Pipe Emptying Rate Calculation
First, determine the rate at which each pipe empties the tank individually.
- Pipe 1 empties the full tank in 6 hours. Its rate is $\frac{1}{6}$ of the tank per hour.
- Pipe 2 empties the full tank in 18 hours. Its rate is $\frac{1}{18}$ of the tank per hour.
Combined Pipe Emptying Rate
When both pipes work together, their rates add up.
Combined rate = Rate of Pipe 1 + Rate of Pipe 2
Combined rate = $\frac{1}{6} + \frac{1}{18}$
To add these fractions, find a common denominator, which is 18.
Combined rate = $\frac{1 \times 3}{6 \times 3} + \frac{1}{18} = \frac{3}{18} + \frac{1}{18}$
Combined rate = $\frac{3+1}{18} = \frac{4}{18}$
Simplify the combined rate:
Combined rate = $\frac{2}{9}$ of the tank per hour.
Combined Time Calculation
The time taken to empty the tank when working together is the reciprocal of the combined rate.
Time = $\frac{1}{\text{Combined Rate}}$
Time = $\frac{1}{\frac{2}{9}}$
Time = $\frac{9}{2}$ hours
Convert this improper fraction to a decimal or mixed number:
Time = $4.5$ hours
Therefore, if both pipes work together, they will empty the full tank in 4.5 hours.
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