The age of a mother today is thrice that of her daughter. After 12 years, the age of the mother will be twice that of her daughter. The present age of the daughter is:

Concept & Logic This involves setting up simultaneous relationships for two distinct timeframes. We express both ages using a single variable based on the present ratio, then add the time elapsed to both expressions to satisfy the future condition. $$ 3x + 12 = 2(x + 12) $$ Step-by-Step Solution * **Given:** The mother's present age is 3 times the daughter's present age. * **Given:** After 12 years, the mother will be twice as old as the daughter. * **Calculation:** Let the daughter's present age be $x$. * The mother's present age is therefore $3x$. * After 12 years, the daughter's age will be $x + 12$. * After 12 years, the mother's age will be $3x + 12$. * Apply the future condition: The mother's future age equals twice the daughter's future age. * $3x + 12 = 2(x + 12)$. * Expand the right side: $3x + 12 = 2x + 24$. * Isolate $x$: $3x - 2x = 24 - 12 \Rightarrow x = 12$. * The daughter's present age is 12 years. Exam Strategy & Shortcut Use the **Age Difference Balancing Method**. Present ratio of Mother : Daughter is $3 : 1$ (Difference = 2 parts). Future ratio (+12 years) is $2 : 1$ (Difference = 1 part). The age difference between two people never changes. Balance the differences by multiplying the future ratio by 2: $(2 \times 2) : (1 \times 2) = 4 : 2$ (Difference = 2 parts). Compare the daughter's ratio parts: Present (1 part) vs. Future (2 parts). The increase is $2 - 1 = 1$ part, which corresponds exactly to the 12 years that passed. So, 1 part = 12 years. The daughter's present age is 1 part, which equals 12 years directly. Common Pitfall Test-takers often forget to add the 12 years to *both* the mother and the daughter when setting up the algebraic equation, incorrectly writing $3x + 12 = 2x$, which severely derails the calculation. Final Answer **Therefore, the correct answer is 12 years.**