A small bus set off from Leeds to pick up Mike and his family. It then went on to pick up Mikeβs parents and grandparents. It then travelled further, dropping them all off at a hotel. The bus then went on a further 10km to pick up another party and it took them back to Leeds. This distance-time graph illustrates the journey. a. How far from Leeds did Mikeβs parents and grandparents live? b. How far from Leeds is the hotel at which they all stayed? c. What was the average speed of the bus on its way back to Leeds? (4) Three friends, Patrick, Araf and Sean, ran a 1000m race. The race is illustrated on the distance-time graph shown here: a. Describe how each of them completed the race. b. i) ii) What is Arafβs average speed in m/s? What is this speed in km/h? 10.1. Distance-Time Graphs (5) Richard and Paul took part in a 5000m race. It is illustrated in this graph. a. Paul ran a steady race. What is his average speed in: i) Metres per minute. ii) Kilometres per hour? b. Richard ran in spurts. What was his highest average speed? c. Who won the race and by how much? (6) A walker sets off at 9.00am from point P to walk along a trail at a steady pace of 6km per hour. 90 minutes later, a cyclist sets off from P on the same trail at a steady pace of 15km per hour. At what time did the cyclist overtake the walker? You may use a graph to help you solve this question. 10.1. Distance-Time Graphs (7) Three school friends set off from school at the same time, 3.45pm. They all lived 12km away from the school. The distance-time graph illustrates their journeys: One of them went by bus, one cycled and one was taken by car. a. i) Explain how you know that Sue used the bus. ii) Who went by car? b. At what times did each of them get home? c. i) When the bus was moving, it covered 2km in 5minutes. What is ii) this speed in kilometres per hour? Overall, the bus covered 12km in 35minutes. What is this speed, in kilometres per hour? iii) How many stops did the bus make before Sue got off? Area and circumference of circles Bronze: Calculate the area and circumference of circles (D) Silver: Use the formula for area and circumference to find the radius or diameter (C) 2. Find the missing lengths in these circles e) f) ? cm Area = 78.5 cm2 ? cm Circumference = 50.3 cm Gold: Solve worded problems involving the area and circumference of circles (C) 1. Trevor needs a new bicycle tire. His tire has a circumference of about 113.04 inches. What diameter tire does he need to buy? 2. A rope is wrapped eight times around a cylindrical post, the diameter of which is 35cm. How long is the rope? 3. a) Daniel is putting a circular flower bed in his garden with a diameter of 2m. He wants to put plastic fencing around the flower bed, how many metres of fencing will he need to buy? b) How many square metres of land are in Danielβs garden? 4. Anna is covering her circular swimming pool with a heavy-duty cover for the winter. The pool has a diameter of 7m. The cover extends 1m beyond the edge of the pool, and a rope runs along the edge of the cover to secure it in place. a) What is the area of the cover? b) What is the length of the rope? 5. How many square centimetres greater is the area of a 30cm diameter circular mirror than an 18cm diameter circular mirror? 