My sister was born 26 years after my mother was born, and I was born 28 years after my father was born. The current average age of my family is 32.5 years. Three years from now, my sister will get married and leave the family, and at that time, the average age of the family will be 41 years. What is the present age of my mother (in years)?

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Accepted Answer

**Solution:**

1. Let the family members be mother, father, sister, and you. So, total 4 members.

2. Given the present average age of the family is 32.5 years.
   - Total present age of family = $32.5 	imes 4 = 130$ years.

3. Let the present age of mother be $M$ years.
   - Sister's age = $M - 26$ (since sister was born 26 years after mother)

4. Let the present age of father be $F$ years.
   - Your age = $F - 28$ (since you were born 28 years after father)

5. Sum of present ages:
   $$M + F + (M - 26) + (F - 28) = 130$$
   Simplify:
   $$M + F + M - 26 + F - 28 = 130$$
   $$2M + 2F - 54 = 130$$
   $$2M + 2F = 184$$
   $$M + F = 92 	ag{1}$$

6. After 3 years, sister leaves the family. So, family members left = 3.

7. Ages after 3 years:
   - Mother: $M + 3$
   - Father: $F + 3$
   - Sister: $(M - 26) + 3 = M - 23$ (but she leaves, so not counted)
   - You: $(F - 28) + 3 = F - 25$

8. Sum of ages of remaining family members after 3 years:
   $$ (M + 3) + (F + 3) + (F - 25) = 41 	imes 3 = 123$$

9. Simplify the sum:
   $$M + 3 + F + 3 + F - 25 = 123$$
   $$M + 2F - 19 = 123$$
   $$M + 2F = 142 	ag{2}$$

10. From equation (1): $M + F = 92$

11. Subtract (1) from (2):
    $$ (M + 2F) - (M + F) = 142 - 92$$
    $$ F = 50$$

12. Substitute $F = 50$ in (1):
    $$ M + 50 = 92 ightarrow M = 42$$

13. Check the options: 42 is option (e), but let's verify the calculations carefully.

14. Re-examining step 7: The ages after 3 years for remaining members are mother, father, and you.

Sum after 3 years:
$$ (M + 3) + (F + 3) + ((F - 28) + 3) = 123$$
$$ M + 3 + F + 3 + F - 25 = 123$$
$$ M + 2F - 19 = 123$$
$$ M + 2F = 142$$

From (1): $M + F = 92$

Subtracting:
$$ (M + 2F) - (M + F) = 142 - 92$$
$$ F = 50$$

Then $M = 92 - 50 = 42$

So, the present age of mother is 42 years.

**Answer:** $42$ years

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