Simple Ratio - League of Learning

Ratio, Proportion & Rates of Change

Ratios

What is a ratio?

A ratio is a way of comparing one part of a whole to another
A ratio can also be expressed as a fraction (of the whole)
We often use a ratio (instead of a fraction) when we are trying to show how things are shared out or in any situation where we might use scale factors
If a pizza were sliced into 8 pieces, and shared in the ratio 6:2, this means that person A receives 6 slices, and person B receives 2 slices

How do I simplify a ratio or find an equivalent ratio?

Using the same example of 8 slices of pizza shared in the ratio 6:2
We could also express this ratio as 3:1, even though we still have 8 slices
- Both sides of the ratio have been divided by 2
  - The amount that each person gets relative to the other is still the same
  - In this case, person A receives 3 times as much as person B
  - 3:1 is “simpler” than 6:2, so we can say that 6:2 simplifies to 3:1
We could also write this ratio as 12:4, 18:6, 24:8, 30:10 and so on
- When finding an equivalent ratio, we must multiply or divide both sides of the ratio by the same number
- For a giant pizza with 800 slices, person A would receive 600 slices and person B would receive 200 slices
  - Both sides of the ratio have been multiplied by 100
- We can keep doing this as long as person A receives 3 times as much as person B
You can think of this process as similar to finding equivalent fractions, or simplifying fractions
- However it is important to note that 1:4 is NOT equivalent to $1 fourth$

How do I use a ratio to find a fraction?

We can use a ratio to find a fraction of the whole amount
Using the same example of 8 slices of pizza shared in the ratio 6:2
Person A receives 6 slices out of 8, or $6 over 8$ of the pizza
- This could be simplified to $3 over 4$ of the pizza
Person B received 2 slices out of 8, or $2 over 8$ of the pizza
- This could be simplified to $1 fourth$ of the pizza
These fractions could also then be converted to percentages if needed

Exam Tip

When finding equivalent ratios, write down what you are doing to both sides, this will help when you come to check your work
- e.g. “×2” or “÷3”
Whilst 3:12 is not the same as $3 over 12$ , you could still type $3 over 12$ into your calculator which would simplify it to $1 fourth$ for you and give you the simplified ratio numbers
- So 3:12 = 1:4

Worked example

Some money is shared between Amber and Beatrice in the ratio 8:12 respectively.

Simplify this ratio.

Divide both sides of the ratio by 4.

$open parentheses 8 divided by 4 close parentheses space colon space open parentheses 12 divided by 4 close parentheses$

2:3

Find the fraction of the money that each person receives. Give your answers in simplest form.

Find the total number of “parts”.

8+12=20

Amber receives 8 parts out of 20. As a fraction this is

$8 over 20$

By dividing the top and bottom by 2, this is equivalent to

$4 over 10$

And dividing the top and bottom by 2 again gives the simplest form.

$2 over 5$

Beatrice receives 12 parts out of 20. Turn this into a fraction and simplify it in the same way as for Amber.

$12 over 20 space equals space 6 over 10 space equals space 3 over 5$

Amber receives $bold 2 over bold 5$ of the money, and Beatrice receives $bold 3 over bold 5$

You could also do this by using the simplified ratio 2:3 from part (a).
That would have given the simplified versions of the fractions directly!

Worked example

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, as 200 ÷ 8 = 25
Some students find it helpful to show this in a simple diagram
- $box enclose $ 25 end enclose space box enclose $ 25 end enclose space box enclose $ 25 end enclose space box enclose $ 25 end enclose space box enclose $ 25 end enclose space colon space box enclose $ 25 end enclose space box enclose $ 25 end enclose space box enclose $ 25 end enclose$
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

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
- $box enclose blank end enclose space box enclose blank end enclose space box enclose blank end enclose space colon space box enclose blank end enclose space box enclose blank end enclose space box enclose blank end enclose space box enclose blank end enclose space box enclose blank end enclose$
The difference in the number of parts of the ratio is 2 (5 – 3 = 2)
- $box enclose blank end enclose space box enclose blank end enclose space box enclose blank end enclose space colon space box enclose blank end enclose space box enclose blank end enclose space box enclose blank end enclose space box enclose blank end enclose bold space box enclose blank end enclose$
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)
- $box enclose blank end enclose space equals space box enclose 6 space cars end enclose$
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
- $box enclose 6 space cars end enclose space box enclose 6 space cars end enclose space box enclose 6 space cars end enclose space colon space box enclose 6 space cars end enclose space box enclose 6 space cars end enclose space box enclose 6 space cars end enclose space box enclose 6 space cars end enclose space box enclose 6 space cars end enclose$
- 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

$box enclose blank end enclose space box enclose blank end enclose space colon space box enclose blank end enclose space box enclose blank end enclose space box enclose blank end enclose$

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

$box enclose 2.5 space litres end enclose space box enclose 2.5 space litres end enclose space colon space box enclose 2.5 space litres end enclose space box enclose 2.5 space litres end enclose space box enclose 2.5 space litres end enclose$
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

Exam Tips

Adding labels to your ratios will help make your working clearer and help you remember which number represents which quantity e.g. $table row straight A colon straight B row 3 colon 4 end table$

Worked example

A large box contains 80 packets of crisps. The packets in the box are in the ratio

ready salted : salt & vinegar : cheese & onion = 4 : 7 : 5

Determine the number of packets of each type of crisp that are in the box.

Add the ratio numbers together to find the total number of parts.

4 + 7 + 5 = 16 parts

Divide 80 by 16 to find the number of packets in 1 part.

80 ÷ 16 = 5 packets per part

Now multiply that answer by the ratio numbers to find the number of packets of each type.

ready salted: 5 × 4 = 20

salt & vinegar: 5 × 7 = 35

cheese & onion: 5 × 5 = 25

20 packets ready salted, 35 packets salt & vinegar, 25 packets cheese & onion

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

Alfred eats 42 cabbage leaves

Find the number eaten by Bob.

5 × 6 = 30

Bob eats 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