A tap can fill a tank completely in 8 hours. After half the tank is filled, one more similar tap is opened. What is the total time taken to fill the tank completely?

Steps to Solve

1. Calculate the time to fill half the tank Since the tap fills the entire tank in 8 hours, it will take half the time to fill half the tank. $$ \frac{1}{2} \times 8 \text{ hours} = 4 \text{ hours} $$

2. Determine the remaining portion of the tank to be filled After the first tap has been running for 4 hours, half of the tank is filled, which means half of the tank remains to be filled.

3. Calculate the combined rate of the two taps If one tap takes 8 hours to fill the tank, its rate is $\frac{1}{8}$ of the tank per hour. With two identical taps, their combined rate is the sum of their individual rates. $$ \frac{1}{8} + \frac{1}{8} = \frac{2}{8} = \frac{1}{4} \text{ of the tank per hour} $$

4. Calculate the time to fill the remaining half of the tank with two taps With the combined rate of $\frac{1}{4}$ of the tank per hour, we can calculate the time to fill the remaining half ($\frac{1}{2}$) of the tank. Let the time be $t$. $$ \frac{1}{4} \times t = \frac{1}{2} $$ $$ t = \frac{1}{2} \div \frac{1}{4} = \frac{1}{2} \times 4 = 2 \text{ hours} $$

5. Calculate the total time taken to fill the tank Add the time it took to fill the first half of the tank (4 hours) to the time it took to fill the second half of the tank with both taps (2 hours). $$ 4 \text{ hours} + 2 \text{ hours} = 6 \text{ hours} $$