Three pumps, A, B, and C are capable of filling a reservoir. A fills in 5 hours, B in 7 hours, and C in 10 hours. When all three pumps are opened together, they operate for 1.5 hours before pump C is closed. How much additional time will it take to completely fill the reservoir after that?

The correct option is $\frac{47}{48}$ hours.

Step 1: Determine the work rates of each pump.
Let the total capacity of the reservoir be represented as 1 unit of work.
- Pump A fills the reservoir in 5 hours, so its rate is $\frac{1}{5}$ of the reservoir per hour.
- Pump B fills the reservoir in 7 hours, so its rate is $\frac{1}{7}$ of the reservoir per hour.
- Pump C fills the reservoir in 10 hours, so its rate is $\frac{1}{10}$ of the reservoir per hour.

Step 2: Find the combined rate of all three pumps working together.

$\text{Combined Rate (A + B + C)}=\frac{1}{5}+\frac{1}{7}+\frac{1}{10}$

To add these fractions, find the least common multiple (LCM) of 5, 7, and 10, which is 70:

$\text{Combined Rate}=\frac{14}{70}+\frac{10}{70}+\frac{7}{70}=\frac{31}{70}$

So, all three pumps together fill $\frac{31}{70}$ of the reservoir per hour.

Step 3: Calculate the fraction of the reservoir filled in 1.5 hours.
Convert 1.5 hours to an improper fraction: $1.5=\frac{3}{2}$ hours.

$\text{Work done in 1.5 hours}=\frac{31}{70}\times \frac{3}{2}=\frac{93}{140}$

Step 4: Calculate the remaining part of the reservoir to be filled.

$\text{Remaining Work}=1-\frac{93}{140}=\frac{47}{140}$

Step 5: Calculate the additional time required for pumps A and B to finish the remaining work.
After pump C is closed, only pumps A and B operate. Their combined work rate is:

$\text{Rate (A + B)}=\frac{1}{5}+\frac{1}{7}=\frac{7}{35}+\frac{5}{35}=\frac{12}{35}$

The additional time needed is given by dividing the remaining work by the combined rate of A and B:

$\text{Additional Time}=\frac{\text{Remaining Work}}{\text{Rate (A + B)}}=\frac{\frac{47}{140}}{\frac{12}{35}}=\frac{47}{140}\times \frac{35}{12}$

Simplifying by dividing 35 and 140 by 35 (since 140 = 35 × 4):

$\text{Additional Time}=\frac{47}{4\times 12}=\frac{47}{48}\text{hours}$

Therefore, it will take an additional $\frac{47}{48}$ hours to completely fill the reservoir.