RATIO AND PROPORTION. All Levels

Transcription

1 RATIO AND PROPORTION All Levels

2 RATIO AND PROPORTION Junior Cert Higher Level

3 2017 JCHL Paper 1 Question 4 (b) (i) Fruitex and Juicy are each made from mixing fruit juice and water. In Fruitex, the ratio of fruit juice to water is 3: 7. Find how many litres of fruit juice are in 20 litres of Fruitex. Fruit Juice: Water 3: = 10 parts = 2 litres in 1 part Add the ratios together to find how many parts there are. Divide the total amount of Fruitex by the sum of the ratios. Fruit Juice: 3 parts 2 3 = 6 Multiply the amount given out per 1 part by the number of parts of the juice. There is 6 litres of Fruit Juice in Fruitex.

4 2017 JCHL Paper 1 Question 4 (b) (ii) 20 litres of Fruitex is mixed with 40 litres of Juicy. In this 60-litre mixture, the ratio of fruit juice to water is 7: 8. Find the ratio of fruit juice to water in Juicy. Give your answer in its simplest form. Divide the 60 litre mixture into its juice and water parts. Fruit Juice: Water 7: = 15 parts = 4 litres in 1 part Fruit Juice: 7 parts 4 7 = 28 There is 28 litres of fruit juice in the 60 litre mixture. There is 6 litres of fruit juice in the 20 litres of Fruitex which leaves 28 6 = 22 litres of fruit juice in Juicy = 18 litres of water in Juicy Ratio of Fruit Juice: Water in Juicy is: = 22 : 18 = 11 : 9 Water: 3 parts 4 3 = 32

5 2015 Supplementary Sample Question 9 (a) A rectangular television screen has a diagonal of length 42 inches. The sides of the television screen are in the ratio 16:9. Find the area of the television screen, correct to the nearest whole number. We don t know the length or breadth of the TV but we know the ratio of length: breadth = 16: 9 Let 16 units be the length and 9 units be the breadth. We need to calculate the diagonal using Pythagoras in terms of units (not inches). c 2 = a 2 + b 2 c 2 = c 2 = c 2 = 337 c = 337 c = inches = units 42 = inches in 1 unit Length = 36.6 Breadth = inches 16 units Area of Rectangle = l b = = inches 2 9 units

6 2012 JCHL Paper 1 Question 4 (a) A soccer team has three strikers John, Paul and Michael. The number of minutes each had played by the end of a particular season is shown on the table. The team divided a bonus of between its strikers in proportion to the time each had played. Calculate the amount each player received. Name Minutes Played John 2250 Paul 2600 Michael 150 John: Paul: Michael = = 30 John = 67,500 Paul = 78,000 Michael = 4,500 Add the ratios together to find how many parts there are. Divide the total bonus by the sum of the ratios to find the bonus per 1 part. Multiply the amount given out per 1 part by the number of parts for each of the players.

7 2012 JCHL Paper 1 Question 4 (b) (i) At the end of the following season a larger total bonus was paid. At that time, John said: The bonus should be paid according to the number of goals scored by the striker. Paul scored 50% more goals than Michael. I scored as many as both of them together. I would get if the team used this method. Calculate the total bonus on offer that season JCHL Paper 1 Question 4 (b) (ii) How much each would Paul and Michael get under John s system? Michael scored the least goals so make his part of the ratio 1. Paul scored 50 more than Michael so his part is 1.5. John scored as many as both so his part is 2.5. Michael:Paul:John 1: 1.5: 2.5 Michael = 56,000 Paul = 84,000 Multiply the amount per 1 part by number of parts for Paul and Michael = in 1 part Divide John s bonus by 2.5 to get the amount in 1 part = = Add the ratios to find the total number of parts. Multiply the amount per 1 part by the total number of parts, 5. The total bonus is 280,000

8 2013* JCHL Paper 1 Question 2 (a) The lengths of two pieces of timber are in a ratio of 5 : 2. The larger piece measures 250 mm. Find the length of the shorter piece. Larger: Smaller 5: 2 5 parts = = 50 is 1 part 5 Larger piece is 5 parts. Divide the length of the larger piece by 5 to get the length of 1 part = 100 Now multiply the share in 1 part by the number or parts in the smaller piece, 2. The smaller piece is 100 mm long.

