A cistern can be filled by one tap in 5 hours and by another in 4 hours. How long will it take to fill if both the taps are opened simultaneously?

Answer

To solve the problem of how long it will take to fill a cistern with two taps opened simultaneously, we can follow these steps:

### Step 1: Determine the filling rates of each tap.
- Tap A fills the cistern in 5 hours. Therefore, its rate of filling is:
\[ \text{Rate of Tap A} = \frac{1 \text{ cistern}}{5 \text{ hours}} = \frac{1}{5} \text{ cistern per hour} \]
- Tap B fills the cistern in 4 hours. Therefore, its rate of filling is:
\[ \text{Rate of Tap B} = \frac{1 \text{ cistern}}{4 \text{ hours}} = \frac{1}{4} \text{ cistern per hour} \]

### Step 2: Calculate the combined filling rate of both taps.
- When both taps are opened, their rates of filling add up:
\[ \text{Combined Rate} = \text{Rate of Tap A} + \text{Rate of Tap B} = \frac{1}{5} + \frac{1}{4} \]
- To add these fractions, we need a common denominator. The least common multiple of 5 and 4 is 20:
\[ \frac{1}{5} = \frac{4}{20}, \quad \frac{1}{4} = \frac{5}{20} \]
- Now, adding these gives:
\[ \text{Combined Rate} = \frac{4}{20} + \frac{5}{20} = \frac{9}{20} \text{ cistern per hour} \]

### Step 3: Calculate the time taken to fill the cistern.
- If both taps together fill \(\frac{9}{20}\) of the cistern in one hour, we can find out how long it will take to fill 1 cistern:
\[ \text{Time} = \frac{1 \text{ cistern}}{\text{Combined Rate}} = \frac{1}{\frac{9}{20}} = \frac{20}{9} \text{ hours} \]

### Conclusion:
- Therefore, the time taken to fill the cistern when both taps are opened simultaneously is:
\[ \frac{20}{9} \text{ hours} \approx 2.22 \text{ hours} \text{ or } 2 \text{ hours and } 13.33 \text{ minutes} \]