Like many ideas in mathematics, the concepts of ratio and proportion involve more than just numbers themselves – they also involve the relationships between numbers.

These relationships produce some fundamental information about the world. Scales and balanced proportions affect our interpretations of beauty in architecture and art, help businesses plan their strategies and even dictate how a bartender mixes their drinks.

Pupils need a working understanding of all this to succeed at maths. That’s why ratios and proportions form part of the KS2 syllabus. But while these core concepts can seem so straightforward to adults that they’re almost not worth thinking about, they can sometimes prove really tricky for younger minds.

Let’s explore the basics of proportions and work through some ratio questions at a Year 6 level.

## Working with ratio and proportion* *

When you follow a recipe (or even just make a glass of squash), you need to mix ingredients in the correct **ratio** to get the right result. The ratio is often given in terms of ‘parts’. The size of a ‘part’ depends on how much of the mixture you need.

Let’s examine this further with some example questions.

**The instructions for mixing purple paint say ‘Mix 1 part red with 4 parts blue’. **As a ratio, we would express this as 1:4 (one to four).

#### 1. How much blue paint should you mix with 500ml of red paint?

If 500ml of red paint is just one part of the whole, then we will need four times this amount of blue paint. We simply multiply 500 by 4 to produce **2000ml **or** 2 litres of blue paint**.

**2. How much red paint should you mix with 500ml of blue paint? **

If 500ml of blue paint makes up **four parts** of the whole, then we will need to **divide 500 by 4** to find the remaining single part. This shows that we need **125ml of red paint**.

### Direct proportion

Sometimes instructions give the amount of each ingredient rather than using ‘parts’.

If you need more (or less) of the mixture, then you need to increase (or decrease) the quantity of each ingredient **in proportion.**

For example:

To **double** the amount, **multiply everything by 2**

To make **five** times as much, **multiply everything by 5**

To make **half** the amount, **divide everything by 2**

To make a **third** of the amount, **divide everything by 3**

### Let’s try out an example with a real-life recipe

**Mushroom Soup (serves 8) **

- 450g mushrooms
- 2 onions
- 2 potatoes
- 1420ml milk
- 100g butter
- 10ml thyme
- 2 sprigs of parsley
- Salt & pepper to taste

#### How much of each ingredient do you need to make mushroom soup for 16 people?

We already know that the recipe serves **8 people**. So to serve twice as many people, you would need to **double** the amount of soup.

That would leave us with a new recipe of:

- 900g mushrooms
- 4 onions
- 4 potatoes
- 2840ml milk
- 200g butter
- 20ml thyme
- 4 sprigs of parsley
- Salt & pepper to taste

## Calculating with ratios

Maths at KS2 requires pupils to be able to look at ratios and simplify them in order to calculate amounts.

As we’ve established, you can express a ratio like ‘1 part plant to 6 parts water’ as **1:6**. This form is especially useful for simplifying ratios with large numbers and units of measurement.

Remember: **to simplify a ratio, divide both sides by the same number**.

### Let’s play around with a KS2 ratio question:

Suppose a recipe for pastry uses **100g** of butter and **200g** flour.

The ratio of butter to flour is **100:200**.

(When the units are the same you do not need to write them in the ratio.)

In this case, you can **divide by 100** (or by 10 then 10 again, or by 5 then 20).

This means that the ratio of butter to flour is **1:2, **and there is twice as much flour as butter.

This ratio can be applied to any quantity in the same way.

## TuitionWorks is here to help

If you or your child is still feeling less than confident about ratios, proportion and other aspects of the primary school maths syllabus, TuitionWorks can provide an intensive course of personalised, one-to-one lessons in KS2 maths from a qualified teacher like me. Just get in touch for a free consultation.

### Andrew Hartshorn

Maths tutor at TuitionWorks

I have over twenty years’ experience of teaching both children and adults. I trained as a primary school teacher after a spending a number of years abroad teaching English as second language.

After qualifying with a PGCE in 2004, I attained my Masters Degree in Education. I believe in keeping my skills sharp and recently completed an online writing course with Harvard University.