Pipe Filling Time Calculation

This problem involves calculating the time taken to fill a tank when two pipes, Pipe A and Pipe B, are working together. We are given the individual times each pipe takes to fill the tank.

Understanding Rates

To solve this, we first determine the rate at which each pipe fills the tank. The rate is the fraction of the tank that a pipe can fill in one unit of time (in this case, one minute).

• Pipe A's Rate: Pipe A fills the entire tank in 20 minutes. Therefore, in 1 minute, Pipe A fills $1/20$ of the tank. Rate of Pipe A = $\frac{1}{20}$ tank per minute.
• Pipe B's Rate: Pipe B fills the entire tank in 16 minutes. Therefore, in 1 minute, Pipe B fills $1/16$ of the tank. Rate of Pipe B = $\frac{1}{16}$ tank per minute.

Calculating Combined Rate

When both pipes are opened together, their rates add up. The combined rate is the sum of the individual rates.

Combined Rate = Rate of Pipe A + Rate of Pipe B

Combined Rate = $\frac{1}{20} + \frac{1}{16}$

To add these fractions, we find a common denominator. The least common multiple (LCM) of 20 and 16 is 80.

Combined Rate = $\frac{1 \times 4}{20 \times 4} + \frac{1 \times 5}{16 \times 5}$

Combined Rate = $\frac{4}{80} + \frac{5}{80}$

Combined Rate = $\frac{4 + 5}{80}$

Combined Rate = $\frac{9}{80}$ tank per minute.

This means that together, the pipes fill $9/80$ of the tank every minute.

Determining Time Taken Together

The time taken to fill the tank together is the reciprocal of the combined rate.

Time Taken = $\frac{1}{\text{Combined Rate}}$

Time Taken = $\frac{1}{\frac{9}{80}}$

Time Taken = $\frac{80}{9}$ minutes.

Approximating the Result

To find the approximate time, we convert the fraction $80/9$ minutes into a mixed number or decimal:

$\frac{80}{9} = 8 \frac{8}{9}$ minutes.

As a decimal, $80 \div 9 \approx 8.89$ minutes.

Comparing this value to the given options:

• 4 minutes
• 9 minutes
• 12 minutes
• 13 minutes

The calculated time, approximately 8.89 minutes, is closest to 9 minutes.