Eight year ago, Poorvi's age was equal to the sum of the present ages of her one son and one daughter. Five years hence, the respective ratio between the ages of her daughter and her son that time will be 7 : 6. If Poorvi's husband is 7 years elder to her and his present age is three times the present age of their son, what is the present age of the daughter? (in years)

### Concept & Strategy While this problem looks like a nightmare of four variables ($P, H, S, D$), it contains a massive **Divisibility Loophole** hidden in the future ratio. Using this loophole allows you to bypass the algebra entirely in a multiple-choice format. ### Step-by-Step Solution * **Standard Algebraic Method (For Understanding):** * Let present ages be $P$ (Poorvi), $S$ (Son), $D$ (Daughter), and $H$ (Husband). * **Equations:** 1. $P - 8 = S + D \implies P = S + D + 8$ 2. $\frac{D + 5}{S + 5} = \frac{7}{6} \implies 6D + 30 = 7S + 35 \implies 7S = 6D - 5$ 3. $H = P + 7$ 4. $H = 3S$ * **Substitution:** Combine Eq 3 and 4: $P + 7 = 3S \implies P = 3S - 7$. * Equate the two expressions for $P$ (Eq 1 and substituted Eq): $$ S + D + 8 = 3S - 7 $$ $$ D + 15 = 2S $$ * Multiply by 7 to make substituting Eq 2 easier: $$ 7(D + 15) = 14S $$ $$ 7D + 105 = 2(7S) $$ * Substitute $7S = 6D - 5$ into the equation: $$ 7D + 105 = 2(6D - 5) $$ $$ 7D + 105 = 12D - 10 $$ $$ 5D = 115 \implies D = 23 $$ ### Exam Strategy & Shortcut **The Divisibility Shortcut (MANDATORY):** Look at the second condition: "Five years hence, the ratio between the ages of her daughter and son... will be 7 : 6." This means: $\frac{\text{Daughter} + 5}{\text{Son} + 5} = \frac{7}{6}$ Therefore, $(\text{Daughter's Present Age} + 5)$ MUST be a multiple of 7. Let's test the given options for the Daughter's present age: (a) $15 + 5 = 20$ (Not divisible by 7) (b) $23 + 5 = 28$ (**Divisible by 7!**) (c) $19 + 5 = 24$ (Not divisible by 7) (d) $27 + 5 = 32$ (Not divisible by 7) (e) $13 + 5 = 18$ (Not divisible by 7) Only option (b) works. You can solve this 4-variable problem in literally 5 seconds by checking options. ### Common Pitfall The biggest mistake students make is diving headfirst into complex substitution algebra without scanning the problem for ratio constraints. Spending 3-4 minutes solving a matrix of four variables destroys your exam pacing when a 5-second divisibility check was available. ### Final Answer **Therefore, the correct answer is 23 years.**