Practice Maths

Ratios and Their Graphs

Key Ideas

Key Terms

ratio
a comparison of two quantities of the same type, written as a:b.
equivalent ratio
ratios that represent the same relationship; found by multiplying or dividing both parts by the same number.
simplest form
a ratio where both parts share no common factor greater than 1; simplify by dividing both parts by their HCF.
HCF
highest common factor — the largest number that divides evenly into all parts of the ratio.
proportional relationship
a relationship where the ratio of two quantities stays constant; graphs as a straight line through the origin with equation y = kx.

Simplifying a Ratio

12:8 → HCF = 4 → 12÷4 : 8÷4 = 3:2

Hot Tip A ratio 3:2 means “for every 3 of the first, there are 2 of the second.” The graph of a proportional relationship is always a straight line through the origin. If it doesn’t pass through (0,0), it is not proportional.

Worked Examples

Example 1 — Simplify 15:25
HCF of 15 and 25 = 5
15÷5 : 25÷5 = 3:5

Example 2 — Find equivalent ratio: 3:4 = ?:20
4 × 5 = 20, so multiply both by 5
3 × 5 = 15 → 15:20

Example 3 — Ratio as fraction: ratio 2:3
Total parts = 2 + 3 = 5
First quantity’s fraction of total = 2/5

What Is a Ratio?

A ratio compares quantities of the same type. It tells you how much of one thing there is for every certain amount of another. Ratios are written with a colon: 3:5, or as a fraction 35.

Example: A smoothie recipe uses 3 parts banana to 5 parts mango. The ratio of banana to mango is 3:5. For every 3 scoops of banana, you use 5 scoops of mango.

Note: Ratios compare same units (both are scoops, both are lengths, both are masses). If units are different, it becomes a rate, not a ratio.

Simplifying Ratios

Simplify a ratio by dividing all parts by their highest common factor (HCF) — just like simplifying a fraction.

Example: Simplify 12:8.
HCF of 12 and 8 is 4. Divide both by 4: 12:8 = 3:2.

Example: Simplify 15:25:10.
HCF is 5. Divide all by 5: 3:5:2.

Ratios with decimals or fractions: multiply through to get whole numbers first. e.g. 0.4:1.2 → multiply by 10 → 4:12 → divide by 4 → 1:3.

Equivalent Ratios and Ratio Tables

Equivalent ratios are like equivalent fractions — you multiply (or divide) both sides by the same number.

Example: 2:3 is equivalent to 4:6, 6:9, 10:15. They all represent the same relationship.

A ratio table lists equivalent ratios in columns, making patterns easy to see:

Banana (cups): 2, 4, 6, 8, 10
Mango (cups): 3, 6, 9, 12, 15

Each column is an equivalent ratio to 2:3. As one goes up, so does the other — at a constant rate.

Graphing Ratios as Straight Lines

When you plot the values from a ratio table on a graph (quantity A on the x-axis, quantity B on the y-axis), you get a straight line through the origin.

For the banana:mango ratio 2:3, the plotted points (2,3), (4,6), (6,9) all lie on a straight line through (0,0).

The gradient of the line equals the ratio: rise ÷ run = 32 = 1.5. This means for every 1 cup of banana, you need 1.5 cups of mango.

A steeper line = a larger ratio. A shallower line = a smaller ratio. Two different ratios appear as two different straight lines with different gradients.

Connecting Ratios to Real Life

Ratios appear in cooking (recipe proportions), art (aspect ratios of screens — 16:9), sport (win/loss records), maps (scale ratios like 1:50 000), and building (mixing concrete 1:2:3 — cement, sand, gravel).

Example: A map has a scale of 1:50 000. A road measures 4 cm on the map. Real length = 4 × 50 000 = 200 000 cm = 2 km.

Example: Orange squash is mixed in ratio 1:4 (squash to water). To make 1 250 mL of drink, squash needed = 15 × 1 250 = 250 mL.

Key tip: When a ratio graph is a straight line through the origin, the gradient is the ratio (as a fraction). A ratio of 3:2 gives a gradient of 32 = 1.5. If you ever see a straight line through the origin on a graph, it is always representing a constant ratio or rate.

Mastery Practice

  1. Simplify each ratio to its simplest form. Fluency

    1. 12:8
    2. 15:25
    3. 100:75
    4. 18:12
    5. 24:36
    6. 50:80
    7. 9:27
    8. 64:48

  2. Find the missing value to make equivalent ratios. Fluency

    1. 3:4 = ?:20
    2. 2:5 = 10:?
    3. 1:3 = ?:18
    4. 5:2 = 25:?
    5. 4:7 = ?:28
    6. 3:8 = 9:?
    7. 6:5 = ?:30
    8. 2:9 = 8:?