9 2012* JCHL Paper 1 Question 2 (a) Fuel consumption in a car is measured in litres per 100 km. Alan s car travels 1250 km on a tank of 68 litres. Calculate his car s fuel consumption in litres per 100 km km = 68 litres = litres in 1 km = 5.44 litres in 100 km 1250

10 2011 JCHL Paper 1 Question 1 (a) Peter and Anne share a lotto prize in the ratio to Peter s share is What is the total prize fund. Peter : Anne : : 5 2 7: = 7 parts Find an equivalent ratio by multiplying by 2. Peter s share is 7 parts = 5000 in one part Divide Peter s share by 7 to find 1 part of the prize fund = Now multiply the amount in 1 part by the total number of parts in the fund, = 12.

11 2009 JCHL Paper 1 Question 2 (a) Eight workers can build a cabin in 60 hours. How many workers are needed if the cabin is to be built in 32 hours? In this type of question find out how long it will take one worker to build a cabin and work your way to the solution. 8 workers = 60 hours 1 worker = worker = 480 hours It will take one worker 8 times as long! For it to be built in 32 hours we will need: = 15 workers

12 2008 JCHL Paper 1 Question 2 (b) (i) Two brands of blackcurrant squash drinks contain concentrated juice and sugar. In brand A, the ratio of concentrated juice to sugar is 19:1. In brand B, the ratio of concentrated juice to sugar is 9:1. What is the volume of concentrated juice in 500 ml of brand A? Brand A Juice: Sugar 19: = 20 parts Add the ratios together to find how many parts there are = 20 ml in 1 part Divide the total amount of Brand A by the sum of the ratios. Juice: 19 parts = 475 Multiply the amount given out per 1 part by the number of parts of the juice. There is 475 ml of concentrated juice in Brand A.

13 2008 JCHL Paper 1 Question 2 (b) (i) What is the volume of sugar in 300 ml of brand B? Brand B Juice: Sugar 9: = 10 parts Add the ratios together to find how many parts there are = 30 ml in 1 part Divide the total amount of Brand B by the sum of the ratios. Sugar: 1 part There is 30 ml of sugar in Brand B.

14 2008 JCHL Paper 1 Question 2 (b) (iii) 500 ml of brand A is mixed with 300 ml of brand B. What is the ratio of the concentrated juice to the sugar in the mixture? 500 ml Brand A Juice 475 ml Sugar = 25 ml 300 ml Brand B Juice = 270 ml Sugar 30 ml Mixture of 500 ml A and 300 ml B Juice = 745 ml Sugar = 55 ml Mixture Ratio Juice: Sugar 745: : 11

15 2004 JCHL Paper 1 Question 1 (a) The area of a house covers 205 m 2. The area of the site for the house covers 1025 m 2. What is the of the area of the house to the area of the site? Give your answer in the form 1: n, where n N. Area of House : Area of Site = = 1 5

16 RATIO AND PROPORTION Leaving Cert Higher Level

17 2015 LCOL Paper 1 Question 2 (a) John, Mary and Eileen bought a ticket in a draw. The ticket cost 50. John paid 25, Mary paid 15 and Eileen paid 10. The ticket won a prize of The prize is divided in proportion to how much each paid. How much prize money does each person receive? John: Mary: Eileen 25: 15: = 50 parts Add the ratios together to find how many parts there are = 400 in 1 part Divide the prize money by the sum of the ratios. John: 25 parts = For each person multiply the amount given out per 1 part by the number of parts. Mary: 15 parts = 6000 Mary: 10 parts = 4000

