Two pipes A and B together can fill a cistern in 4 hours. Had they been opened separately, then B would have taken 6 hours more than A to fill the cistern. How much time will be taken by A alone to fill the cistern?

### Concept & Work Equation When two entities work together, their combined rate is the sum of their individual rates. We can set up an algebraic equation based on their rates and solve it, or use the options to find the correct fit much faster. $$ \frac{1}{T_A} + \frac{1}{T_B} = \frac{1}{T_{Total}} $$ ### Step-by-Step Solution 1. **Set Up the Variables:** * Let the time taken by pipe A alone = $x$ hours. * Time taken by pipe B alone = $(x + 6)$ hours. 2. **Formulate the Equation:** * The combined rate is the sum of individual rates: * $\frac{1}{x} + \frac{1}{x + 6} = \frac{1}{4}$ 3. **Solve Using Options (Highly Recommended):** * **Test Option (a) $x = 1$:** $\frac{1}{1} + \frac{1}{7} \neq \frac{1}{4}$ * **Test Option (b) $x = 2$:** $\frac{1}{2} + \frac{1}{8} = \frac{5}{8} \neq \frac{1}{4}$ * **Test Option (c) $x = 6$:** $\frac{1}{6} + \frac{1}{12}$ * Common denominator is 12: $\frac{2}{12} + \frac{1}{12} = \frac{3}{12} = \frac{1}{4}$. * This perfectly matches the right side of our equation. ### Exam Strategy & Shortcut For questions forming quadratic equations (like $\frac{1}{x} + \frac{1}{x+k} = \frac{1}{t}$), **always plug in the options**. Solving the quadratic $x^2 - 2x - 24 = 0$ takes considerably more time and is prone to factorization errors under pressure. ### Common Pitfall Attempting to solve the quadratic equation manually often leads to sign errors ($x^2 - 2x - 24 = 0 \Rightarrow (x-6)(x+4)=0 \Rightarrow x=6$). While correct, it wastes precious time. ### Final Answer Therefore, the correct answer is **6 hrs**.