How to work out ratios

Ratios are a basic mathematical function that you're likely using regularly without knowing. Despite this, you may still feel a sense of panic when directly asked to work out the ratio of something. You may even have flashbacks to sitting at the kitchen table trying to do your maths homework, dreading the thought of having to solve complex problems.

However, while there may be aspects of the mathematics curriculum that you don't need to use regularly, ratios are incredibly helpful in all sorts of everyday situations, from business to baking. And thankfully, when it comes to understanding and working out ratios, there's no need to panic — they're simpler than they look.

In this article, we'll cover the basics, from what ratios are and what they are used for, to how to calculate them. And to make things even easier when you come to calculating ratios, we'll also run through some commonly made mistakes to avoid. So, let's jump in!

Ratios are mathematical expressions that compare two or more numbers and show how much of each value there is present. They can be used to compare absolute values and quantities, or they can be used to compare portions of a whole value. For this reason, ratios can be classified into the following two types:

• Part-to-part ratios: these show how two or more distinct groups or entities are related to one another. For example, you may want to show how many boys and girls there are in a classroom. So, say you have a class of 20 with 14 boys and 6 girls, then the ratio of males to females is 14:6. If the number of girls in the class then changed, the ratio could be altered to reflect this.
• Part-to-whole ratios: this type of ratio shows the relationship between a smaller value and a whole entity or group. For example, if you have a group of 15 people and 5 of them like to read. In this case, the part-to-whole ratio would be 5:15 to show that five out of 15 people like reading. This can then be simplified to 1:3 because 5:15 and 1:3 are equivalent ratios.

Usually, when writing ratios, a colon (:) denotes the ratio formula. For example, you could write a ratio as 2:1. With this format, the first number is known as the antecedent, and the second is the consequent.

However, ratios can also be written in fraction form or by writing 'to' between the numbers. So, say you are cooking, and the recipe you're using requires you to mix 3 cups of flour and one cup of water. This ratio can be written using one of the following ratio formats:

• 3:1
• 3 to 1
• 3/1

When using ratios, it is important that both the antecedent and consequent are the same unit of measurement. In this example, the ratio is being used in a recipe, so depending on what ingredients are required for the recipe, you could use units like millilitres, litres, or grams — just ensure that both ingredients are measured using the same unit. It is also worth remembering that when working with ratios, the numbers should be non-zero so that the comparison between them has meaning.

Ratios crop up in daily life, and more so than you likely realise. They are frequently used to simplify numbers and put them into perspective. Real-life examples in which ratios are used include:

• Map scales: maps will use a ratio to denote the scale it has been reduced to. For example, if a map has a scale of 1:25 000, 1 unit is the equivalent of 25,000 units in real life. Therefore, 1 cm on the map denotes 25,000 cm in real life, which is 0.25 kilometres.
• Baking: as previously mentioned, ratios are used regularly in cooking to show you how much of an ingredient to use, particularly when mixing them together.
• Business: ratios can be across plenty of different aspects within businesses. One example is the use of ratios to identify the level at which products are returned to a business by customers. Businesses can keep track of the number of products sold, and the number of products returned. This can then be expressed as a ratio. For example, if the ratio of bought to returned products is 300:1, it means that for every three hundred products sold, only one is returned.
• Construction: in building and construction, ratios can be used for several things. For example, mixing cement requires construction workers to use ratios to identify how much of each component they need to use. Alternatively, ratios are used when making scale model of a building. In both these scenarios, ratios are crucial to avoid making costly mistakes, whether that's in terms of safety or money.

The first thing you'll need to understand when working out ratios is the ratio formula. This is written as:

a:b = a/b

You can then use this formula to work out other ratio problems — we'll explain how to do this in the next few sections.

Finding equivalent ratios

Equivalent ratios are ratios that can be simplified or reduced by the same number. For example, 9:4 is an equivalent ratio of 18:8 because the values in the second ratio both have the common factor 2.

So, if you have been given the two ratios 10:8 and 30:24 and you need to find out whether they are equivalent, here are the steps you can take:

1. Write down the ratios in fractional form: this would be 10/8 and 30/24.
2. Cross-multiply the values: in this case, you'd multiply 10 by 24 and 8 by 30.
3. Identify whether the products are equal: if both results are the same, it means the ratios are equivalent. These two ratios are equal as 10 × 24 = 8 × 30 = 240.

Simple ratio calculation

Say you're given a problem where you need to share £50 between two people in the ratio of 2:3. The easiest method of calculating such a ratio is to do the following:

1. Calculate how many parts are in the ratio: when using ratio formula, the sum of 'a' and 'b' would give you the total number of parts. In this example, the sum of these two values is 5.
2. Calculate the value of one part: to do this, you'd divide the whole value, by the number of parts. Therefore, you'd divide £50 by 5 parts. Each part is then worth £10.
3. Scale the ratio: you'd then multiply the values of 'a' and 'b' by the value of one part. This is 2 x £10 and 3 x £10, so one person gets £20, and the other gets £30.

Finding unknown quantities from an existing ratio

If you have a problem where you need to find an unknown quantity, and you have a ratio where you know one of the values, you can use the following steps to solve it:

1. Identify the different values in the equation: for example, if you're given two values, and the ratio of 'a' to 'b' is 3:5. If you know that the value of 'a' is 54, you'll need to calculate the value of 'b'.
2. Input the values into the formula: in this case, you'd write the formula as 54:b = 3/5
3. Solve the equation: to solve this equation, you'd need to convert both sides of the equation into fractions, which would be 54/b = 3/5. The simplest way to then solve this equation is to work out the size increase in the value of 'a'. In this equation, this is the increase from 3 to 54. You can use division to work this out, so the calculation is 54/3 = 18. As each side of the equation is equal, you can then multiply 5 by 18 to get the value of 'b', which would be 90. This makes the workings out for this equation:
   1. 54/b = 3/5
   2. increase in 'a' = 54/3
   3. 54/3 = 18
   4. therefore 'b' = 5 x 18
   5. b = 90
   6. final ratio is 54:90 = 3:5
4. Check to see whether the ratio can be shown in a simplified form: when calculating ratios, you'd always finish with this step. However, in this example, the ratio is already in its simplest form.
   

If you're working with ratios, here are some things to be mindful of:

• Ensure the values in the ratio are written in the right order: For example, if there are 24 boys in a room and 12 girls, it's important that when writing the ratio of boys to girls, the numbers are the right way round. In this example, it should be 24:12 rather than 12:24.
• Use the same units: when working with ratios, it's important to use the same unit of measurement, even if the quantities are different. In some cases, you may need to convert units before calculating the ratio.
• Reading the ratio in the right order: this is similar to ensuring that ratios are written in the correct order. If you're working from an existing ratio, check you have read the values correctly before getting stuck into calculating anything.

Ratios are a way of comparing values and putting them into perspective. They're incredibly useful, not only across a variety of different professions but also just in everyday life. So, while it may have been a while since you last actively sat down and had to calculate a ratio problem, it is worth brushing up on your maths skills.

The best way to do this is to start with the basics, such as how to simplify ratios using division. From here, you can move on to calculating ratios and finding missing values using the steps we covered earlier. And rest assured, it's never too late to learn how to do these things!