This problem involves calculating the time it takes for two pipes, Pipe A and Pipe B, working together to fill half of an empty cistern. We are given the individual times each pipe takes to fill the cistern.

Understanding Pipe Rates

First, let's determine the rate at which each pipe fills the cistern. The rate is the fraction of the cistern that can be filled in one hour.

• Pipe A fills the cistern in 26 hours. So, the rate of Pipe A is $ \frac{1}{26} $ of the cistern per hour.
• Pipe B fills the cistern in 36 hours. So, the rate of Pipe B is $ \frac{1}{36} $ of the cistern per hour.

Calculating Combined Filling Rate

When both pipes operate together, their rates add up. We need to find the combined rate.

Combined Rate = Rate of Pipe A + Rate of Pipe B

Combined Rate = $ \frac{1}{26} + \frac{1}{36} $

To add these fractions, we find a common denominator. The Least Common Multiple (LCM) of 26 and 36 is 468.

• $26 = 2 \times 13$
• $36 = 2^2 \times 3^2$
• LCM(26, 36) = $2^2 \times 3^2 \times 13 = 4 \times 9 \times 13 = 468$

Now, we convert the fractions:

Combined Rate = $ \frac{1 \times 18}{26 \times 18} + \frac{1 \times 13}{36 \times 13} = \frac{18}{468} + \frac{13}{468} $

Combined Rate = $ \frac{18 + 13}{468} = \frac{31}{468} $ cistern per hour.

Determining Time to Fill Full Cistern

The time taken to fill the entire cistern together is the reciprocal of their combined rate.

Time to fill the full cistern = $ \frac{1}{\text{Combined Rate}} = \frac{1}{\frac{31}{468}} $ hours

Time to fill the full cistern = $ \frac{468}{31} $ hours.

Calculating Time for Half Cistern

The question asks for the time taken to fill only half of the cistern. Therefore, we need half the time it takes to fill the full cistern.

Time to fill half cistern = $ \frac{1}{2} \times (\text{Time to fill the full cistern}) $

Time to fill half cistern = $ \frac{1}{2} \times \frac{468}{31} $ hours

Time to fill half cistern = $ \frac{468}{2 \times 31} = \frac{234}{31} $ hours.

Converting to Mixed Fraction

To express the answer in the format given in the options, we convert the improper fraction $ \frac{234}{31} $ into a mixed fraction.

Divide 234 by 31:

$ 234 \div 31 = 7 $ with a remainder of $ 234 - (7 \times 31) = 234 - 217 = 17 $.

So, the time taken is $ 7 \frac{17}{31} $ hours.

Thus, the time taken by Pipe A and Pipe B to fill half of the cistern together is $ 7 \frac{17}{31} $ hours.