Australian Curriculum v9 / ACiQ Year 7 Mathematics - Unit 1 - Ratios and equivalent ratios

Ratios and equivalent ratios

Represent part-to-part and part-to-whole ratios, then generate and simplify equivalent ratios using multiplicative reasoning.

Updated 2026-07-24 - 5 min read

A ratio is an ordered comparison. It tells us how much of one quantity there is relative to another quantity. Before calculating, name the quantities, keep their order and decide whether you are comparing two parts or one part with the whole.

1. The order carries meaning

Suppose a tray has 3 red counters and 5 blue counters.

  • red to blue is $3:5$
  • blue to red is $5:3$

These ratios use the same counters but answer different questions. Reversing a ratio is not a harmless formatting change.

2. Part-to-part and part-to-whole

A part-to-part ratio compares two subgroups. In the tray, red to blue is $3:5$.

A part-to-whole ratio compares one subgroup with the total. There are $3+5=8$ counters, so red to all counters is $3:8$.

| Question | First quantity | Second quantity | Ratio | | --- | ---: | ---: | ---: | | red to blue | 3 | 5 | $3:5$ | | blue to red | 5 | 3 | $5:3$ | | red to all | 3 | 8 | $3:8$ | | blue to all | 5 | 8 | $5:8$ |

3. A ratio describes relative group size

The ratio $2:3$ does not always mean exactly 2 objects and 3 objects. It could describe:

  • 2 red and 3 blue counters
  • 4 red and 6 blue counters
  • 10 red and 15 blue counters

Each pair has the same multiplicative relationship. The total scale can change while the comparison stays constant.

4. Equivalent ratios

Equivalent ratios express the same comparison. Generate one by multiplying or dividing every term by the same non-zero number.

$ 2:3 = (2\times4):(3\times4)=8:12 $

The scale factor is 4. It must be applied to both terms.

5. Simplifying a ratio

Simplifying reverses the scaling process. Divide all terms by a common factor. A whole-number ratio is in simplest form when its terms share no factor greater than 1.

You can also simplify in more than one step:

$ 18:24 \div 2 = 9:12,\qquad 9:12 \div 3 = 3:4 $

Both methods are valid because both terms are divided together at every step.

6. Faded example: you complete the reasoning

A drink uses cordial to water in the ratio $1:4$. The recipe is scaled to use 3 cups of cordial.

  1. The scale factor is $3\div1=\underline{\hspace{1cm}}$.
  2. The water amount is $4\times\underline{\hspace{1cm}}=\underline{\hspace{1cm}}$ cups.
  3. The equivalent ratio is $\underline{\hspace{1cm}}:\underline{\hspace{1cm}}$.

Check: the missing values are 3, 3, 12, 3 and 12. The completed ratio is $3:12$.

7. A reliable problem routine

  1. Write the quantity labels in the requested order.
  2. Write the known values beneath the labels.
  3. Decide whether the comparison is part-to-part or part-to-whole.
  4. If scaling, find one multiplicative factor and apply it to every term.
  5. If simplifying, divide every term by the same common factor.
  6. Interpret the result using the original quantities.

Glossary

| Term | Meaning | | --- | --- | | ratio | an ordered comparison of quantities | | term | one number in a ratio | | part-to-part | comparison between two subgroups | | part-to-whole | comparison between a subgroup and the total | | equivalent ratios | ratios with the same multiplicative relationship | | scale factor | the common multiplier or divisor applied to all terms | | simplest form | a whole-number ratio whose terms share no factor greater than 1 |

Quick check

  1. A bag has 6 green and 9 yellow beads. Write green:yellow in simplest form.
  2. In the same bag, write green:all beads in simplest form.
  3. Complete $4:7=20:\square$.
  4. Is $5:8$ equivalent to $15:24$? Justify with a scale factor.

Answers:

  1. $6:9=2:3$.
  2. There are 15 beads, so $6:15=2:5$.
  3. The scale factor is 5, so $\square=35$.
  4. Yes. Both terms were multiplied by 3.

Teach-back and self-check

Close the worked examples and explain ratios as if you were helping another Year 7 student. Name the two quantities in order, state whether the comparison is part-to-part or part-to-whole, and show why one common multiplicative scale factor preserves the relationship.

Before calling the method secure, identify the quantities, represent the relationship, justify the scale factor and verify the result in the original context. If the explanation depends on “I just divided”, return to the ratio table or diagram and name what each number represents.

Transfer task

Sources