This problem involves calculating the time required to fill a partially filled cistern when multiple pipes with different functions (filling and draining) are operating simultaneously.

Pipe Rates Calculation

First, determine the rate at which each pipe works. The rate is the fraction of the cistern filled or drained per hour.

• Pipe A fills the cistern in 4 hours. Rate of Pipe A = $ \frac{1}{4} $ cistern/hour.
• Pipe B fills the cistern in 10 hours. Rate of Pipe B = $ \frac{1}{10} $ cistern/hour.
• Pipe C drains the cistern in 6 hours. Rate of Pipe C = $ -\frac{1}{6} $ cistern/hour (negative because it drains).

Combined Rate Calculation

When all three pipes are opened simultaneously, their rates add up. Calculate the net rate of filling:

Combined Rate = Rate(A) + Rate(B) + Rate(C)

Combined Rate = $ \frac{1}{4} + \frac{1}{10} - \frac{1}{6} $

To add/subtract these fractions, find a common denominator. The Least Common Multiple (LCM) of 4, 10, and 6 is 60.

Combined Rate = $ \frac{1 \times 15}{4 \times 15} + \frac{1 \times 6}{10 \times 6} - \frac{1 \times 10}{6 \times 10} $

Combined Rate = $ \frac{15}{60} + \frac{6}{60} - \frac{10}{60} $

Combined Rate = $ \frac{15 + 6 - 10}{60} = \frac{11}{60} $ cistern/hour.

This means that when all pipes are open, the cistern fills at a net rate of $ \frac{11}{60} $ of its capacity every hour.

Time Calculation for Half-Full Cistern

The cistern is initially half-full. This means only half of the cistern needs to be filled.

Work to be done = $ \frac{1}{2} $ cistern.

The formula relating time, work, and rate is: Time = Work / Rate.

Time = $ \frac{\text{Work to be done}}{\text{Combined Rate}} $

Time = $ \frac{1/2}{11/60} $

Time = $ \frac{1}{2} \times \frac{60}{11} $

Time = $ \frac{60}{22} = \frac{30}{11} $ hours.

Therefore, it will take $ \frac{30}{11} $ hours to fill the remaining half of the cistern.