6. GCSE question Averages Finding Mode, Median, Mean and Range Mode= The value that occurs the most Example: 4, 9, 4, 6,10,6,3,5,4, 9 Arranged in order Median= The middle value when the values are put in order from smallest to biggest Mean= Sum of all values in the set Number of values in the set (sometimes referred to as the βaverageβ) Range= Biggest value- smallest value 3, 4, 4, 4, 5, 6, 6, 9, 9, 10 Mode= 4 Median= 3, 4, 4, 4, 5, 6, 6, 9, 9, 10 If there are 2 numbers left in the middle, add them up and divide by 2! Tip: So median= Putting the values in order from smallest to biggest will make it easier to work out your averages! Mean= π+π π = 5.5 π+π+π+π+π+π+π+π+π+ππ ππ Range= 10-3=7 1) The set of values shows the number of goals scored by a school football team in their first 10 matches: 2, 4, 1, 0, 2, 3, 2, 6, 2, 4 Find: a) Mode= c) Median= a) Mean= d) Range Find the mode, mean, median and range for each set of values given: 2) 7,2,5,3,8,5,5 Mode = Median = Mean = Range = 3) 1,7,2,5,3,1,1,4 Mode = Median = Mean = Range = 4) 6,9, 9,1,5 Mode = Median = Mean = Range = =6 5) 9,3,2,4,6,5,4,8,4 Mode = Median = Mean = Range = 6) 6,3,8,1,9,3 Mode = Median = Mean = Range = Section B- Using the mean to find unknown values 1) The mean of 4 numbers is 9. Three of the numbers are 6, 8 and 11. What is the fourth number? 2) The results of a Maths test scores were 15,17,22,19 and X. The mean test score was 18. Find the value of X. 3) In his first 11 basketball games, John scored on average (mean) 18 points. His mean increases to 19 points after his 12th game. How many points did he score in his 12th game? 4) The mean of the number of ice lollies sold each hour for 7 hours is 4. The number of ice lollies sold in the first fist 6 hours were 2, 10, 4, 2, 4, 1 5) There are 15 employees in an office. Their mean age is 35. A new employee joins the office. The mean age decreases to 31. How old is the new Employee? 6) Ricky has a mean score of 23 runs this season in 6 cricket matches. His mean score must be at least 25 runs for him to win a scholarship. How many runs must he score in his next match? 2 Averages from Frequency tables 1) The table shows the number of minutes taken by students in a year 5 class to solve a Maths problem Number of Minutes Frequency 2 3 4 5 6 7 1 6 7 10 3 1 Find the: a) Mode c) Range b) Median d) Mean 2) A die is thrown and the scores are recorded. The results are shown in this table Find the: a) Mode b) Range c) Median Die Score Frequency 1 15 2 12 3 8 4 14 5 10 6 13 d) Mean 3) The weights of people in a fitness club are measured. The results are shown in the table. Weight in Kg Frequency 45β€WΛ50 6 50β€WΛ55 4 55β€WΛ60 5 60β€WΛ65 2 65β€WΛ70 8 a) How many people are in the fitness class? b) What is the modal class? c) In which class is the median? Answer=________________________ Answer=___________________________ Answer=__________________________ 3 ANSWERS Averages Finding Mode, Median, Mean and Range 1) Mode= 2 Mean=2.6 Median=2 Range=6-0=6 2) Mode=5 Median-5 Mean=5 Range=5 3) Mode= 1 Median=2.5 Mean=3 Range=6 4) Mode=9 Median=6 Mean=6 Range=8 5) Mode=4 Median=4 Mean=5 Range=7 6) Mode=3 Median=4.5 Mean=5 Range=5 Section B 1) 6+8+11=25 36-25=11 4x9=36 2) X=17 3) 11x18=198 228-198=30 12x19=228 4) 5 ice lollies 5) 29 Years old 6) 37 runs Averages from a frequency table 1) Mode=5 Median=4 Range=5 Mean=4.39 (2dp) 2) Mode=1 Median=4 Range=5 Mean=3.43 (2dp) 3) a) 25 People b) 65β€WΛ 70 c) 55β€WΛ 60 4 Name : Score : ES1 Surface Area - Cylinder Find the exact surface area of each cylinder. 