How to Calculate Ratios: A Step-By-Step Guide
Updated 5 September 2021
What Are Ratios?
A ratio is a mathematical term used to describe how much of one thing there is in comparison to another thing.
Ratios are usually written in the following formats:
- 2 to 1
Used in mathematics and everyday life, you may have come across ratios without knowing it – for example in scale drawings or models, in baking and cooking, and even when converting currency for a holiday abroad.
Ratios are useful when you need to know how much of one thing there needs to be in comparison with another thing.
In a bag of 20 sweets, the ratio of blue to pink might be 2:3
The use of ratio in this example will inform us that there would be 8 blue sweets and 12 pink sweets. (This question and the way to work it out is detailed below).
If you are making a cake, and you require 3 cups of flour and 2 cups of sugar to make enough to feed 10 people, then you can express that as the ratio 3:2.
To scale the ingredients up to feed 20 people (to double the recipe size) you need to double the ingredients – so you would need 6 cups of flour and 4 of sugar (or 6:4).
Key Ratio Facts
Ratios can describe quantity, measurements or scale.
When describing a ratio, the first number is known as the ‘antecedent’ and the second is the ‘consequent’. So, in the ratio 3:1, the antecedent is 3 and the consequent is 1.
Ratios should always be presented in their simplified form. You can simplify a ratio by dividing both sides by the highest common factor. For example, 12:4 simplified would be 3:1 – both sides of the ratio divided by 4.
Equivalent ratios can be divided and/or multiplied by the same number on both sides, so as above, 12:4 is an equivalent ratio to 3:1.
Ratios can inform you of the direct proportion of each number in comparison to the other. For example, when a pair of numbers increase or decrease in the same ratio, they are directly proportional.
When expressing ratios, you need to ensure that both the antecedent and the consequent are the same units – whether that be cm, mm, km. This makes it easier to use the ratio to inform your calculations.
Ratios are used in maps to provide scale. Usually expressed as 1:10,000 or similar, this tells you that for every 1 unit on the map, the real distance is 10,000 units. If you measure 1 cm on the map, the real distance would be 10,000 cm (or 100 m).
Ratios are also used in drawings, such as architectural designs, to show perspective and relative size on a smaller scale, and in models. For example, a model car might have a ratio of 1:20 – so 1 cm on the model would be 20 cm on the actual car.
Sample Questions and Their Solutions
Understanding how to work out ratios is an important skill and can be particularly useful when applying for jobs where a good understanding of mathematics is required.
Here are the key ratio skills that you need to master:
1. How to Find the Ratio of Two Things
In a bag of 20 sweets, there are 8 blue sweets and 12 pink sweets. What is the ratio of blue to pink sweets?
This problem gives us all the information we need to express the ratio:
8 blue : 12 pink = 8:12
Remember, all ratios should be simplified where possible, so divide both the antecedent and consequent by the highest common factor – in this case, the highest number that goes into both 8 and 12 is 4.
8 divided by 4 = 2
12 divided by 4 = 3
Therefore the correct answer is: 2:3
2. How to Convert Ratios With Different Units
How should the scale factor of 3 cm : 15 m be expressed as a simplified ratio?
Firstly, we need to ensure that the units we are using are the same. Ratios need to presented with no denomination, so both the antecedent and the consequent must be shown as either centimeters or meters.
Because whole numbers are preferred, it is easier to present this scale factor as centimeters.
So, the ratio would be 3:1,500
The highest common factor in this ratio is 3. Both numbers can be divided by three with none remaining, so the simplified ratio would be:
3 divided by 3 = 1
1,500 divided by 3 = 500
So, the correct answer is: 1:500
3. Working With Ratios That Include Decimals
To easily work with ratios, whole numbers are necessary. The easiest way to make this ratio include whole numbers is to multiply both sides by the same number – in this case, 2 makes sense.
10 x 2 = 20
2.5 x 2 = 5
Our whole number ratio, therefore, will be 20:5
The highest common factor is 5 – both sides can be equally divided by 5:
20 divided by 5 = 4
5 divided by 5 = 1
Therefore, the correct answer is: 4:1
4. Using Ratios to Work out the Direct Proportions of Quantity
If you go to the shop and buy 4 apples for £0.64, how much would 11 apples cost?
This might not look like a problem where ratios could help but considering this problem by expressing the given numbers as a ratio will help you to solve the problem.
The numbers that are directly proportionate increase in the same ratio.
Here, we know that 4 apples cost £0.64
£0.64 divided by 4 = £0.16
So, we know that 1 apple costs £0.16
You now have a ratio that you can use to find your answer:
To keep the proportions correct, we need to multiply both parts of the ratio by the required number – in this case, 11.
1 x 11 = 11
16 x 11 = 176
The ratio is: 11:176
Therefore, the correct answer is: 11 apples cost 176p or £1.76
5. How to Divide a Number by a Ratio
Andrew and James have 400 sweets and they need to share them in the ratio 5:3. How many sweets does each of them receive?
Sharing these sweets in the given ratio means we need to find out exactly how many each person will get.
We know that the sweets will be divided into 8 equal portions (5 + 3 = 8). For every 5 sweets that Andrew gets, James will get 3.
As 8 is the total number of portions, we need to divide 400 by 8 – which gives us 50.
Our ratio was 5:3, so we need to multiply both parts by 50 to get our answer:
5 x 50 = 250
3 x 50 = 150
The correct answer is: Andrew would get 250 sweets and James would get 150
6. How to Use Ratios to Find an Unknown Number
The sweet company likes to put uneven numbers of sweets in their bags. They are currently creating a bag of blue and pink sweets in the ratio 4:6.
If you get a bag with 12 blue sweets in it, how many will there be in total?
To work this one out, it is necessary to divide the number of blue sweets by the ratio, to find the common denominator.
12 divided by 4 = 3
We can then multiply the consequent number by 3 to find the unknown quantity:
6 x 3 = 18
Therefore, if there are 12 blue sweets, there will be 18 pink ones (or 12:18)
To find the total number of sweets, we need to add both sides of the scaled-up ratio 12:18
12 + 18 = 30
The correct answer is: There are 30 sweets in total in the bag.
Ratios are a mathematical expression to compare units.
They can be used as equivalent ratios to help you scale up numbers – for example, quantities of ingredients for baking a cake.
In mathematical terms, they can be used to work out problems relating to direct proportion, where the increase or decrease in units occurs in the same ratio.
Ratios can be simplified and, in most cases, it is preferable to give a simplified ratio as an answer. Like fractions, you can simplify a ratio by dividing it by the highest common factor.
When using scales on drawings or models, ratios help to describe the relationship between the real-life item and the created one – allowing for accurate measurements as well as an idea of proportion.
When trying to understand ratios, it is easiest to work with the same units.
Remember, to fully explore a ratio, you need to use a whole number, so try to avoid creating any decimals when you are transforming units to match.
Practicing ratio problems will make them much easier to understand.
It is likely you will use ratios throughout your life and might be tested on math skills like these when applying for jobs in technical industries.