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RATIO AND PROPORTIONSOLUTIONS
Ratio and Proportion – Table of Contents
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2017 Paper 1 – Q4 (b) (i)
2017 Paper 1 – Q4 (b) (ii)
2015 SUPPLEMENTARY – Q9 (a)
2012 Paper 1 – Q4 (a)
2012 Paper 1 – Q4 (b)
JCHL Old Course
2013 Paper 1 – Q2 (a)
JCHL New Course
2012 Paper 1 – Q2 (a)
2011 Paper 1 – Q1 (a)
2009 Paper 1 – Q2 (a)
2008 Paper 1 – Q2 (b) (i)
2008 Paper 1 – Q2 (b) (ii)
2008 Paper 1 – Q2 (b) (iii)
2004 Paper 1 – Q1 (a)
LCOL New Course
2015 Paper 1 – Q2 (a)
LCOL Old Course
2012 Paper 1 – Q1 (a)
2011 Paper 1 – Q1 (a)
2009 Paper 1 – Q1 (a)
2007 Paper 1 – Q1 (a)
2006 Paper 1 – Q2 (a)
2005 Paper 1 – Q1 (a)
2005 Paper 1 – Q1 (b) (ii)
2002 Paper 1 – Q1 (a)
2001 Paper 1 – Q1 (a)
2000 Paper 1 – Q1 (a)
3 + 7 = 10 parts
20
10= 2 litres in 1 part
Fruit Juice: Water3: 7
Add the ratios together to find how many ‘parts’ there are.
Divide the total amount of Fruitex by the sum of the ratios.
Multiply the amount given out per 1 part by the number of parts of the juice.
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.
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Fruit Juice: 3 parts 3 × 2 = 6
There is 6 litres of Fruit Juice in Fruitex.
10 Marks for (a) and (b) (i)
Juicy
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Divide the 60 litre mixture into its juice and water parts.
7 + 8 = 15 parts
60
15= 4 litres in 1 part
Fruit Juice: Water7: 8
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.
40 − 22 = 18 litres of water in Juicy
Ratio of Fruit Juice: Water in Juicy= 22 : 18= 11 : 9
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.
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Fruit Juice: 7 parts 7 × 4 = 28
Water: 3 parts 8 × 4 = 32
5 Marks Main Menu
= 36.6 × 20.6= 753.96≈ 754 inches2
We need to calculate the diagonal using Pythagoras in terms of units (not inches).
𝑐2 = 𝑎2 + 𝑏2
𝑐2 = 162 + 92
𝑐2 = 256 + 81𝑐2 = 337
𝑐 = 337𝑐 = 18.3576
42 inches = 18.3576 units42
18.3576= 2.2879 inches in 1 unit.
Length
16 2.2879 = 36.6 inches
Breadth
9 2.2879 = 20.6 inches
We don’t know the length or breadth of the TV but we know the ratio of length: breadth = 16: 9Let 16 units be the length and 9 units be the breadth.
42 inches
16 units
9 units
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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.
𝐀𝐫𝐞𝐚 𝐨𝐟 𝐑𝐞𝐜𝐭𝐚𝐧𝐠𝐥𝐞
= Length × Breadth
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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 €150 000 between its strikers in proportion to the time each had played.Calculate the amount each player received.
10 Marks
Name Minutes Played
John 2250
Paul 2600
Michael 150
2250 + 2600 + 150 = 5000
John2250 × 30 = €67,500Paul2600 × 30 = €78,000Michael 150 × 30 = €4,500
150 000
5000= 30
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.
John: Paul: Michael2250 ∶ 2600 ∶ 150
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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 €140 000 if the team used this method.”
Calculate the total bonus on offer that season.
5+2 Marks
(ii)
How much each would Paul and Michael get under John’s system?
Michael:Paul:John1: 1.5: 2.5
140000
2.5= 56000 in 1 part
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.
Divide John’s bonus by 2.5 to get the amount in 1 ‘part’.
1 + 1.5 + 2.5 = 5
56000 × 5 = 280000
The total bonus is €280,000
Add the ratios to find the total number of parts.
