The sum of the ages of a daughter and her mother is 56 years. After four years, the age of the mother will be three times that of the daughter. At present their ages are

### Concept & Logic The total sum of ages for a group increases proportionally with time. For two people, the total sum increases by $2 \times (\text{years passed})$. Calculate the future total sum first, then distribute it according to the future ratio condition to find the individual ages. ### Step-by-Step Solution * **Given:** * Present sum (Daughter + Mother) = $56$ years. * After $4$ years: Mother = $3 \times$ Daughter. * **Calculation / Deduction:** * In $4$ years, both the mother and the daughter will age by $4$ years. * Total sum after $4$ years = $56 + 4 + 4 = 64$ years. * Let the future daughter's age be $x$. The future mother's age is $3x$. * $$x + 3x = 64$$ * $$4x = 64$$ * $$x = 16$$ * Future daughter = $16$ years. Future mother = $3(16) = 48$ years. * To find their present ages, subtract the $4$ years back: * Present Daughter = $16 - 4 = 12$ years. * Present Mother = $48 - 4 = 44$ years. ### Exam Strategy & Shortcut Use Option Elimination by adding $4$ to both numbers in the choices and checking if the mother is exactly thrice the daughter's age. (a) $10+4=14$, $46+4=50$ (Here, $50 \neq 3 \times 14$) (b) $12+4=16$, $44+4=48$ (Here, $48 = 3 \times 16$) - Immediate match! You don't need to test the rest. ### Common Pitfall Students often set up the algebraic equation directly as $x + 3x = 56$, completely forgetting to advance the total sum by $8$ years before applying the future ratio. Always synchronize your timeframes. ### Final Answer **Therefore, the correct answer is 12 years, 44 years.**