Question

A pipe can fill a tank in 6 hours, and another pipe can empty it in 8 hours. If both are opened together, how many hours will it take to fill the tank? (Enter the answer as a fraction or decimal.)

Hint:

Calculate net rate by subtracting emptying rate from filling rate and take reciprocal for time.

Answer 1

We need the time to fill the tank with both pipes open.
- Step 1: Determine rates. Filling pipe rate = \( \frac{1}{6} \) tank/hour. Emptying pipe rate = \( \frac{1}{8} \) tank/hour.
- Step 2: Calculate net rate. Net rate = filling rate - emptying rate:
\[ \frac{1}{6} - \frac{1}{8} = \frac{4 - 3}{24} = \frac{1}{24} \, \text{tank/hour} \] - Step 3: Find time. Time = \( \frac{1}{\text{net rate}} = \frac{1}{\frac{1}{24}} = 24 \) hours.
- Step 4: Verify. Work done in 24 hours: \( 24 \times \frac{1}{6} = 4 \) tanks filled, \( 24 \times \frac{1}{8} = 3 \) tanks emptie(d) Net = \( 4 - 3 = 1 \) tank. Correct.
- Step 5: Alternative approach. LCM of 6 and 8 = 24. In 24 hours:
- Fill: \( \frac{24}{6} = 4 \) tanks.
- Empty: \( \frac{24}{8} = 3 \) tanks.
- Net: 1 tank in 24 hours.
- Step 6: Answer. Non-MCQ: 24 hours.
Thus, the answer is 24.