If you know two group averages, their combined average and the total number of pupils, you can work backwards to find the group sizes. Start by pretending everyone has the lower group average. The extra points needed to reach the actual total tell you how many pupils belong to the higher-average group.
Suppose there are 20 pupils. Group A averages 12 points, Group B averages 18, and all 20 pupils average 14.4. The higher-average group has 8 pupils, and the other group has 12.
- Multiply the overall average by the total number of pupils.
- Work out the total if everyone were counted at the lower average.
- Divide the extra points by the difference between the group averages.
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Use averages to rebuild totals
An average lets you calculate a group’s total. If 5 pupils average 12 points, their total is 5 × 12 = 60 points. Their individual scores can vary.
In the group-size problem, think of each pupil as carrying their group’s average for the purpose of counting the total. Reclassifying one pupil from the 12-point group to the 18-point group adds 6 points to this model.
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Worked example | Find the number in each group
The actual total is 20 × 14.4 = 288 points. If all 20 pupils were in Group A, the total would be 20 × 12 = 240 points. The actual total is 288 − 240 = 48 points above that starting total.
Each pupil counted in Group B adds 18 − 12 = 6 points. Group B therefore has 48 ÷ 6 = 8 pupils. Group A has 20 − 8 = 12 pupils.
Check the answer with the group totals. Group A contributes 12 × 12 = 144 points. Group B contributes 8 × 18 = 144 points. Their combined total is 288 points, and 288 ÷ 20 = 14.4.
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Try the method with another pair of groups
There are 30 pupils in two groups. The group averages are 10 and 15 points, and the combined average is 12 points. The actual total is 30 × 12 = 360 points. If everyone averaged 10, the total would be 300 points.
The extra 60 points come in groups of 15 − 10 = 5 points per pupil. So 12 pupils belong to the higher-average group and 18 pupils belong to the lower-average group.
Ask your child what the 5 represents. It is the extra number of points contributed when a pupil is counted in the higher-average group.
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Frequently asked questions
Why can I start with the lower average?
It gives a total that is easy to calculate. Each pupil in the higher-average group then contributes the same extra amount in the model.
What if both group averages are the same?
The overall average is the same regardless of group sizes. You need more information to find how the pupils are divided.
Can I start with the higher average?
Yes. Work out how far the real total falls below that starting total. Divide the shortfall by the gap between the averages to find the size of the lower-average group.
Contents · Official syllabus and next practice
Official syllabus and next practice
Read the official MOE Primary Mathematics syllabus
Try solving the first example by starting with 20 pupils at 18 points each. Check whether the shortfall gives 12 pupils in the lower-average group.