2) 1) 3) 7 in 13 ft 6 yd 12 10 ft 4 in Surface Area = Surface Area = Surface Area = 5) 4) 6) 11 yd 13 ft 8 ft 13 in 18 yd 12 in Surface Area = Surface Area = Surface Area = 8) 7) 9) 4i n 10 ft Surface Area = n 9i 6 yd 15 ft 15 yd Surface Area = Printable Math Worksheets @ www.mathworksheets4kids.com Surface Area = yd Name : Score : Answer key ES1 Surface Area - Cylinder Find the exact surface area of each cylinder. 2) 1) 3) yd 7 in 13 ft 6 yd 12 10 ft 4 in Surface Area = Surface Area = 36Ο in2 5) 4) Surface Area = 460Ο ft2 216Ο yd2 6) 11 yd 13 ft 8 ft 13 in 18 yd 12 in Surface Area = Surface Area = 360Ο yd2 8) 7) 336Ο ft2 9) n 9i 6 yd 15 ft 15 yd 4i 500Ο ft2 n 10 ft Surface Area = Surface Area = 228Ο in2 Surface Area = 252Ο yd2 Printable Math Worksheets @ www.mathworksheets4kids.com Surface Area = 104Ο in2 Mr Martin Maths Simplifying Ratio Green β these questions are not too tricky so try these to get you started. Simplify: e.g. 8 : 12 = 2 : 3 1) 4 : 8 = 2) 12 : 8 = 3) 5 : 10 = 4) 5) 6) 7) 6:9= 20 : 30 = 7 : 14 = 6 : 12 = 8) 6 : 18 = 9) 10 : 100 = 10) 9 : 18 = 11) 4 : 16 = 12) 13) 14) 15) 16 : 8 = 11 : 22 = 24 : 12 = 10 : 20 = Orange β these questions are a little trickier than the green questions. Simplify: e.g. 4 : 8 : 12 = 1 : 2 : 3 1) 6 : 8 : 12 = 2) 4 : 8 : 12 = 3) 9 : 9 : 27 = 4) 1 hour : 20 mins = 5) 2 hours : 30 mins = 6) 50p : £2.50 = 7) 20cm : 15cm = 8) 1 day : 6 hours = 9) £1 : 70p = 10) 70p : £1 = 11) 4km : 12km = Red β these questions are the trickiest and may require you to interpret worded questions. e.g. 4 : 8 : 12 : 24 = 1 : 2 : 3 : 6 1) 3 : 9 : 15 : 24 = 4) 6 : 18 : 9 : 15 = 2) There are 32 pupils in a class. 20 of them are girls. What is the ratio of boys to girls in its simplest form? 5) There are 50 sweets in a mixed pack. 25 are jellies, 10 are fizzy cola bottles and the rest are boiled. Write the ratio of each type of sweet in its simplest form. 3) A fruit drink is made by mixing 60ml of orange juice with 180ml of pineapple juice. What is the ratio of orange juice to pineapple juice in its simplest form? 6) Concrete is made by mixing sand, water and concrete mix in the ratio of 6 parts sand, 3 parts water and 3 parts concrete mix. What ratio is this in its simplest form? Ratio Bingo Card 3:5 4:1 £20 : £30 4:5 5:3 4:3 3:4 1:4 5:6 £40 : £10 £30 : £20 £10 : £40 1:3 5:4 6:5 3:1 Full horizontal row or vertical column wins Mr Martin Maths Dividing Ratio Steps 1. 2. 3. 4. to Success Add together the total number of parts of the ratio Divide the given amount by the total number of parts Multiply the answer by each ratio amount Check that your answers add up to the same total given in the question Green β these questions are not too tricky so try these to get you started. 1) Divide £50 into the ratio 1 : 4 2) Divide 40 litres into the ratio 1 : 3 3) Divide £10 into the ratio 2 : 3 4) Divide 500g into the ratio 2 : 3 5) Divide 100ml into the ratio 3 : 7 6) Divide 30cm into the ratio 2 : 4 Orange β these questions are a little trickier than the green questions. 1) Divide 49p in the ratio 4 : 2 : 1 2) Divide £350 in the ratio 4 : 2 : 1 3) Divide £250 in 4) Divide 120cm in the ratio 1 : 2 : 3 the ratio 3 : 6 : 1 5) Divide 51g in the ratio 1 : 2 6) Divide 68p in the ratio 1 : 2 : 1 Red β these questions are the trickiest and will require you to interpret worded questions. 1) Mr Martin has 120 CDs. The ratio of indie CDs to dance CDs is 5 : 7. How many of each type of CD does he have? 3) There are 1400 students in Plantsbrook School with a ratio of 4 : 3 boys to girls. How many boys and how many girls are there? 