The ratio between the ages of Neelam and Shiny is 5 : 6 respectively. If the ratio between the one-third age of Neelam and half of Shiny's age is 5 : 9, then what is Shiny's age?

### Concept & Logic For a system of linear equations to have a unique numerical solution, the equations must be mathematically independent. If the second given condition is just a manipulated restatement of the first condition, you have infinite possible solutions. $$ \text{If Eq } A \equiv \text{Eq } B, \text{ the exact value cannot be found.} $$ ### Step-by-Step Solution * **Given:** $\text{Neelam} : \text{Shiny} = 5 : 6$. Also, $(\frac{1}{3} \text{ Neelam}) : (\frac{1}{2} \text{ Shiny}) = 5 : 9$. * **Calculation:** Let Neelam's age be $N$ and Shiny's age be $S$. * The first condition states: $\frac{N}{S} = \frac{5}{6}$ * Let's analyze the second condition mathematically: $\frac{\frac{N}{3}}{\frac{S}{2}} = \frac{5}{9}$ * Simplify the complex fraction by multiplying by the reciprocal: $\frac{N}{3} \times \frac{2}{S} = \frac{5}{9}$ * This becomes: $\frac{2N}{3S} = \frac{5}{9}$ * Now, isolate $\frac{N}{S}$ by multiplying both sides by $\frac{3}{2}$: $\frac{N}{S} = \frac{5}{9} \times \frac{3}{2} = \frac{15}{18}$ * Simplify the fraction $\frac{15}{18}$ by dividing numerator and denominator by 3: $\frac{N}{S} = \frac{5}{6}$ * The second condition simply yields $\frac{N}{S} = \frac{5}{6}$, which is the exact same information provided in the first sentence. We have no absolute time shifts or fixed numerical anchors to determine their actual ages. ### Exam Strategy & Shortcut **Information Audit:** Skim the problem for any real absolute numbers (like "5 years ago", "sum is 50", etc.). Both conditions provided here are purely relative ratios. Without at least one absolute numerical anchor tied to time or a sum, it is fundamentally impossible to extract a specific age. You can instantly mark "Cannot be determined". ### Common Pitfall Blindly substituting $5x$ and $6x$ into the second equation and thinking you made a mistake when everything cancels out to $1 = 1$ or $5 = 5$. This cancellation is the mathematical proof that the data is insufficient. ### Final Answer Therefore, the correct answer is **Cannot be determined**.