Practice · Grade 7 Percentages
Average Percentage — Practice
Training problems on averaging percentages in rising difficulty, plus a boss question on the weighted average. With a hint and full working for each, free.
Q1 of 7
0 correct
Two equally sized classes sit a test. Average the two pass rates.
(30% + 50%) ÷ 2
Quick answer
What is the best way to practise averaging percentages?
For every problem, first decide whether the groups are equal in size. Equal → use the simple mean (p₁ + p₂) ÷ 2. Different sizes → use the weighted average: sum all parts and all wholes separately, then divide, (Σ parts) ÷ (Σ wholes) · 100. Work through five or six problems in rising difficulty and, for each one, check whether the simple mean is even allowed — that is where most mistakes happen.
HowTo
A 4-step solving strategy
This order works for every average-percentage problem — whether the groups are equal or different in size.
- 1Step 1 of 4
Check the group sizes
Are the groups equal? If the text only says "40% and 60%" with no sizes, you can usually use the simple mean. If parts and wholes are given (e.g. "20 out of 50"), think weighted.
- 2Step 2 of 4
Equal groups: simple mean
Add both percentages and divide by 2: (40% + 60%) ÷ 2 = 50%. This is only valid when the groups are genuinely equal in size.
- 3Step 3 of 4
Unequal groups: sum parts and wholes
Add all parts (hits) and, separately, all wholes (group sizes): (20 + 30) hits out of (50 + 150) = 50 out of 200.
- 4Step 4 of 4
Divide, times 100 — then sanity-check
50 ÷ 200 · 100 = 25%. Finally ask: does the result fall between the two individual rates? If not, you slipped somewhere.
Examples
Worked practice examples with full working
Four typical problem types. Try each one yourself first, then compare with the working.
Easy
Average 40% and 60% from equally sized groups
(40% + 60%) ÷ 2
= 100% ÷ 2
= 50%
Check: 50% lies exactly between 40% and 60% ✓
Equal groups — the simple mean is allowed and correct.
Medium
Weighted: 20 of 50 and 30 of 150
(20 + 30) ÷ (50 + 150) · 100
= 50 ÷ 200 · 100
= 25%
Check: 50 out of 200 = 1/4 = 25% ✓
Sum parts and wholes separately first, then divide. The simple mean (40% + 20%) ÷ 2 = 30% would be wrong.
Medium
The classic: 90% of 10 and 1% of 90
(9 + 1) ÷ (10 + 90) · 100
= 10 ÷ 100 · 100
= 10%
Check: 10 out of 100 = 10% ✓
The large group (90 items, only 1%) pulls the result right down. The simple mean would be 45.5% — way off.
Hard
Boss: 48 of 60 and 12 of 40
(48 + 12) ÷ (60 + 40) · 100
= 60 ÷ 100 · 100
= 60%
Check: 60 out of 100 = 60% ✓
The individual rates are 80% and 30%. Their simple mean (55%) is wrong — the bigger group (60) gives more weight to the high rate.
Pitfalls
Common mistakes — and how to avoid them
These five traps come up again and again when averaging percentages.
Using the simple mean for unequal groups
(p₁ + p₂) ÷ 2 is only valid for equally sized groups. If parts and wholes are given, compute the weighted average — otherwise the result is wrong.
Treating the percentages as the weights
The weights are the group sizes (the wholes), not the percentages. 20 out of 50 is weighted by 50, not by 40%.
Mixing parts and wholes
Sum hits with hits and group sizes with group sizes only: (20 + 30) ÷ (50 + 150) — never (20 + 150) or similar.
Adding percentages directly
40% + 60% "= 100%" is meaningless as a final answer. For the simple mean you still divide by the count afterwards.
Not sanity-checking the result
The average must lie between the smallest and the largest individual rate. If it lands outside, there is an arithmetic slip.
Study
Practise with a plan — three quick tips
Choose the method before you calculate
For every problem ask first: equal groups or not? That decision is the real learning content — the arithmetic afterwards is routine.
Keep the counterexample in mind
Remember the 90/10 and 1/90 problem: simple mean 45.5%, but the correct answer is 10%. Knowing this extreme case keeps you from forgetting to weight.
For every wrong answer: why?
Wrong method chosen? Parts and wholes swapped? Note the cause — next time you will spot the mistake straight away.
FAQ
Frequently asked practice questions
Glossary
Terms in one sentence
- Average percentage
- The mean percentage of several percentages — either simply averaged or weighted.
- Simple mean
- Sum of the percentages divided by their count; correct only for equally sized groups.
- Weighted average
- (Σ parts) ÷ (Σ wholes) · 100 — accounts for differing group sizes.
- Part
- The amount that belongs to a percentage (e.g. the hits).
- Whole (base)
- The group size that equals 100%.
- Weight
- In a weighted average, the size of a group — that is, its whole.
- Percentage point
- The absolute difference between two percentages (e.g. from 40% to 60% is 20 percentage points).