A couple has a son and a daughter. The age of the father is four times that of the son and the age of the daughter is one-third of that of her mother. The wife is 6 years younger to her husband and the sister is 3 years older than her brother. The mother's age is

### Concept & Logic This is a simultaneous equation problem involving four variables. The key is to express all family members' ages in terms of a single variable, typically the mother's or father's age, using substitution. $$ M = F - 6 $$ $$ D = S + 3 $$ ### Step-by-Step Solution * **Given:** Father ($F$) = $4 \times$ Son ($S$). Daughter ($D$) = Mother ($M$) / $3$. Mother ($M$) = $F - 6$. Daughter ($D$) = $S + 3$. * **Calculation:** Substitute $D$ and $S$ in terms of the parents' ages into the sibling equation. * We know $S = F / 4$ and $D = M / 3$. * Substituting into $D = S + 3$, we get $M / 3 = (F / 4) + 3$. * Multiply the entire equation by 12 to clear fractions: $4M = 3F + 36$. * Substitute $F = M + 6$ into the equation: $4M = 3(M + 6) + 36$. * Expand and solve: $4M = 3M + 18 + 36 \implies 4M = 3M + 54$. * Therefore, $M = 54$. ### Exam Strategy & Shortcut **Working Backwards from Options:** The mother's age must be divisible by 3 (since the daughter's age is $M/3$ and usually an integer in these problems). Options (a) 42, (b) 48, (c) 54, (d) 63 are all divisible by 3. Test $M = 54$: $D = 18$. $F = 54 + 6 = 60$. $S = 60 / 4 = 15$. Is $D = S + 3$? $18 = 15 + 3$. Yes! It matches perfectly. ### Common Pitfall Misinterpreting "younger to" or "older than" relationships leads to reversed signs (e.g., writing $M = F + 6$ instead of $F - 6$). Always double-check who is the older person. ### Final Answer Therefore, the correct answer is **54 years**.