Calculating Tap Filling Time
This problem involves calculating the time taken to fill a cistern by two taps with different filling rates.
Step 1: Determine Individual Tap Rates
Find the rate at which each tap fills the cistern per hour.
- Tap A fills the cistern in 6 hours. Rate of A = \(\frac{1}{6}\) cistern/hour.
- Tap B fills the cistern in 7.5 hours (which is \(\frac{15}{2}\) hours). Rate of B = \(\frac{1}{15/2} = \frac{2}{15}\) cistern/hour.
Step 2: Calculate Combined Rate and Work Done
Calculate how much work both taps complete when open together.
- Combined Rate (A + B) = Rate A + Rate B = \(\frac{1}{6} + \frac{2}{15}\).
- To add these fractions, find a common denominator (30): \((\frac{1 \times 5}{6 \times 5}) + (\frac{2 \times 2}{15 \times 2}) = \frac{5}{30} + \frac{4}{30} = \frac{9}{30} = \frac{3}{10}\) cistern/hour.
- Work done in the first 2 hours = Combined Rate \(\times\) Time = \(\frac{3}{10} \times 2 = \frac{6}{10} = \frac{3}{5}\) of the cistern.
Step 3: Calculate Remaining Work
Determine the portion of the cistern left to be filled.
- Total cistern capacity = 1.
- Remaining Work = Total Capacity - Work Done = \(1 - \frac{3}{5} = \frac{2}{5}\) of the cistern.
Step 4: Calculate Time for Tap A to Fill Remaining Work
Find the time Tap A needs to fill the remaining portion alone.
- Time = Remaining Work / Rate of A
- Time = \((\frac{2}{5}) \div (\frac{1}{6})\)
- Time = \(\frac{2}{5} \times 6 = \frac{12}{5}\) hours.
- Converting to decimal: \(\frac{12}{5} = 2.4\) hours.
Therefore, Tap A will take 2.4 hours to fill the remaining cistern.
Similar Questions
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