# Expressing Relationships

## Multiplicative relationships as a ratio

Multiplicative relationships as ratios are when numbers are connected by multiplication. For example, to go from list A to list B you need to multiply the A’s by 8

List A => List B

5 => 40

8 => 64

-1 => -8

0 =>0

The numbers have a multiplicative relationship. When it comes to ratio being able to spot and understand multiplicative relationships is very important. It often allows us to scale up or scale down amounts.

For example:

Fred uses yellow and blue paint to make a specific colour green which requires a ratio of 2:5

How many tins of yellow paint will he need to make 10 tins of yellow?

So firstly let’s think about what information we have.

Yellow:Blue

We want to find out how many blue we will need if we have 10 tins of yellow.

Therefore we need to think about what we multiply 2 by to get to 10? It’s 5 therefore we must also multiply 5 by 5. The total number of Blue tins is therefore 25.

## Using proportion as ratio

Using proportion as ratio is a very important skill and is one that we use a lot in real life- especially in the kitchen!

When ratios are__ in proportion__ it means that the ratios are **equal**.

Therefore a:b = c:d or a/b = c/d

A real life example of this is changing recipe amounts.

This recipe is for 10 people:

- 5 eggs
- 50 g of butter
- 250g of flour
- 100g of sugar

I want to find out much butter I would need in a recipe for 6 people.

Given the multiplicative relationships between ration I just need to multiply both sides by 6/10 or 0.6.

It is important to set you work out carefully and think about what you are multiplying by!

## Relate ratios to fractions and linear functions

- When we have a ratio between two things we can also show it as an equation.

__Example 1: __the ratio of Blue to Red tiles is always in the ratio 1:6.

That means for every 1 blue tile we have we will have 6 red tiles.

Therefore the number of Red tiles = the number of blue tiles x 6

R=6B

__Example 2: __Sarah and Jess are paid in a ratio of £2:£6 find a linear equation to represent the relationship of their pay?

Step 1: Simplify the ratio 2:6 = 1:3

Step 2: Now we know Sarah gets £1 for every £3 Jess gets.

Step 3: Create an equation J = 3S. Jess gets paid= 3 times as much as Sarah

Step 4: Check using the first ratio if S= 2 how much will J get? 3x2=£6 so we are correct!

- 1. Theo and Reiss share biscuits in a ratio of 3:4. How many biscuits will Reiss has if Theo has 9?
- 12
- A recipe require 6 eggs for 12 people. How many eggs will be needed in a recipe for 4 people?
- 2