Two pipes A and B can fill a tank in $20$ and $30$ minutes respectively. If both the pipes are used together, how long will it take to fill the tank?

### Concept & Combined Work Rate When two pipes work together, their combined rate of filling is the sum of their individual rates. If Pipe A takes $x$ time and Pipe B takes $y$ time, their combined 1-unit work is $\frac{1}{x} + \frac{1}{y}$. $$ \text{Combined Time} = \frac{xy}{x + y} $$ ### Step-by-Step Solution * **Given:** Time taken by Pipe A = $20$ minutes. Time taken by Pipe B = $30$ minutes. * **Calculate 1-minute work:** Part filled by A in $1$ minute = $\frac{1}{20}$ Part filled by B in $1$ minute = $\frac{1}{30}$ * **Calculate combined 1-minute work:** Part filled by $(A + B)$ in $1$ minute = $\frac{1}{20} + \frac{1}{30}$ $= \frac{3 + 2}{60} = \frac{5}{60} = \frac{1}{12}$ * **Find total time:** Since $\frac{1}{12}$ of the tank is filled in $1$ minute, the entire tank is filled in $12$ minutes. ### Exam Strategy & Shortcut Use the direct formula for two pipes filling: $\frac{a \times b}{a + b}$. Calculation: $\frac{20 \times 30}{20 + 30} = \frac{600}{50} = 12$ minutes. This is much faster than adding fractions manually. ### Common Pitfall A common blunder is taking the average of the times ($\frac{20 + 30}{2} = 25$ minutes). Two pipes working together will always fill the tank faster than the fastest individual pipe. ### Final Answer Therefore, the correct answer is **$12$ minutes**.