2) Mr Martin has 36 calculators in a box. The ratio of ordinary calculators to scientific is 5 : 1. How many of each calculator does he have? 4) 180 people go bowling, the ratio of adults to children is 5 : 4. How many adults and how many children go bowling? Area and Volume of Cylinders (A) Calculate the surface area and volume for each cylinder. Surface Area = (Οr2 × 2) + (Οd × h) Volume = Οr2 × h d = 2r 1. 2. r d h h r = 1.2 km d = 12.6 cm h = 3.6 km h = 7.5 cm Surface Area = Surface Area = Volume = Volume = 3. 4. d r h h d = 12 m h = 18.6 m Surface Area = r = 18 ft h = 27.2 ft Volume = Surface Area = Volume = Math-Drills.com Area and Volume of Cylinders (A) Answers Calculate the surface area and volume for each cylinder. Surface Area = (Οr2 × 2) + (Οd × h) Volume = Οr2 × h d = 2r 1. 2. r d h h r = 1.2 km d = 12.6 cm h = 3.6 km h = 7.5 cm Surface Area = 36.19 km2 Surface Area = 546.26 cm2 Volume = 16.29 km3 Volume = 935.17 cm3 3. 4. d r h h d = 12 m h = 18.6 m Surface Area = 927.4 m2 r = 18 ft h = 27.2 ft Volume = 2103.61 m3 Surface Area = 5112 ft2 Volume = 27,686.23 ft3 Math-Drills.com Mean and Mode from Frequency Tables 1. Majid carried out a survey of the number of school dinners students had in one week. The table shows this information. Copy and complete the table. a) Write down the mode. b) How many students were there altogether? c) How many school dinners were served during the week? d) Calculate the mean number of school dinners Number of school dinners 0 1 2 3 4 5 Frequency 0 8 12 6 4 2 2. Josh asked some adults how many cups of coffee they each drank yesterday. The table shows his results. Copy and complete the table a) Write down the mode. b) How many adults were asked altogether? c) How many cups of coffee were drank altogether yesterday by the people asked? d) Calculate the mean number of cups of coffee drank yesterday. Number of cups 0 1 2 3 4 5 Frequency 5 9 7 4 3 2 3. The table gives some information about the number of tracks on each CD. Copy and complete the table. Number of Frequency tracks a) Write down the mode. 11 1 b) How many CDs were there? 12 3 c) How many tracks were there on these CDs? 13 0 d) Calculate the mean number of tracks per CD. 14 2 15 4 4. Sarah works in a post office. She recorded the number of parcels posted on each of 16 days. Here are her results. 2 3 2 6 5 4 3 6 2 2 4 2 2 3 2 3 a) Copy and complete the frequency table to show Sarahβs results. b) c) d) Write down the mode. Work out the range. Calculate the mean. Number of parcels 2 3 4 5 6 Tally Frequency 5. Some students were asked how many pets they owned. The results are given in the bar chart below. a) Use the graph to copy and complete the table Number of pets 0 1 2 3 4 5 Frequency 16 14 12 10 8 6 4 2 0 0 1 2 3 4 5 b) Write down the mode c) How many students were asked altogether? d) How many pets are owned, in total, by all the students. e) Find the mean number of pets owned. 6. The table gives information about the number of goals Number of goals 0 1 2 3 4 scored by a football team in each match last season. a) Write down the modal number of goals scored. b) Work out the total number of goals scored by the team last season. c) Work out the mean number of goals scored last season. Frequency 4 5 4 7 4 6. The table gives information about the number of cars sold by a company each day over a period of 20 days. a) Write down the modal number of cars scored. c) Work out the mean number of cars sold during this time. Number of cars sold daily 6 5 Frequency b) Work out the total number of cars sold over the 20 day period. 7 4 3 2 1 0 0 1 2 3 Cars sold in one day 4 5