Multiply the amount per 1 part by the total number of parts, 𝟓.
Michael56000 × 1 = €56,000
Paul56000 × 1.5 = €84,000
Multiply the amount per 1 part by number of parts for Paul and Michael.
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RATIO AND PROPORTIONSOLUTIONS
Larger: Smaller5: 2
5 parts = 250250
5= 50 is 1 part
2 × 50 = 100
The smaller piece is 100 mm long.
Larger piece is 5 ‘parts’.
Divide the length of the larger piece by 5 to get the length of 1 ‘part’.
Now multiply the share in 1 part by the number or ‘parts’ in the smaller piece, 2.
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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.
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1250 km = 68 litres
68
1250= 0.0544 litres in 1 km
68
1250× 100 = 5.44 litres in 100 km
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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.
Find out how many litres for 1 km by dividing 68 by 1250
Multiply by 100 to find car's fuel consumption for 100 km
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Peter : Anne
31
2: 2
1
2
7
2:5
2
7: 5
Find an equivalent ratio by multiplying by 2.
35000 = 7 parts
35000
7= 5000 in one part
Now multiply the amount in 1 part by the total number of parts in the fund, 𝟕 + 𝟓 = 𝟏𝟐.
Divide Peter’s share by 7 to find 1 part of the prize fund.
5000 × 12 = €60000
The total prize fund is €60,000
Peter’s share is 7 parts.
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2011 JCHL Paper 1 – Question 1 (a)
Peter and Anne share a lotto prize in the ratio 31
2to 2
1
2.
Peter’s share is €35 000.
What is the total prize fund.
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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 = 60 × 81 worker = 480 hours
For it to be built in 32 hours we will need:
480
32= 15 workers
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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?
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It will take one worker 8 times as long!
19 + 1 = 20 parts
500
20= 20 ml in 1 part
Juice: 19 parts 25 × 19 = 475
There is 475 ml of concentrated juice in Brand A.
Brand AJuice: Sugar19: 1
Add the ratios together to find how many ‘parts’ there are.
Divide the total amount of Brand A by the sum of the ratios.
Multiply the amount given out per 1 part by the number of parts of the juice.
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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?
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9 + 1 = 10 parts
300
10= 30 ml in 1 part
Sugar: 1 part
There is 30 ml of sugar in Brand B.
Brand BJuice: Sugar9: 1
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2008 JCHL Paper 1 – Question 2 (b) (ii)
What is the volume of sugar in 300 ml of brand B?
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Add the ratios together to find how many ‘parts’ there are.
Divide the total amount of Brand B by the sum of the ratios.
500 ml Brand AJuice475 mlSugar500 − 475 = 25 ml
300 ml Brand BJuice300 − 30 = 270 mlSugar30 ml
Mixture of 500 ml A and 300 ml BJuice475 + 270 = 745 mlSugar25 + 30 = 55 ml
Mixture RatioJuice: Sugar745: 55149: 11
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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?
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Area of House : Area of Site= 205 ∶ 1025= 1 ∶ 5
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2004 JCHL Paper 1 – Question 1 (a)
The area of a house covers 205 m2.The area of the site for the house covers 1025 m2.
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.
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RATIO AND PROPORTIONSOLUTIONS
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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 €20 000. The prize is divided in proportion to how much each paid. How much prize money does each person receive?
15 Marks
John: Mary: Eileen25: 15: 10
25 + 15 + 10 = 50 parts
20000
50= €400 in 1 part
John: 25 parts 400 × 25 = €10000
Mary: 15 parts 400 × 15 = €6000
Mary: 10 parts 400 × 10 = €4000
Add the ratios together to find how many ‘parts’ there are.
Divide the prize money by the sum of the ratios.
For each person multiply the amount given out per 1 part by the number of parts.
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RATIO AND PROPORTIONSOLUTIONS
140 = 4 parts
140
4= 35 in one part
Now multiply the distance in 1 part by the total number of parts in zinc, 𝟗.
Divide the amount travelled so far by 4 to find 1 part of the journey.
