Percentage comparisons | Year 8

01 / LEARN

Understand the idea.

Compare shares, not just counts

When totals differ, compare rates, not just successful counts. Divide each success count by its own number of attempts, then convert to a percentage. This puts both shares on the same out-of-one-hundred scale without pretending the groups had equal sizes.

A higher observed percentage describes the recorded results, but does not remove differences in difficulty or conditions. Check that the quantities measure the same thing before drawing a broader conclusion. More successes can still represent a lower success rate.

A percentage-point difference is not a percentage increase

Subtract two percentages to find a difference in percentage points. A rise from $30 %$ to $36 %$ is six percentage points. To find the relative percentage increase, divide that difference by the original rate: $\frac{6}{30} \times 100 % = 20 %$ .

State which comparison you mean: a difference between rates, a change relative to the original rate, or an amount of money. A larger discount percentage need not save more dollars when the starting prices differ.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Different group totals

Team A makes 12 of 40 shots. Team B makes 18 of 75 shots. Compare their success rates.

Worked solution

1
Calculate Team A's rate.
$\frac{12}{40} \times 100 % = 30 %$
Team A's twelve successes must be compared with its own forty attempts.
2
Calculate Team B's rate.
$\frac{18}{75} \times 100 % = 24 %$
Team B's eighteen successes came from a different total of seventy-five attempts.
3
Compare the percentages.
$30 - 24 = 6 percentage points$
Team A has the larger successful share, even though Team B scored more times.

Answer

A: 30%; B: 24%; Team A has the higher success rate.

EXAMPLE 02

Two ways to describe a rise

A survey result rises from 30% to 36%. Find the percentage-point rise and relative percentage increase.

Worked solution

1
Find the percentage-point difference.
$36 - 30 = 6 percentage points$
Subtracting the two percentage rates measures their direct gap.
2
Compare the gap with the original rate.
$\frac{6}{30} = 0.2$
The original thirty-percent rate is the reference whole for relative change.
3
Convert to a relative percentage increase.
$0.2 \times 100 % = 20 %$
The six-point increase is one fifth of the original rate.

Answer

6 percentage points; 20% relative increase.

03 / YOUR TURN

Try it yourself.

Check your understanding

Which is greater: 8 successes out of 20, or 15 out of 50?

Show answer & reasoning

8 out of 20.

The rates are $\frac{8}{20} \times 100 % = 40 %$ and $\frac{15}{50} \times 100 % = 30 %$ . Therefore eight successes out of twenty is the larger proportion.

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6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Write 18 out of 60 as a percentage.
2 marks
Compare 14 out of 20 with 18 out of 30. Which rate is higher?
2 marks
A rate rises from 40% to 50%. Find the difference in percentage points.
2 marks
For the rise from 40% to 50%, find the relative percentage increase.
2 marks
Shop A takes 15% off a 200 AUD item. Shop B takes 20% off an 80 AUD item. Which discount saves more dollars?
3 marks
A class of 25 has 20 completed projects; another class of 40 has 30. Compare completion rates. Give one reason the rates alone cannot prove which class worked harder.
3 marks

Read the worked solutions

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

Write 18 out of 60 as a percentage.
2 marks
$18 \div 60 \times 100 % = 30 %$ .
30%
Compare 14 out of 20 with 18 out of 30. Which rate is higher?
2 marks
$14 \div 20 = 0.7$ .
$18 \div 30 = 0.6$ .
14 out of 20: 70%, compared with 60%.
A rate rises from 40% to 50%. Find the difference in percentage points.
2 marks
$50 - 40 = 10$ percentage points.
10 percentage points
For the rise from 40% to 50%, find the relative percentage increase.
2 marks
The change is 10 percentage points.
$10 \div 40 \times 100 % = 25 %$ .
25%
Shop A takes 15% off a 200 AUD item. Shop B takes 20% off an 80 AUD item. Which discount saves more dollars?
3 marks
Shop A: $0.15 \times 200 = 30$ .
Shop B: $0.20 \times 80 = 16$ .
Shop B has the larger percentage, but Shop A saves more dollars.
Shop A saves 30 AUD; Shop B saves 16 AUD.
A class of 25 has 20 completed projects; another class of 40 has 30. Compare completion rates. Give one reason the rates alone cannot prove which class worked harder.
3 marks
$20 \div 25 \times 100 % = 80 %$ .
$30 \div 40 \times 100 % = 75 %$ .
Rates compare completed shares, not effort directly.
80% and 75%; the first rate is higher, but task difficulty, time or support may differ.

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Curriculum reference & lesson scope

Australian Curriculum v9 / SA Curriculum

Percentage increase/decrease; finding an original amount; discounts and mark-ups; percentage comparisons; budgets and purchase decisions; modelling and checking financial problems with digital tools. Simple interest can be offered here as a school-dependent application and developed in Year 9.

Official curriculum source ↗

In this lesson

Understand the idea.

Compare shares, not just counts

A percentage-point difference is not a percentage increase

See the method in action.

Different group totals

Calculate Team A's rate.

Calculate Team B's rate.

Compare the percentages.

Two ways to describe a rise

Find the percentage-point difference.

Compare the gap with the original rate.

Convert to a relative percentage increase.

Try it yourself.

Check your understanding

Keep practising