Equivalent Ratios

Learn about ratios that represent the same relationship

1. What are Equivalent Ratios?

The Big Idea

Equivalent ratios are ratios that show the same relationship even though they use different numbers.

Example 1: Egg and Cheese Sandwiches

There are 8 egg sandwiches and 4 cheese sandwiches.

Ratio = 8 : 4

Let's group them in twos:

Ratio = 4 : 2 (same relationship!)

Let's group them in fours:

Ratio = 2 : 1 (simplest form!)

8 : 4 = 4 : 2 = 2 : 1

These are all equivalent ratios!

2 : 1 is the ratio in its simplest form.

Example 2: Squares and Circles

There are 12 squares and 8 circles. Find the ratio of the number of squares to the number of circles in its simplest form.

Original ratio = 12 : 8

Divide both by 2: 12 : 8 → 6 : 4

Equivalent ratio = 6 : 4

Divide both by 2 again: 6 : 4 → 3 : 2

Simplest form = 3 : 2

12 : 8 = 6 : 4 = 3 : 2

3 : 2 means that for every 3 squares there are 2 circles

2. Simplifying Ratios

How to Simplify

To simplify a ratio, divide every term by their highest common factor (HCF).

Example 3: Motorcycles and Bicycles

There are 5 motorcycles and 15 bicycles in a car park. Find the ratio of the number of motorcycles to the number of bicycles in its simplest form.

HCF of 5 and 15 = 5

Divide both by 5: $5 : 15 \rightarrow 1 : 3$

5 : 15 → ÷5 → 1 : 3

Example 4: Brown and Yellow Buttons

Jane has 21 brown buttons and 6 yellow buttons. Find the ratio of the number of brown buttons to the number of yellow buttons in its simplest form.

HCF of 21 and 6 = 3

Divide both by 3: $21 : 6 \rightarrow 7 : 2$

7 : 2 means that for every 7 brown buttons there are 2 yellow buttons

21 : 6 → ÷3 → 7 : 2

Example 5: Raju's and Anna's Stickers

Raju has 72 stickers and Anna has 48 stickers. Find the ratio of the number of Raju's stickers to the number of Anna's stickers in its simplest form.

Method 1 (Step by step):

72 : 48 → ÷8 → 9 : 6 → ÷3 → 3 : 2

Method 2 (Using the HCF directly):

HCF of 72 and 48 = 24

72 : 48 → ÷24 → 3 : 2

Ratio in simplest form = 3 : 2

Example 6: Three Parts A, B, and C

A piece of ribbon is cut into 3 parts A, B, and C. A is 15 cm, B is 24 cm, and C is 18 cm. Find the ratio of the length of A to the length of B to the length of C in its simplest form.

HCF of 15, 24, and 18 = 3

$15 : 24 : 18 \rightarrow 5 : 8 : 6$

15 : 24 : 18 → ÷3 → 5 : 8 : 6

Key Points to Remember

✓ Equivalent ratios show the same relationship
✓ To find equivalent ratios: multiply or divide EVERY term by the same number
✓ Simplest form: when the terms have no common factor except 1
✓ Use the highest common factor (HCF) to simplify in one step
✓ The actual numbers change, but the relationship stays the same!