Ratios

Ratios compare the values of things. They tell us how much of one thing is made up of another thing, or how much of one thing there is compared to another thing.

For example,

in the picture, there are 3 red jelly babies and 2 green jelly babies. The ratio of red to green jelly babies is \(3:2\), but we can soon fix that (I love red jelly babies)!

How about in this picture?

There are 6 red jelly babies to 4 red jelly babies, so the ratio of red to green jelly babies is \(6:4\). But this simplifies to \(3:2\) as both \(6\) and \(4\) have a factor of \(2\), so we can divide both sides of the ratio by \(2\).

Simplifying (or complicating) Ratios

We can make equivalent ratios in much the same way as we make equivalent fractions. We just have to make sure we multiply (or divide) both parts of the ratio by the same amount.

For example, \(6:9\) is the same as \(6 \times 3: 9 \times 3 = 18:27\) and the same as \(6 \div 3 : 9 \div 3 = 2: 3\).

Example: A Recipe for Puffed Rice Crackles

Christo and Sam have a lot of allergies, so we need to make our own, allergy-free, versions of many foods. Here's a recipe they particularly like for a treat.

So, why have I put this recipe in the middle of an article on ratios? I'm glad you asked. The reason is that we can use it to find some ratios.

Example

What is the ratio of butter to marshmallows in the Puffed Rice Crackles Recipe?

Solution:

There are \(80\) g of butter and \(200\) g of marshmallows, so the ratio is

Example

What is the ratio of marshmallows to puffed rice in the Puffed Rice Crackles Recipe?

Solution:

There are \(200\) g of marshmallows and \(120\) g of puffed rice, so the ratio is

We can combine these two ratios to give a ratio of the three ingredients butter : marshmallows : puffed rice of \(2:5:3\).

This ratio can be used to work out the quantities of butter, marshmallows and puffed rice if you only have a limited amount of one of these ingredients.

For example, if I only have \(100\) g of marshmallows, I know that I need \(100 \div 5 = 20 \) g for each part of each ingredient in the recipe, so I need \(2 \times 20 = 40) g of butter and \(3 \times 20 = 60\) g of puffed rice to make some puffed rice crackles.

Of course, if I want to make a triple quantity of the recipe, I can multiply the original quantities by \(3\) to give requirements of \(600\) g of marshmallows, \(360\) g of puffed rice and \(240\) g of butter, and I can also multiply each part of the ratio \(80:200:120\) by \(3\). I still keep my ingredients in the same ratio, so my puffed rice crispies will taste the same.

Ratio and Proportion

In our previous examples, we've only found ratios of part of something to another part of something.

A proportion is a ratio of a part to a whole. In our rice bubble crackle recipe, the proportion of marshmallows is \(200: 200 + 120 + 80 = 200: 400 = 1:2\).

In the jelly baby examples, the proportion of red jelly babies is \(3:3 +2 = 3:5\). Yes, I like red jelly babies the best.

One for You to Do

Scale Models and Scale Drawings

We can use ratios to make scale drawings or models of real-life objects. This has the advantage that we can fit them on paper, or in our display cabinet, but we still maintain the proportions of the original objects.

Example

Let's look at one last example. This one is about the models that Sam and Christo like to build.

Example