18 2012 LCOL Paper 1 Question 1 (a) When Katie had travelled 140 km, she had completed 4 of her journey. 9 Find the length of her journey. 140 = 4 parts = 35 in one part Divide the amount travelled so far by 4 to find 1 part of the journey = 315 km Now multiply the distance in 1 part by the total number of parts in zinc, 9. Alternate Method Let x be the total distance. Then: 4 x = x = x = 315 km

19 2011 LCOL Paper 1 Question 1 (a) Aoife and Brian share a prize fund in the ratio 4 : 3. Aoife gets 56. (i) Find the total prize fund. (ii) How much does Brian get? Aoife: Brian 4: 3 4 parts = = 14 is 1 part Total Prize Fund 14 7 = 98 Aoife gets 4 parts Divide Aoife s share by 4 to find the amount of 1 part. Now multiply the share in 1 part by the total number of parts, = 7. Brian 14 3 = 42 Brian gets 3 parts so multiply the share in 1 part by 3.

20 2009 LCOL Paper 1 Question 1 (a) Conor and Alice share 50 apples in the ratio 3 : 7. (i) How many apples does Conor get? (ii) How many apples does Alice get? Conor: Alice 3: = 10 parts = 5 apples in 1 part Add the ratios together to find how many parts there are. Divide the total number of apples by the sum of the ratios. Conor: 3 parts 5 3 = 15 For each person multiply the amount given out per 1 part by the number of parts. Alice: 7 parts 5 7 = 35

21 2007 LCOL Paper 1 Question 1 (a) Convert 164 miles to kilometres, taking 5 miles to be equal to 8 kilometres. 8 km = 5 miles 8 5 = 1.6 km in 1 mile = km in 164 miles

22 2006 LCOL Paper 1 Question 2 (a) 320 is 4 of a prize fund. Find the total prize fund = 4 parts = 80 in one part Divide the amount travelled so far by 4 to find 1 part of the prize fund = 720 Now multiply the amount in 1 part by the total number of parts in the fund, 9. Alternate Method Let x be the total prize fund. Then: 4 x = x = x = 720

23 2005 LCOL Paper 1 Question 1 (a) Express 35 cm as a fraction of 1 m. Give your answer in its simplest form. 1 m = 100 cm = 7 20

24 2005 LCOL Paper 1 Question 1 (b) (ii) Express the ratio as a ratio of natural numbers Divide 325 in the ratio Find an equivalent ratio by multiplying each ratio by the lowest common denominator, = 13 parts Add the ratios together to find how many parts there are = 25 in 1 part Divide the 325 by the sum of the ratios = = 100 For each ratio multiply the amount given out per 1 part by the number of parts = 75

25 2002 LCOL Paper 1 Question 1 (a) Copper and zinc are mixed in the ratio 19 : 6. The amount of copper used is 133 kg. How many kilogrammes of zinc are used? Copper: Zinc 19: = 7 kg in 1 part Divide the amount of copper by the number of parts in copper, 19, to find the amount in 1 part. Zinc has 6 parts 7 6 = 42 kg Now multiply the amount given out per 1 part by the number of parts in zinc, 6.

26 2001 LCOL Paper 1 Question 1 (a) A cookery book gives the instruction for calculating the amount of time for which a turkey should be cooked: Allow 15 minutes per 450 grammes plus an extra 15 minutes. For how many hours and minutes should a turkey weighing 9 kilogrammes be cooked? 9 kg = 9,000 grammes minutes or 5 hours 15 minutes Divide 9000 grammes by 450 to find out how many 15 minutes we need to cook the turkey. We must also add an extra 15 minutes.

27 2000 LCOL Paper 1 Question 1 (a) Express 400 grammes as a fraction of 1 kilogramme. Give your answer in its simplest form. 1 kg = 1000 g = 2 5

28 1999 LCOL Paper 1 Question 1 (a) 40 is divided between 2 pupils in the ratio 7:3. How much does each pupil get? Pupil 1: Pupil 2 7: = 10 parts = 4 in 1 part Pupil = 28 Add the ratios together to find how many parts there are. Divide the 40 by the sum of the ratios to find how much is in 1 part. For each person multiply the amount given out per 1 part by the number of parts. Pupil = 12