RATIO AND PROPORTION - mathspoints.iemathspoints.ie/wp-content/uploads/2018/03/Ratio-and-Proportion... · RATIO AND PROPORTION Junior Cert Higher Level. Fruitexand Juicy are each

RATIO ANDPROPORTION

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RATIO ANDPROPORTION

Junior Cert Higher Level

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.

2017 JCHL Paper 1 – Question 4 (b) (i)

3 + 7 = 10 parts

20

10= 2 litres in 1 part

Fruit Juice: 3 parts 2 × 3 = 6

There is 6 litres of Fruit Juice in Fruitex.

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.

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.

2017 JCHL Paper 1 – Question 4 (b) (ii)

Divide the 60 litre mixture into its juice and water parts.

7 + 8 = 15 parts

60

15= 4 litres in 1 part

Fruit Juice: 7 parts 4 × 7 = 28

Water: 3 parts 4 × 3 = 32

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 is:= 22 : 18= 11 : 9

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.

2015 Supplementary Sample – Question 9 (a)

Area of Rectangle= 𝑙 × 𝑏= 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

Breadth

9 2.2879 = 20.6

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

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

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.

2012 JCHL Paper 1 – Question 4 (a)


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

Michael:Paul:John1: 1.5: 2.5

140000

2.5= 56000 in 1 part

Michael56000 × 1 = €56,000

Paul56000 × 1.5 = €84,000

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.


How much each would Paul and Michael get under John’s system?

2012 JCHL Paper 1 – Question 4 (b) (ii)

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.

Multiply the amount per 1 part by number of parts for Paul and Michael.

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.

2013* JCHL Paper 1 – Question 2 (a)

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.

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.

2012* JCHL Paper 1 – Question 2 (a)

1250 km = 68 litres

68

1250= 0.0544 litres in 1 km

68

1250× 100 = 5.44 litres in 100 km

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.


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

Peter’s share is 7 parts.

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 = 60 × 81 worker = 480 hours

It will take one worker 8 times as long!

For it to be built in 32 hours we will need:

480

32= 15 workers

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?


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


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.

What is the volume of sugar in 300 ml of brand B?


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


Divide the total amount of Brand B by the sum of the ratios.

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?

2008 JCHL Paper 1 – Question 2 (b) (iii)

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

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.


Area of House : Area of Site= 205 ∶ 1025= 1 ∶ 5

RATIO ANDPROPORTION

Leaving Cert Higher Level

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?

2015 LCOL Paper 1 – Question 2 (a)

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


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.

When Katie had travelled 140 km, she had completed 4

9of her journey.

Find the length of her journey.


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

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: 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

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?


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


Divide the total number of apples by the sum of the ratios.


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

1.6 × 164 = 262.4 km in 164 miles

€320 is 4

9of a prize fund. Find the total prize fund.


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

Express 35 cm as a fraction of 1 m. Give your answer in its simplest form.


1 m = 100 cm

35

100

=7

20

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.

2005 LCOL Paper 1 – Question 1 (b) (ii)

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


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.

Copper: Zinc19: 6

133

19= 7 kg in 1 part

Zinc has 6 parts7 × 6 = 42 kg

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?


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.

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 kilogrammesbe cooked?


9 kg = 9,000 grammes

9000

450× 15 + 15

20 × 15 + 15315 minutesor5 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.

Express 400 grammes as a fraction of 1 kilogramme. Give your answer in its simplest form.


1 kg = 1000 g

400

1000

=2

5

€40 is divided between 2 pupils in the ratio 7:3. How much does each pupil get?


7 + 3 = 10 parts

40

10= €4 in 1 part

Pupil 17 × 4 = €28

Pupil 23 × 4 = €12


Divide the €40 by the sum of the ratios to find how much is in 1 ‘part’.


Pupil 1: Pupil 27: 3

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RATIO AND PROPORTION - mathspoints.iemathspoints.ie/wp-content/uploads/2018/03/Ratio-and-Proportion... · RATIO AND PROPORTION Junior Cert Higher Level. Fruitexand Juicy are each