Two pieces A and B can fill a tank in 18 hrs and 6 hrs respectively. If both the pipes are opened simultaneously, how much time will be taken to fill the tank?

### Concept & Formula When two pipes work together to fill a tank, their combined filling rate is the sum of their individual rates. If pipe A takes $x$ hours and pipe B takes $y$ hours, their individual rates are $\frac{1}{x}$ and $\frac{1}{y}$ of the tank per hour. $$ \text{Combined Rate} = \frac{1}{x} + \frac{1}{y} $$ $$ \text{Time Taken} = \frac{x \times y}{x + y} $$ ### Step-by-Step Solution 1. **Identify Individual Rates**: - Pipe A takes 18 hours. Its rate is $\frac{1}{18}$ tank/hour. - Pipe B takes 6 hours. Its rate is $\frac{1}{6}$ tank/hour. 2. **Calculate Combined Rate**: - $\text{Combined Rate} = \frac{1}{18} + \frac{1}{6}$ - Find a common denominator, which is 18: - $\text{Combined Rate} = \frac{1}{18} + \frac{3}{18} = \frac{4}{18} = \frac{2}{9}$ tank/hour. 3. **Determine Total Time**: - Time taken is the reciprocal of the combined rate. - $\text{Time} = \frac{9}{2} \text{ hours} = 4 \frac{1}{2} \text{ hours}$. ### Exam Strategy & Shortcut For two pipes filling together, use the direct formula $T = \frac{a \times b}{a + b}$. $T = \frac{18 \times 6}{18 + 6} = \frac{108}{24}$. Dividing numerator and denominator by 12 gives $\frac{9}{2}$, which is $4.5$ or $4 \frac{1}{2}$ hours. ### Common Pitfall A common mistake is adding the times directly ($18 + 6 = 24$ hours) instead of adding their rates (inverses). Always add rates when combining work. ### Final Answer Therefore, the correct answer is **$4 \frac{1}{2}$ hrs**.