There are a number of occasions when we may see a **relationship** expressed as a **multiple**.

For example, we could say,* "this bottle holds three times as much as that one."*

This is an example of a **multiplicative relationship**.

One of the essential skills you need to master to be successful with questions involving multiplicative relationships, is to know how to write them as either a **ratio **or a** fraction**.

Let's look at this in action now.

**e.g. One jug of water is three times larger than another. Express this relationship as a ratio.**

We know that all ratios are written as the form **a:b**, where **a **and **b **are whole numbers.

In this case, we can write this relationship as **1:3** or **3:1**.

This question doesn't specify which one is bigger out of the jugs, so it is acceptable to use either order in our ratio here.

**e.g. I have two pieces of wood. Piece A is 60 cm and piece B is 80 cm long.**

**a) Write the lengths of A:B as a ratio in its simplest form.**

We start by writing each number simply as a ratio, in the order expressed in the question:

**60:80**

We can then cancel this ratio down into its simplest form by finding the Highest Common Factor (HCF) of both numbers.

Here the HCF is **20**:

60 ÷ 20 = **3**

80 ÷ 20 =** 4**

So the simplest possible form of this ratio is **3:4**.

**b) Find the length of B as a fraction of A.**

In this question, we are asked to find B as a fraction of A, so B must go on the top of the fraction and A on the bottom, like this:

80 |

60 |

This fraction can then be cancelled down if we divide by a HCF of **20**:

80 ÷ 20 |
= | 4 |

60 ÷ 20 | 3 |

In this activity, we will express multiplicative relationships in terms of ratios and fractions, cancelling them into their simplest by finding the HCF or using them to find missing variables.