Equivalent ratios

Same comparison, different numbers

Equivalent ratios describe the same relationship. A juice recipe uses 1 cup of concentrate to 3 cups of water. A double batch uses 2:6, and a triple batch uses 3:9. All three taste the same, so 1:3, 2:6 and 3:9 are equivalent.

Making equivalent ratios

Multiply or divide every term by the same non-zero number.

• 2:5 × 3 → 6:15
• 12:18 ÷ 6 → 2:3

Just like equivalent fractions!

Ratio tables

A ratio table lists equivalent ratios in columns:

You can scale up by multiplying, scale down by dividing, or add columns together (2:1 + 4:2 = 6:3).

Finding a missing term

Example: 3:4 = ?:20

• 4 × 5 = 20, so multiply 3 by 5 too: 15.
• 3:4 = 15:20

Example: 9:12 = 3:?

• 9 ÷ 3 = 3, so 12 ÷ 3 = 4.

Checking if ratios are equivalent

Method 1: Simplify both. 6:8 → 3:4 and 9:12 → 3:4. Equivalent ✓.

Method 2: Cross-multiply. Write as fractions: 6/8 and 9/12. 6 × 12 = 72 and 8 × 9 = 72. Equal ✓.

Using equivalent ratios

Problem: A recipe for 4 people uses 300 g of pasta. How much for 10 people?

Halve to get 2 people (150 g), then multiply by 5: 750 g.

Problem: Paint is mixed 2 parts blue to 5 parts white. You use 15 L of white. How much blue?

2:5 = ?:15 → × 3 → 6 L of blue.

Worked examples

Example 1: Write two ratios equivalent to 4:7.

8:14 and 12:21.

Example 2: Are 5:8 and 15:24 equivalent?

5 × 3 = 15, 8 × 3 = 24. Yes.

Example 3: 7:2 = 35:?

× 5 → 10.

Common mistakes

• Adding the same number to each term: 2:3 is not equivalent to 4:5.
• Changing only one term.
• Mixing up the order of terms.

Try it yourself

1. 3:5 = 12:?
2. 18:24 = ?:4
3. Are 4:6 and 10:15 equivalent?
4. A map uses 2 cm for 5 km. How many km does 14 cm show?
5. Complete the table: 3:7 → 6:? → ?:28

Answers: 1) 20 2) 3 3) yes 4) 35 km 5) 6:14 and 12:28

Quick recap

• Equivalent ratios make the same comparison.
• Multiply or divide all terms by the same number.
• Use ratio tables and cross-multiplication to find and check them.