  3. Write each ratio as a fraction of the total, then convert to a percentage. Fluency

    1. Ratio 2:3 — what fraction is the first quantity?
    2. Ratio 1:4 — what percentage is the first quantity?
    3. Ratio 3:7 — what fraction is the second quantity?
    4. Ratio 3:2 — what percentage is the second quantity?
    5. Ratio 1:1 — what percentage is each quantity?
    6. Ratio 3:1 — what fraction is the first quantity?
    7. Ratio 5:3 — what fraction is the first quantity?
    8. Ratio 2:8 — what percentage is the first quantity?

  4. Solve each real-world ratio problem. Understanding

    1. A map has a scale of 1:50 000. A distance of 3 cm on the map equals what real distance (in km)?
    2. A map has a scale of 1:25 000. A real distance of 5 km equals what distance on the map (in cm)?
    3. Paint is mixed in a ratio of 3 parts blue to 2 parts white. How many litres of white paint are needed to mix with 9 litres of blue?
    4. Concrete is mixed in a ratio of 1 part cement : 2 parts sand : 3 parts gravel. How much of each ingredient is needed for 24 kg of concrete?
    5. A recipe calls for flour and sugar in ratio 4:1. If 120 g of flour is used, how much sugar is needed?
    6. Two lengths are in ratio 5:3. The longer length is 40 cm. Find the shorter length.
    7. A school has students and teachers in ratio 20:1. There are 500 students. How many teachers are there?
    8. Orange juice and lemonade are mixed in ratio 3:2 to make 500 mL of punch. How much of each drink is used?

  5. Extended ratio problems. Show full working. Problem Solving

    1. A recipe for 12 cupcakes uses 180 g of flour, 90 g of sugar and 60 g of butter. Scale the recipe up to make 20 cupcakes.
    2. A scale drawing of a room uses 1 cm to represent 2.5 m. The room is drawn as 6 cm × 4.8 cm. What are the real dimensions of the room?
    3. A business earns revenue and pays costs in ratio 5:3. If the total revenue is $40 000, how much are the costs?
    4. Three friends invest money in ratio 2:3:5. The total investment is $20 000. How much does each person invest?

  6. The bar diagram below shows a total amount split into two parts. Each small segment represents 1 part. Understanding

    1 1 1 1 1 Part A Part B
    1. Write the ratio of Part A (blue) to Part B (orange) in simplest form.
    2. If the total amount is $150, what is the value of each small segment?
    3. Use your answer to find Part A’s share and Part B’s share.
    4. What fraction of the total does Part A represent?

  7. A grocery store sells apples in the ratio: cost ($) to mass (kg) = 3.50 : 1. Complete the table. Understanding

    Mass (kg)123510
    Cost ($)3.50????
    1. Complete all missing values in the table.
    2. Write the equation relating Cost (C) to Mass (m).
    3. Describe the shape of the graph of this relationship and where it starts.
    4. Using the equation, find the cost of 7.5 kg of apples.

  8. Two graphs show proportional relationships. Read the descriptions and answer the questions. Understanding

    Graph A

    x y 1 2 3 2 4 6

    Graph B

    x y 1 2 3 6
    1. What is the ratio x:y for Graph A? (Use the point (1, 2).)
    2. What is the ratio x:y for Graph B? (Use the point (1, 3).)
    3. Which graph represents the relationship “3 cups of water for every 1 cup of cordial”?
    4. Write the equation of the line in Graph A and Graph B.

  9. A builder mixes mortar using sand and cement in ratio 5:1 by mass. Problem Solving

    1. If the builder uses 2 kg of cement, how much sand is needed?
    2. Complete the table for different batches of mortar:
    Cement (kg)12346
    Sand (kg)?????
    Total (kg)?????
    1. Write the equation linking sand (S) to cement (C).
    2. The builder needs a total of 42 kg of mortar. How much cement and sand does he need?
    3. Sand costs $0.40/kg and cement costs $1.20/kg. What is the total cost of materials for 42 kg of mortar?

  10. Decide whether each relationship is proportional (represents a ratio). Explain your reasoning. Problem Solving

    1. x: 1, 2, 3, 4  |  y: 4, 8, 12, 16
    2. x: 0, 1, 2, 3  |  y: 3, 5, 7, 9
    3. A graph of a straight line passing through (0, 0) and (5, 15).
    4. A graph of a straight line crossing the y-axis at (0, 4).
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