Averages and ages
An average is the total divided by the number of items, so almost every average question is solved by working with totals. Age questions are the same idea: each statement becomes a short equation, and everyone's age moves by the same number of years.
Formulas and when they apply
- Average = total ÷ number of items, so total = average × number of items
- Combined average of two groups = (n₁a₁ + n₂a₂) ÷ (n₁ + n₂)
- Replacing one member: new − old = change in average × number of membersApplies when: group size unchanged
- Adding one item: its value = new average + (new average − old average) × old number of itemsApplies when: one item added
- Average of n consecutive numbers (evenly spaced) = the middle value, or the mean of the first and last
- Age after t years = present age + t, and age t years ago = present age − tApplies when: everyone's age changes by the same t
- Boys : girls = (class average − girls' average) : (boys' average − class average)Applies when: two groups, overall average between the two group averages
Worked examples
Example 1. 15 pupils have an average mark of 40 and 25 other pupils have an average mark of 56. What is the average of all 40 pupils?
Solution. Total of the first group = 15 × 40 = 600. Total of the second group = 25 × 56 = 1,400. Together 2,000 marks for 40 pupils, so the average is 2,000 ÷ 40 = 50.
Example 2. The average age of 6 friends is 50 years. One friend aged 44 is replaced and the average becomes 52. How old is the new friend?
Solution. The average rises by 2 for 6 friends, so the total rises by 6 × 2 = 12 years. The new friend is 44 + 12 = 56 years old.
Example 3. A father is 30 years older than his son. In 5 years he will be 3 times as old as his son. How old is the son now?
Solution. Let the son be S now, so the father is S + 30. In 5 years: S + 35 = 3(S + 5) = 3S + 15. So 20 = 2S and S = 10. Check: in 5 years the son is 15 and the father is 45, which is 3 times 15.
Shortcuts and their limits
- To find the combined average, measure how far the averages are apart and share that gap in the ratio of the group sizes.Does not apply when: It works for two groups at a time and needs the sizes of both groups; it does not work if one size is missing.
- When one value in a group changes, only the change matters: total change = change in value, and new average = old average + change ÷ number of items.Does not apply when: Group size must stay the same; if members are added or removed, recompute the totals.
- In multiple-choice age questions, test the options against each condition.Does not apply when: Check every condition in the question, because an option may satisfy just one of them.
Common mistakes
- Averaging two averages without weighting by group size.
- Using the old number of items when the group size has changed.
- Adding years to only one person in an age problem, instead of everyone.
- Forgetting that the difference between two ages never changes with time.
- Reading a ratio of ages as the ages themselves.
Independent practice made by EduVistaHQ with original questions. It is not an official examination paper, it is not affiliated with or endorsed by any examination body or company, and a practice score does not predict a real result.