Rajan got married 8 years ago. His present age is $\frac{6}{5}$ times his age at the time of his marriage. Rajan's sister was 10 years younger to him at the time of his marriage. The age of Rajan's sister is

### Concept & Strategy There are two critical principles at play here: 1. Formulating an equation that connects present age and a past life event using a fractional multiplier. 2. The core rule of age problems: **Age difference between two individuals is permanent and constant.** If a sister was $10$ years younger a decade ago, she is still exactly $10$ years younger today. ### Step-by-Step Solution * **Given:** * Married $8$ years ago. * Present Age = $\frac{6}{5} \times$ (Age at Marriage) * Sister is $10$ years younger. * **Calculation / Deduction:** * Let Rajan's present age be $x$. * His age at the time of his marriage ($8$ years ago) was $x - 8$. * Set up the fractional equation as given in the problem: * $$x = \frac{6}{5}(x - 8)$$ * $$5x = 6(x - 8)$$ * $$5x = 6x - 48$$ * $$x = 48$$ * Rajan's present age is $48$ years. * His sister is $10$ years younger than him. This gap remains constant throughout their entire lives. * Sister's present age = $48 - 10 = 38$ years. ### Exam Strategy & Shortcut Analyze the fraction $\frac{6}{5}$ directly. This means Present Age is $6$ ratio parts, and Marriage Age is $5$ ratio parts. The difference between these states is exactly $1$ ratio part. This $1$ part represents the actual time passed since the marriage, which is $8$ years. Therefore, $1 \text{ part} = 8 \text{ years}$. Rajan's present age is $6 \text{ parts} = 6 \times 8 = 48$ years. Sister's present age = $48 - 10 = 38$ years. ### Common Pitfall Students often overcomplicate finding the sister's age. They calculate Rajan's past marriage age ($40$), find the sister's past marriage age ($30$), and then add the $8$ years back to get the present age ($38$). While technically correct, recognizing that the gap is permanent bypasses these extra steps entirely. ### Final Answer **Therefore, the correct answer is 38 years.**