The average age of the mother and her six children is 12 years which is reduced by 5 years if the age of the mother is excluded. How old is the mother?

### Concept & Strategy When a member leaves and the average drops dramatically, that member held a disproportionately high value. The departing member's value is the new average plus the total decrease they "took away" from the rest of the group. $$Total = Average \times Number of Entities$$ ### Step-by-Step Solution * **Calculate initial total age:** The group consists of the mother + 6 children = 7 people. $$Sum_{initial} = 7 \times 12 = 84 \text{ years}$$ * **Calculate new total age (without mother):** There are now 6 children left. The average is reduced by 5 years ($12 - 5 = 7$ years). $$Sum_{new} = 6 \times 7 = 42 \text{ years}$$ * **Find the mother's age:** Subtract the children's total from the entire group's total: $$Mother = 84 - 42 = 42 \text{ years}$$ ### Exam Strategy & Shortcut Use the deviation method to bypass calculating totals. When the mother leaves, she takes her share of the old average (12 years). Her departure also causes the remaining 6 children to lose 5 years each from their average. She was providing those extra years. $$Age = \text{Old Average} + (\text{Remaining Members} \times \text{Decrease in Average})$$ $$Age = 12 + (6 \times 5)$$ $$Age = 12 + 30 = 42 \text{ years}$$ ### Common Pitfall Miscounting the initial number of people. Students often read "six children" and use 6 as the total initial group size instead of 7 (6 children + 1 mother). Always count the specific entities making up the average. ### Final Answer **Therefore, the correct answer is 42 years.**