Practice Maths

L16.1 — Solving Ratio Problems

Understanding Ratios

A ratio is a comparison of two or more quantities of the same kind. Ratios use a colon (:) to separate values.

Simplifying Ratios: Ratios should always be expressed in simplest form. Divide both sides by the Highest Common Factor (HCF).

Double Number Lines: These are used to verify and solve equivalent ratios. If a multiplier works for the first set of values, it must work for the second.

Key Terms

divide in a ratio
to share a quantity so that each share matches a given ratio (e.g. divide $60 in the ratio 2 : 3 gives $24 and $36)
total parts
the sum of the numbers in a ratio, used to find the value of one part (e.g. ratio 2 : 3 has 2 + 3 = 5 total parts)
one part
the value of each share, found by dividing the total quantity by the total number of parts
scale model
a physical or drawn model where all dimensions are reduced (or enlarged) by the same ratio (e.g. 1 : 20 means 1 cm on the model = 20 cm in real life)
Ratio vs Fraction
If a ratio is 3 : 5 (e.g. Red to Blue), the fraction of Red is 38 (part over total: 3 + 5 = 8).

Worked Example

Question: Divide $60 in the ratio 2 : 3.

Step 1 — Find total parts: 2 + 3 = 5 parts.

Step 2 — Value of 1 part: $60 ÷ 5 = $12.

Step 3 — Calculate each share: First = 2 × $12 = $24. Second = 3 × $12 = $36.

Dividing a Quantity in a Given Ratio

A common ratio problem is: "Share $60 between two people in the ratio 2:3." Here's how to solve it step by step:

  • Step 1: Add the parts of the ratio: 2 + 3 = 5 parts total
  • Step 2: Find the value of one part: $60 ÷ 5 = $12 per part
  • Step 3: Multiply each share: 2 × $12 = $24, and 3 × $12 = $36
  • Check: $24 + $36 = $60 ✓

This method works for any two-part or three-part ratio.

Three-Part Ratios (a:b:c)

Three-part ratios work the same way. For example: "Share 90 lollies among three friends in the ratio 1:2:3."

  • Total parts: 1 + 2 + 3 = 6
  • Value of one part: 90 ÷ 6 = 15 lollies
  • Friend 1: 1 × 15 = 15 lollies
  • Friend 2: 2 × 15 = 30 lollies
  • Friend 3: 3 × 15 = 45 lollies
  • Check: 15 + 30 + 45 = 90 ✓

Scale Models and Maps

A scale ratio like 1:200 means "1 unit on the model = 200 units in real life." To find a real-life measurement, multiply the model measurement by the scale factor. To find a model measurement, divide the real measurement by the scale factor.

  • A model car uses a scale of 1:20. If the model is 15 cm long, the real car is 15 × 20 = 300 cm = 3 m long.
  • A map has a scale of 1:50 000. If two towns are 8 cm apart on the map, the real distance is 8 × 50 000 = 400 000 cm = 4 km.

Worked Example: A Recipe Problem

A smoothie recipe uses banana and mango in a ratio of 3:2. You want to use 450 g of banana. How much mango do you need?

  • Banana = 3 parts = 450 g, so 1 part = 450 ÷ 3 = 150 g
  • Mango = 2 parts = 2 × 150 = 300 g
Key tip: When dividing in a ratio, always find the value of ONE part first, then multiply. The most common mistake is forgetting to add the ratio parts before dividing. For a ratio of 3:7, there are 10 parts total — not 7 and not 3.

Practice Questions

  1. Foundation: Writing Ratios Fluency

    A fruit bowl contains 8 apples and 12 bananas. Calculate:

    1. The ratio of apples to bananas.
    2. The ratio of bananas to apples.
    3. The ratio of apples to total fruit.
    4. The fraction of apples in the bowl.
    5. The fraction of bananas in the bowl.
    6. The simplified ratio of apples to bananas.

  2. Foundation: Simplifying Ratios Fluency

    Express the following ratios in their simplest form:

    1. 5 : 15
    2. 12 : 18
    3. 24 : 36
    4. 45 : 10
    5. 100 : 250
    6. 14 : 49

  3. Foundation: Units and Ratios Fluency

    Simplify these ratios (Ensure units are the same first!):

    1. 20cm : 1m
    2. 500g : 2kg
    3. 30 min : 2 hours
    4. 40c : $2.00
    5. 250mL : 1L
    6. 15mm : 3cm

  4. Foundation: Equivalent Ratios Fluency

    Find the missing value (x) to make the ratios equivalent:

    1. 1 : 3 = 4 : x
    2. 2 : 5 = 10 : x
    3. 3 : 7 = x : 21
    4. 8 : 12 = 2 : x
    5. x : 9 = 20 : 45
    6. 5 : x = 25 : 50

  5. Foundation: Equivalent Verification Understanding

    Are the following pairs equivalent? (Yes/No):

    1. 2:6 and 3:9
    2. 4:5 and 12:20
    3. 7:3 and 21:9
    4. 10:2 and 50:10
    5. 1:9 and 9:81
    6. 13:2 and 26:6

  6. Foundation: Multi-part Ratios Understanding

    Simplify these three-term ratios:

    1. 2 : 4 : 6
    2. 10 : 20 : 50
    3. 12 : 18 : 24
    4. 5 : 15 : 25
    5. 8 : 16 : 32
    6. 21 : 14 : 7

  7. Worded Problem: Mixing Cordial Understanding

    Kane makes cordial using a ratio of 2 parts cordial to 7 parts water.

    1. How many parts are there in the total mixture?
    2. If Kane uses 500mL of cordial, how much water does he need?
    3. If Kane wants to make 1.8L of the total mixture, how much cordial is needed?
    4. If he accidentally uses 14 scoops of water, how much cordial should he add to keep the ratio?

  8. Worded Problem: Sharing Money Understanding

    Mr J divided $4,200 between Jenna and Jamie in the ratio 5:2.

    1. How many "shares" or "parts" is the money being divided into?
    2. What is the value of one single share?
    3. How much money did Jenna receive?
    4. Jamie receives the smaller amount. If Jamie received $1,256 in a different sale with the same ratio, how much did Jenna receive?

  9. Worded Problem: Construction & Recipes Problem Solving

    1. Tanya mixes concrete using sand and cement in a ratio of 2:3. If she has 12L of cement, how much sand is needed?
    2. A biscuit recipe requires 300g of flour to 350g of sugar. If Jo has 1.4kg of sugar, how much flour is needed?
    3. To make mortar, a bricklayer mixes 3 parts sand to 1 part cement. To make 20kg of total mortar, how many kg of each is needed?

  10. Worded Problem: Populations Problem Solving

    1. A camp has a boy-to-girl ratio of 9:11. If there are 99 boys, what is the total number of children at the camp?
    2. A fish breeder has 200 catfish. 40 of these are the rare peppermint strain. What is the simplified ratio of peppermint to non-peppermint fish?
    3. In a school, the ratio of teachers to students is 1:25. If there are 20 teachers, how many people are in the school altogether?
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