A swimming pool is filled by three pipes with uniform flow. The first two pipes operating simultaneously fill the pool in the same time during which the pool is filled by the third pipe alone. The second pipe fills the pool 5 hours faster than the first pipe and 4 hours slower than the third pipe. The time required by the first pipe is

### Concept & Relative Time Equations When given relationships in time between three entities where the combined work of two equals the third, express all times in terms of a single variable and test the given options against the work equation. ### Step-by-Step Solution 1. **Define the Variables:** * Let the time taken by the first pipe = $x$ hours. * The second pipe fills 5 hours faster than the first, so time for second pipe = $(x - 5)$ hours. * The second pipe fills 4 hours slower than the third pipe, which means the third pipe is 4 hours faster than the second. Time for third pipe = $(x - 5) - 4 = (x - 9)$ hours. 2. **Formulate the Equation:** * The problem states: Rate of first + Rate of second = Rate of third. * $\frac{1}{x} + \frac{1}{x - 5} = \frac{1}{x - 9}$ 3. **Use Option Elimination:** * **Test Option (a) 6:** $\frac{1}{6} + \frac{1}{1} = \frac{7}{6}$. RHS = $\frac{1}{6 - 9} = -\frac{1}{3}$. (Invalid, negative time). * **Test Option (b) 10:** $\frac{1}{10} + \frac{1}{5} = \frac{3}{10}$. RHS = $\frac{1}{10 - 9} = 1$. (Not equal). * **Test Option (c) 15:** * LHS: $\frac{1}{15} + \frac{1}{10} = \frac{2 + 3}{30} = \frac{5}{30} = \frac{1}{6}$. * RHS: $\frac{1}{15 - 9} = \frac{1}{6}$. * LHS = RHS. This is the correct value. ### Exam Strategy & Shortcut Any question asking to solve $\frac{1}{x} + \frac{1}{x-a} = \frac{1}{x-b}$ must be solved via options. Solving algebraically yields $x^2 - 18x + 45 = 0 \Rightarrow (x-15)(x-3) = 0$. Since $x$ cannot be 3 (as $x-5$ and $x-9$ would be negative), $x=15$. While factorable, testing 10 and 15 is vastly quicker. ### Common Pitfall Setting up the time relations backwards. Saying the second pipe is "5 hours faster" means it takes *less* time, so $(x - 5)$, not $(x + 5)$. Misinterpreting "faster/slower" as addition/subtraction incorrectly ruins the equation. ### Final Answer Therefore, the correct answer is **15 hrs**.