Practice Maths

L15 — Equivalent Ratios

What are Equivalent Ratios?

Equivalent ratios are ratios that name the same comparison. Even though the numbers are different, the proportion remains the same.

Key Terms

equivalent ratios
ratios that represent the same comparison, obtained by multiplying or dividing both parts by the same number (e.g. 1 : 2 = 2 : 4 = 5 : 10)
scaling
multiplying or dividing both parts of a ratio by the same number to produce an equivalent ratio
scale factor
the number used to multiply or divide both parts of a ratio when scaling
ratio table
a table showing a series of equivalent ratios, useful for solving proportional problems
proportion
a statement that two ratios are equal (e.g. 1 : 2 = 5 : 10)
The Multiplication Rule You can find an equivalent ratio by multiplying or dividing both parts of the ratio by the same number.
Never add or subtract!

Think of it like a recipe: If you want to make double the amount of food, you double every ingredient. The taste (the ratio) stays the same!

Worked Example

Question: Find an equivalent ratio to 3 : 4.

Step 1 — Choose a multiplier: multiply both parts by the same number, e.g. ×3.

Step 2 — Multiply: 3×3 : 4×3 = 9 : 12.

Step 3 — Verify: So 3 : 4 = 9 : 12 = 15 : 20 etc.

What Are Equivalent Ratios?

Equivalent ratios represent the same relationship between quantities, just written with different numbers. They work exactly like equivalent fractions. For example, 1:2, 2:4, 5:10, and 50:100 are all equivalent ratios — they all express the same "one for every two" relationship.

Think of mixing paint: a ratio of 1 part red to 2 parts blue gives the same colour whether you use 1 cup and 2 cups, or 5 buckets and 10 buckets. The relationship is the same.

Creating Equivalent Ratios

To make an equivalent ratio, multiply or divide both parts by the same number. This is called scaling.

  • 2:3 → multiply both by 4 → 8:12 (equivalent)
  • 10:15 → divide both by 5 → 2:3 (equivalent)
  • 3:7 → multiply both by 3 → 9:21 (equivalent)

You must do the same operation to both parts. If you multiply one side and divide the other, the ratio changes.

Using a Ratio Table

A ratio table is a great way to organise equivalent ratios, especially when solving real-life problems. For example, if a recipe uses flour and sugar in a ratio of 3:1:

Flour (cups)36912
Sugar (cups)1234

Each column is an equivalent ratio, obtained by multiplying the original 3:1 by 1, 2, 3, 4, etc.

Checking If Two Ratios Are Equivalent

To check if two ratios are equivalent, simplify both to their lowest form and see if they match. Alternatively, use cross-multiplication: for a:b and c:d, multiply a × d and b × c — if the products are equal, the ratios are equivalent.

  • Are 4:6 and 6:9 equivalent? 4 × 9 = 36, 6 × 6 = 36 → Yes
  • Are 3:5 and 4:7 equivalent? 3 × 7 = 21, 5 × 4 = 20 → No
Key tip: If you need to find a missing value in an equivalent ratio (like 3:5 = ?:20), ask "what did I multiply 5 by to get 20?" The answer is 4. So multiply 3 by 4 as well → the missing value is 12. Always find the scale factor first, then apply it to the other part.

Practice Questions

  1. The Ratio Table Fluency

    Complete the following table where the ratio of Blue to Red counters is 1:4.

    Blue (Part A)1234? (v)10
    Red (Part B)48? (i)? (ii)20? (vi)
    1. What is the missing value for (i)?
    2. What is the missing value for (ii)?
    3. What is the missing value for (v)?
    4. What is the missing value for (vi)?

  2. Identifying the Multiplier Fluency

    To create an equivalent ratio, we multiply both sides by a "scaling factor". Identify the factor used in these examples:

    1. 2:5 became 6:15. What was the multiplier?
    2. 1:9 became 7:63. What was the multiplier?
    3. 12:20 became 3:5. What was the divisor (number divided by)?
    4. 0.5 : 1 became 5 : 10. What was the multiplier?

  3. Completing the Series Fluency

    Fill in the blanks to keep the ratio chain equivalent:

    1. 1 : 2 = 2 : ___ = 5 : ___ = ___ : 20
    2. 3 : 4 = 6 : ___ = ___ : 12 = 30 : ___
    3. 10 : 5 = 2 : ___ = ___ : 10 = 100 : ___
    4. 7 : 3 = 14 : ___ = ___ : 9 = 70 : ___

  4. True or False Mastery Understanding

    Are the following pairs of ratios equivalent? Write True or False:

    1. 4:5 and 12:15
    2. 1:8 and 5:45
    3. 10:2 and 5:1
    4. 3:7 and 30:700

  5. Finding the Unknown Understanding

    Solve for the letter to make the ratios equivalent:

    1. 2 : 3 = 10 : x
    2. 5 : 1 = y : 6
    3. a : 12 = 1 : 4
    4. 100 : 50 = 2 : b

  6. Tropical Punch Recipe Understanding

    A recipe uses 200ml of Orange Juice and 500ml of Lemonade.

    1. What is the ratio of Orange Juice to Lemonade in simplest form?
    2. If you double the recipe, how much Lemonade do you need?
    3. If you use 1000ml (1L) of Orange Juice, how much Lemonade is needed?
    4. What is the ratio of Orange Juice to Lemonade if you make 5 batches?

  7. Weed Killer Mixture Understanding

    The instructions say to mix 15ml of weed killer with 1 Litre (1000ml) of water.

    1. Write the ratio of weed killer to water (in ml).
    2. How much weed killer is needed for 3 Litres of water?
    3. If you have a 10 Litre spray tank, how much weed killer is needed?
    4. If you accidentally put in 45ml of weed killer, how much water should you use to keep the ratio correct?

  8. Units and Ratios Understanding

    Decide if these comparisons represent the same ratio:

    1. Is 1cm : 5cm equivalent to 10mm : 50mm?
    2. Is $2 : $10 equivalent to 20c : $1?
    3. Is 1kg : 2kg equivalent to 500g : 1000g?
    4. Is 1 minute : 60 seconds equivalent to 1 : 1?

  9. School Proportions Problem Solving

    In a school, the ratio of boys to girls is 4:5.

    1. If there are 8 boys, how many girls are there?
    2. If there are 25 girls, how many boys are there?
    3. How many total "parts" are in the ratio 4:5?
    4. If there are 90 students in total, how many are boys? (Hint: Use the total parts).

  10. Challenge: Triple Ratios Problem Solving

    Equivalent ratios also work with three numbers! (Example: 1:2:3 = 2:4:6)

    1. 1 : 2 : 5 = 2 : ___ : ___
    2. 3 : 1 : 4 = ___ : 5 : ___
    3. 10 : 20 : 30 = 1 : ___ : ___
    4. ___ : ___ : 12 = 1 : 2 : 3
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