Equivalent ratios and simplest form | Year 7

01 / LEARN

The idea, step by step

Scale every entry by the same factor

Equivalent ratios describe the same relative amounts. Multiplying both entries of $2 : 3$ by 4 gives $8 : 12$ . Dividing both entries by the same non-zero number also preserves the ratio. Every complete group is enlarged or reduced in the same way. This differs from adding an equal amount to each entry.

Simplest form has no common factor above one

For $18 : 30$ , divide both entries by 6 to get $3 : 5$ . You can divide in smaller stages, but continue until no common factor greater than 1 remains. A three-part ratio uses the same rule: the same operation must be applied to every entry, not just two of them.

A common trap

Adding 3 to both entries of $2 : 5$ and calling $5 : 8$ equivalent.

Instead: Equivalent ratios use a common multiplying or dividing factor. The fractions $2 / 5$ and $5 / 8$ are unequal, so these ratios are not equivalent.

02 / WORKED EXAMPLES

See the thinking

Example 1

Simplify a ratio

Simplify $24 : 36$ .

Both entries are divisible by 12.
$24 / 12 = 2$ and $36 / 12 = 3$ .
There is no further common factor above one.

2 : 3

Example 2

Find a missing entry

Complete $5 : 8 = 20 : x$ .

The first entry was multiplied by 4.
Multiply the second entry by 4 as well: $8 (4) = 32$ .
Check that $20 : 32$ simplifies to $5 : 8$ .

x = 32

Check your understanding

Is $6 : 15$ equivalent to $2 : 5$ ?

Show answer

Yes.

Divide both entries by 3.

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Simplify $12 : 20$ .
2 marks
Simplify $42 : 18$ .
2 marks
Complete $3 : 7 = 12 : x$ .
2 marks
Simplify $15 : 25 : 35$ .
2 marks
Simplify 2 m:50 cm.
3 marks
A student says $4 : 6$ and $7 : 9$ are equivalent because 3 was added to each entry. Explain the error.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Simplify $12 : 20$ .
2 marks
Divide both entries by 4.
$3 : 5$ .
Simplify $42 : 18$ .
2 marks
Divide both entries by 6.
$7 : 3$ .
Complete $3 : 7 = 12 : x$ .
2 marks
The first entry is multiplied by 4, so multiply 7 by 4.
$x = 28$ .
Simplify $15 : 25 : 35$ .
2 marks
Divide all three entries by 5.
$3 : 5 : 7$ .
Simplify 2 m:50 cm.
3 marks
Convert 2 m to 200 cm.
Divide both entries of $200 : 50$ by 50.
$4 : 1$ .
A student says $4 : 6$ and $7 : 9$ are equivalent because 3 was added to each entry. Explain the error.
3 marks
$4 : 6$ simplifies to $2 : 3$ .
To scale 4 to 7 requires multiplying by $7 / 4$ ; that would change 6 to 10.5, not 9.
The same addition does not preserve a ratio.

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Dividing an amount in a ratio

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum Mathematics

Year 7 ratio comparisons and practical proportional models, including units, part-to-whole meaning and assumptions.

Official curriculum source ↗

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