35 × 9 = 315 km
Let 𝒙 be the total distance. Then:
4
9𝑥 = 140
𝑥 =140
49
𝑥 = 315 km
Alternate Method
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2012 LCOL Paper 1 – Question 1 (a)
When Katie had travelled 140 km, she had completed 4
9of her journey.
Find the length of her journey.
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Aoife: Brian4: 3
4 parts = €56
56
4= €14 is 1 part
Total Prize Fund14 × 7 = €98
Aoife gets 4 parts
Divide Aoife’s share by 4 to find the amount of 1 part.
Brian gets 3 parts so multiply the share in 1 part by 3.
Now multiply the share in 1 part by the total number of parts, 𝟒 + 𝟑 = 𝟕.
Brian14 × 3 = €42
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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?
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3 + 7 = 10 parts
50
10= 5 apples in 1 part
Conor: 3 parts 5 × 3 = 15
Alice: 7 parts 5 × 7 = 35
Conor: Alice3: 7
Add the ratios together to find how many ‘parts’ there are.
Divide the total number of apples by the sum of the ratios.
For each person multiply the amount given out per 1 part by the number of parts.
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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?
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8 km = 5 miles
8
5= 1.6 km in 1 mile
1.6 × 164 = 262.4 km in 164 miles
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2007 LCOL Paper 1 – Question 1 (a)
Convert 164 miles to kilometres, taking 5 miles to be equal to 8 kilometres.
Find out how many km there are in 1 mile.
Then multiply by 164.
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320 = 4 parts
320
4= 80 in one part
Now multiply the amount in 1 part by the total number of parts in the fund, 𝟗.
Divide the amount travelled so far by 4 to find 1 part of the prize fund.
80 × 9 = €720
Let 𝒙 be the total prize fund. Then:
4
9𝑥 = 320
𝑥 =320
49
𝑥 = €720
Alternate Method
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2006 LCOL Paper 1 – Question 2 (a)
€320 is 4
9of a prize fund. Find the total prize fund.
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35
100
=7
20
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2005 LCOL Paper 1 – Question 1 (a)
Express 35 cm as a fraction of 1 m. Give your answer in its simplest form.
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1 m = 100 cm
Express 35 as a fraction of 100.
Simplify Fraction
6 + 4 + 3 = 13 parts
325
13= 25 in 1 part
6 × 25 = 150
4 × 25 = 100
3 × 25 = 75
1
2∶
1
3∶
1
4
121
2∶ 12
1
3∶ 12
1
4
6 ∶ 4 ∶ 3
Add the ratios together to find how many ‘parts’ there are.
Divide the 325 by the sum of the ratios.
For each ratio multiply the amount given out per 1 part by the number of parts.
Find an equivalent ratio by multiplying each ratio by the lowest common denominator, 𝟏𝟐.
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2005 LCOL Paper 1 – Question 1 (b) (ii)
Express the ratio 1
2∶
1
3∶
1
4as a ratio of natural numbers.
Divide 325 in the ratio 1
2∶
1
3∶
1
4.
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Copper: Zinc19: 6
133
19= 7 kg in 1 part
Zinc has 6 parts7 × 6 = 42 kg
Divide the amount of copper by the number of parts in copper, 19, to find the amount in 1 part.
Now multiply the amount given out per 1 part by the number of parts in zinc, 𝟔.
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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?
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9000
450× 15 + 15
20 × 15 + 15
315 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.
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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?
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9 kg = 9,000 grammes
400
1000
=2
5
1 kg = 1000 g
Express 400 as a fraction of 1000.
Simplify Fraction
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2000 LCOL Paper 1 – Question 1 (a)
Express 400 grammes as a fraction of 1 kilogramme. Give your answer in its simplest form.
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7 + 3 = 10 parts
40
10= €4 in 1 part
Pupil 17 × 4 = €28
Pupil 23 × 4 = €12
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 1: Pupil 27: 3
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1999 LCOL Paper 1 – Question 1 (a)
€40 is divided between 2 pupils in the ratio 7:3. How much does each pupil get?
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RATIO AND PROPORTIONSOLUTIONS