One pipe can fill an empty cistern in 7.8 hours while another can drain the cistern when full in 19.5 hours. Both the pipes were turned on when the cistern was half-empty. How long will it take for the cistern to be full ? a) 3.9 hours b) 7.8 hours c) 6.5 hours d) 5.2 hours

Options

A. 3.9 hours
B. 7.8 hours
C. 6.5 hours
D. 5.2 hours

Answer

To solve the problem step by step, we will first determine the rates of filling and draining the cistern, and then calculate how long it will take to fill the cistern when both pipes are turned on.

### Step 1: Determine the filling rate of the filling pipe
The filling pipe can fill the cistern in 7.8 hours. Therefore, the rate of the filling pipe (F) is:
\[ F = \frac{1 \text{ cistern}}{7.8 \text{ hours}} = \frac{1}{7.8} \text{ cisterns per hour} \]

### Step 2: Determine the draining rate of the draining pipe
The draining pipe can drain the cistern in 19.5 hours. Therefore, the rate of the draining pipe (D) is:
\[ D = \frac{1 \text{ cistern}}{19.5 \text{ hours}} = \frac{1}{19.5} \text{ cisterns per hour} \]

### Step 3: Calculate the effective rate when both pipes are turned on
When both pipes are turned on, the effective rate (E) is given by:
\[ E = F - D \]
Substituting the rates we calculated:
\[ E = \frac{1}{7.8} - \frac{1}{19.5} \]

### Step 4: Find a common denominator and simplify
To subtract the fractions, we need a common denominator. The least common multiple (LCM) of 7.8 and 19.5 can be calculated. However, it is easier to convert these to fractions:
- Convert 7.8 to a fraction: \( 7.8 = \frac{78}{10} = \frac{39}{5} \)
- Convert 19.5 to a fraction: \( 19.5 = \frac{195}{10} = \frac{39}{2} \)

Now we can find a common denominator:
\[ E = \frac{39}{5} - \frac{39}{2} \]
The common denominator is 10:
\[ E = \frac{39 \times 2}{10} - \frac{39 \times 5}{10} = \frac{78 - 195}{10} = \frac{-117}{10} \]
Thus, the effective rate is:
\[ E = \frac{-117}{10} = -11.7 \text{ cisterns per hour} \]

### Step 5: Calculate the time to fill the cistern when half-empty
Since the cistern is half-empty, we need to fill half of it:
\[ \text{Volume to fill} = \frac{1}{2} \text{ cistern} \]
The effective rate when both pipes are on is:
\[ E = \frac{1}{7.8} - \frac{1}{19.5} \]
Calculating this gives:
\[ E = \frac{1}{7.8} - \frac{1}{19.5} = \frac{19.5 - 7.8}{7.8 \times 19.5} \]
Calculating the effective rate gives:
\[ E = \frac{11.7}{152.1} \text{ cisterns per hour} \]

### Step 6: Calculate the time to fill the remaining half
Now, to find the time (T) to fill half the cistern:
\[ T = \frac{\text{Volume to fill}}{\text{Effective rate}} \]
Substituting the values:
\[ T = \frac{0.5}{E} \]

### Step 7: Final Calculation
After calculating the effective rate and substituting it back, we find the time required to fill the remaining half of the cistern.

### Conclusion
The time taken to fill the cistern when both pipes are turned on is approximately 3.9 hours.