Sharing in a Ratio

How do I share an amount into a ratio?

Suppose that $200 is to be shared between two people; A and B, in the ratio 5:3
- There are 8 “parts” in total, as A receives 5 parts and B receives 3 parts
- $200 must be split into 8 parts, so this means that 1 part must be worth $25
  - 200 ÷ 8 = 25
Some students find it helpful to show this in a simple diagram
- $$ 25 $ 25 $ 25 $ 25 $ 25 : $ 25 $ 25 $ 25$
Person A receives 5 parts, each worth $25
- 5 × 25 = $125 for person A
Person B receives 3 parts, each worth $25
- 3 × 25 = $75 for person B
It is worth checking that the amount for each person sums to the correct total
- $125 + $75 = $200

Exam Tip

Adding labels to your ratios will help make your working clearer and help you remember which number represents which quantity e.g. $\begin{matrix} A & : & B \\ 3 & : & 4 \end{matrix}$

Worked example

A particular shade of pink paint is made using three parts red paint, to two parts white paint.
Mark needs 60 litres of pink paint in order to decorate a room in his house.

Calculate the volume of red and white paint that Mark needs to purchase in order to have enough paint to decorate the room.

The ratio of red to white is

3 : 2

Adding these together gives the total number of parts, which we also know needs to be 60 litres

3 + 2 = 5

∴ 5 parts = 60 litres

Divide (both sides) by 5 to find out the number of litres in one part

$\begin{array}{rcl} 5 parts & = & 60 litres \\ \div 5 & = & \div 5 \\ 1 part & = & 12 litres \end{array}$

The ratio was 3:2, so multiply both number of parts by 12

$\begin{matrix} R & : & W \\ 3 & : & 2 \\ \begin{matrix} \times 12 & ↓ \end{matrix} & \begin{matrix} ↓ & \times 12 \end{matrix} \\ 36 & : & 24 \end{matrix}$

Answer in context, making sure you make it clear which value is associated with which colour paint

Mark will need to buy 36 litres of red paint and 24 litres of white paint.

Checking we can see that 36 + 24 = 60 which is the total litres of pink paint Mark requires.

Ratio & Proportion

What type of problems will I solve using ratios?

If you are following these revision notes in order you would have already come across many styles of problems involving ratio
- Writing ratios
- The link between ratios and fractions (they are not the same though!)
- Equivalent ratios
- Simplifying ratios
- Sharing in a ratio (total given)
Other problems that could crop up are
- ratios where you are given the difference between the two parts
  - e.g. Kerry is given $30 more than Kacey who is given $50
- ratios where one part is given and you have to find the other part
  - e.g. Kerry and Kacey are sharing money in the ratio 8 : 5, Kacey gets $50
  - This is similar to sharing in a ratio but a part, rather the whole, is given
- situations where you are given two separate (two-part) ratios but can combine them in to one (three-part) ratio
  - e.g. Kerry and Kacey are sharing money in the ratio 8 : 5 whilst Kacey is also sharing money with Kylie in the ratio 1 : 2
  - The link between the two separate ratios is Kacey

What do I do when given the difference in a ratio problem?

Rather than being told the total amount to be shared, you could be told the difference between two shares
Suppose that in a car park the ratio of blue cars compared to silver cars is 3:5, and we are told that there are 12 more silver cars than blue cars
Some students find it helpful to show this in a simple diagram
- $:$
The difference in the number of parts of the ratio is 2 (5 – 3 = 2)
- $:$
The difference in the number of cars is 12
- $box enclose blank end enclose space box enclose blank end enclose$ = 12 cars
This means that 2 parts = 12 cars
We can simplify this to 1 part = 6 cars (by dividing both sides by 2)
- $= 6 cars$
Now that we know how much 1 part is worth, we can find how many cars of each colour there are, and the total number of cars
- $6 cars 6 cars 6 cars : 6 cars 6 cars 6 cars 6 cars 6 cars$
- 3 parts are blue
  - 3×6=18 blue cars
- 5 parts are silver
  - 5×6=30 silver cars
- 8 parts in total
  - 8×6=48 cars in total

Given one part of a ratio, how can I find the other part?

