Practice Maths

L16 — Simplifying Ratios

What is Simplifying?

Simplifying a ratio means expressing the relationship using the smallest possible whole numbers. It is exactly like simplifying fractions!

Key Terms

simplest form
a ratio written using the smallest possible whole numbers, where no common factor greater than 1 divides both parts
highest common factor (HCF)
the largest number that divides exactly into both parts of a ratio; dividing by the HCF simplifies a ratio in one step
common factor
a number that divides exactly into two or more numbers (e.g. 4 is a common factor of 8 and 12)
lowest terms
another name for simplest form; a ratio is in lowest terms when the only common factor of both parts is 1
The Division Rule To simplify a ratio, you must divide both sides by the same number (a common factor).
A ratio is in its simplest form when the only number you can divide both sides by is 1.

Highest Common Factor (HCF): If you divide both sides by the HCF, you will reach the simplest form in just one step.

Worked Example

Question: Simplify the ratio 12 : 18.

Step 1 — Find the HCF: HCF of 12 and 18 = 6.

Step 2 — Divide both parts by the HCF: 12÷6 : 18÷6

Step 3 — Simplified ratio: 2 : 3

What Does It Mean to Simplify a Ratio?

Simplifying a ratio means writing it using the smallest possible whole numbers while keeping the same relationship. Just like simplifying the fraction 68 to 34, we simplify the ratio 6 : 8 to 3 : 4 by dividing both parts by their highest common factor (HCF).

A simplified ratio is also called a ratio in its simplest form or lowest terms. It is the most useful form for comparing ratios and solving problems.

How to Simplify a Ratio

Step 1: Find the HCF of both parts. Step 2: Divide both parts by the HCF.

  • Simplify 12:8 → HCF of 12 and 8 is 4 → 12÷4 : 8÷4 = 3:2
  • Simplify 15:25 → HCF is 5 → 15÷5 : 25÷5 = 3:5
  • Simplify 6:9:12 → HCF is 3 → 6÷3 : 9÷3 : 12÷3 = 2:3:4

If you're unsure of the HCF, divide by any common factor and keep going until you can't divide anymore.

Ratios with Different Units: Convert First!

If a ratio involves two different units, you must convert both to the same unit before simplifying. For example:

  • Write the ratio of 50 cm to 2 m in simplest form. First, convert 2 m to 200 cm. Ratio = 50:200. HCF = 50. Simplified: 1:4.
  • Write the ratio of 30 minutes to 2 hours. Convert 2 hours to 120 minutes. Ratio = 30:120. HCF = 30. Simplified: 1:4.

Simplifying Ratios Involving Fractions or Decimals

To simplify a ratio like 0.5:1.5, multiply both parts by 10 (or 100) to make them whole numbers first, then simplify:

  • 0.5:1.5 → ×10 → 5:15 → ÷5 → 1:3
  • 12 : 34 → multiply both by 4 (the LCD) → 2 : 3 → already simplified → 2 : 3
Key tip: The simplest form of a ratio must contain whole numbers only. A ratio like 1.5:2 or 1/2:1 is not fully simplified even if the numbers look small. Always check: are both parts whole numbers? If not, multiply through to make them whole, then divide by the HCF.

Practice Questions

  1. Ratios vs Fractions Fluency

    A tray of fruit contains 8 Green Apples and 12 Red Apples.

    1. What is the total number of apples in the tray?
    2. What fraction of the total apples are Green? (Simplify the fraction)
    3. What is the ratio of Green Apples to Red Apples?
    4. Simplify the ratio of Green Apples to Red Apples to its simplest form.

  2. Direct Simplification Fluency

    Simplify the following ratios by finding a common factor:

    1. 10 : 15
    2. 18 : 6
    3. 21 : 14
    4. 40 : 100

  3. Using the HCF Fluency

    For each ratio, identify the Highest Common Factor (HCF) and then simplify:

    1. 12 : 30 (HCF is ___; Simplest form is ___)
    2. 45 : 20 (HCF is ___; Simplest form is ___)
    3. 16 : 48 (HCF is ___; Simplest form is ___)
    4. 24 : 36 (HCF is ___; Simplest form is ___)

  4. Handling Units Understanding

    Convert the quantities to the same unit first, then simplify the ratio:

    1. 50c to $2.00
    2. 200m to 1km
    3. 15 minutes to 1 hour
    4. 400g to 2kg

  5. Sports Field Dimensions Understanding

    A school basketball court is 28 metres long and 15 metres wide. A mini-court is 14 metres long and 7.5 metres wide.

    1. What is the ratio of Length to Width for the school court?
    2. What is the ratio of Length to Width for the mini-court?
    3. Are these two ratios equivalent? (Explain why)
    4. If a third court has a ratio of 56 : 30, what is that in simplest form?

  6. The Perfect Orange Paint Understanding

    To make a certain shade of orange, a painter mixes 600ml of Yellow paint with 400ml of Red paint.

    1. Write the ratio of Yellow to Red paint.
    2. Simplify this ratio to its simplest form.
    3. What is the total volume of the mixture?
    4. What fraction of the total mixture is Red paint? (Simplify the fraction)

  7. Simplifying Three Parts Understanding

    Simplify these three-part ratios by dividing all three numbers by the same factor:

    1. 10 : 20 : 30
    2. 12 : 18 : 6
    3. 25 : 50 : 75
    4. 14 : 28 : 21

  8. Survey Results Understanding

    In a class of 30 students, 18 students prefer dogs, 9 prefer cats, and 3 prefer birds.

    1. What is the ratio of Dog-lovers to Cat-lovers to Bird-lovers?
    2. Simplify this three-part ratio.
    3. What is the ratio of Bird-lovers to the total number of students?
    4. Simplify the Bird-lovers to total students ratio.

  9. Step-by-Step Simplification Problem Solving

    Sometimes you might simplify in steps. Start by dividing by 2, then see if you can go further:

    1. 80 : 64
    2. 120 : 160
    3. 48 : 72
    4. 144 : 96

  10. The Concrete Mix Problem Solving

    A builder mixes 4 buckets of cement, 8 buckets of sand, and 12 buckets of gravel.

    1. What is the ratio of cement to sand to gravel?
    2. Simplify the ratio to its simplest form.
    3. If the builder wants to make a double batch, what is the new ratio (not simplified)?
    4. Does doubling the batch change the simplest form of the ratio? (Explain)
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