This question involves calculating the time it takes for two pipes, A and B, working together to fill a tank of a specific capacity. We are given the individual filling times for a certain volume and need to determine the combined filling time for a different volume.

Determining Pipe Filling Rates

First, let's find the rate at which each pipe fills the tank. The rate is the amount of liquid filled per unit of time (litres per hour in this case).

• Pipe A's Rate: Pipe A fills 7800 litres in 12 hours. Rate of Pipe A ($R_A$) = $\frac{\text{Volume}}{\text{Time}} = \frac{7800 \text{ litres}}{12 \text{ hours}}$ $R_A = 650$ litres/hour
• Pipe B's Rate: Pipe B fills 7800 litres in 10 hours. Rate of Pipe B ($R_B$) = $\frac{\text{Volume}}{\text{Time}} = \frac{7800 \text{ litres}}{10 \text{ hours}}$ $R_B = 780$ litres/hour

Calculating Combined Filling Rate

When both pipes are opened together, their rates add up. This gives us the combined filling rate.

• Combined Rate ($R_{AB}$): $R_{AB} = R_A + R_B$ $R_{AB} = 650 \text{ litres/hour} + 780 \text{ litres/hour}$ $R_{AB} = 1430$ litres/hour

Calculating Time to Fill the New Tank

Now, we need to find out how long it will take for both pipes working together to fill a new tank with a capacity of 8800 litres.

• Time Calculation: Time = $\frac{\text{New Tank Capacity}}{\text{Combined Rate}}$ Time = $\frac{8800 \text{ litres}}{1430 \text{ litres/hour}}$

Simplifying the Result

To match the format of the options, we simplify the fraction:

• Time = $\frac{8800}{1430}$ hours Divide both the numerator and the denominator by 10: Time = $\frac{880}{143}$ hours We notice that $143 = 11 \times 13$. Let's check if 880 is divisible by 11: $880 \div 11 = 80$. So, we can simplify the fraction by dividing both parts by 11: Time = $\frac{880 \div 11}{143 \div 11}$ hours Time = $\frac{80}{13}$ hours

Therefore, it will take $\frac{80}{13}$ hours for pipes A and B to fill an empty tank of 8800 litres when opened together.