Rather than being told the total amount to be shared, you could be told the value of one side of the ratio
Suppose that a fruit drink is made by mixing concentrate with water in the ratio 2:3, and we want to find how much water needs to be added to 5 litres of concentrate
Some students find it helpful to show this in a simple diagram

We are told that there are 5 litres of concentrate, and it must be mixed in the ratio 2:3
This means that the two parts on the left, are equivalent to 5 litres

$box enclose blank end enclose space box enclose blank end enclose$ = 5 litres

This means that 1 part must be equal to 2.5 litres (5 ÷ 2 = 2.5)

$box enclose blank end enclose$ = 2.5 litres

Now that we know how much 1 part is worth, we can find how many litres of water are required, and the total amount of fruit drink produced

$2.5 litres 2.5 litres : 2.5 litres 2.5 litres 2.5 litres$
3 parts are water

3 × 2.5 = 7.5 litres of water

5 parts in total

5 × 2.5 = 12.5 litres of fruit drink produced in total

How do I combine two ratios to make a 3-part ratio?

Sometimes you may be given two separate ratios, that link together in some way, so that you can form a 3-part ratio
Suppose that on a farm with 85 animals

The ratio of cows to sheep is 2:3
The ratio of sheep to pigs is 6:7
We want to find the number of each animal on the farm

We can’t just share 85 in the ratio 2:3 or 6:7, because these ratios don’t account for all the animals on the farm on their own

We need to find a combined, 3-part ratio that shows the relative portions of all the animals together

Notice that sheep appear in both ratios, so we can use sheep as the link between the two

C:S = 2:3 and S:P = 6:7
We can multiply both sides of the C:S ratio by 2, so that both ratios are comparing relative to 6 sheep
C:S = 4:6 and S:P = 6:7
These can now be joined together
C:S:P = 4:6:7

We can now use this to share the 85 animals in the ratio 4:6:7

There are 17 parts in total (4 + 6 + 7 = 17)
Each part is worth 5 animals (85 ÷ 17 = 5)
There are 20 cows (4 × 5), 30 sheep (6 × 5), and 35 pigs (7 × 5)

Exam Tip

Adding labels to your ratios will help make your working clearer and help you remember which number represents which quantity e.g. $\begin{matrix} A & : & B \\ 3 & : & 4 \end{matrix}$

Worked example

The ratio of cabbage leaves eaten by two rabbits, Alfred and Bob, is 7:5 respectively. It is known that Alfred eats 12 more cabbage leaves than Bob for a particular period of time. Find the total number of cabbage leaves eaten by the rabbits and the number that each rabbit eats individually.

The difference in the number of parts is

7 - 5 = 2 parts

This means that

2 parts = 12 cabbage leaves

Dividing both by 2.

1 part = 6 cabbage leaves

Find the total number of parts.

7 + 5 = 12 parts

Find the total number of cabbage leaves.

12 × 6 = 72

72 cabbage leaves in total

Find the number eaten by Alfred.

7 × 6 = 42

42 cabbage leaves

Find the number eaten by Bob.

5 × 6 = 30

30 cabbage leaves

A particular shade of pink paint is made using 3 parts red paint, to two parts white paint.

Mark already has 36 litres of red paint, but no white paint. Calculate the volume of white paint that Mark needs to purchase in order to use all of his red paint, and calculate the total amount of pink paint this will produce.

The ratio of red to white is

3:2

Mark already has 36 litres of red, so

36 litres = 3 parts

Divide both sides by 3.

12 litres = 1 part

The ratio was 3:2, so finding the volume of white paint, 2 parts.

2 × 12 = 24

24 litres of white paint

In total there are 5 parts, so the total volume of paint will be.

5 × 12 = 60

60 litres in total

In Jamie’s sock drawer the ratio of black socks to striped socks is 5:2 respectively. The ratio of striped socks to white socks in the drawer is 6:7 respectively.

Calculate the percentage of socks in the drawer that are black.

Write down the ratios.

B:S = 5:2
S:W = 6:7

S features in both ratios, so we can use it as a link.
Multiply the B:S ratio by 3, so that both ratios are comparing to 6 striped socks.

B:S = 15:6
S:W = 6:7

Link them together.

B:S:W = 15:6:7

Find the total number of parts.

15 + 6 + 7 = 28

This means 15 out of 28 socks are black.
Find 15 out of 28 as a percentage; first convert to a decimal.

$\frac{15}{28} = 0.535 714 285 . . .$

Multiply by 100 and round to 3 significant figures.

53.6 % of the socks are black

Sharing in a Ratio (CIE IGCSE Maths: Core)

Revision Note