Question

It takes 6 hours for pump A, used alone, to fill a tank of water. Pump B used alone takes 8 hours to fill the same tank. We want to use three pumps A, B and C to fill the tank in 2 hours. What should be the rate of pump C? How long would it take pump C, used alone, to fill the tank?

Answer 1

Hint: For solving this problem we will use a simple unitary method. First find how much water each pump can fill in the tank in 6 hours, 8 hours and 1 hour. Use it according to the question to reach a final answer.

Complete step-by-step answer:
Given:
Pump A takes 6 hours to fill the tank when used alone. Similarly, pump B takes 8 hours to fill the tank when used alone.

Let, the capacity of the tank is $L$ litres.
Then,
In 6 hours pump A fills the $L$ litres of water. So, in 1-hour pump A can fill $\dfrac{L}{6}$ litres of water.
Then,
In 1-hour amount of water pump A can fill $=\dfrac{L}{6}\text{ litres}...................\left( 1 \right)$

Similarly, in 8 hours pump B fills the $L$ litres of water. So, in 1-hour pump B can fill $\dfrac{L}{8}$ litres of water.
Then,
In 1-hour amount of water pump B can fill $=\dfrac{L}{8}\text{ litres}...................\left( 2 \right)$

Now, let pump C can fill the tank alone in $x$ hours. Then,
As in $x$ hours pump C fill the $L$ litres of water. So, in 1-hour pump C can fill $\dfrac{L}{x}$ litres of water.
Then,
In 1-hour amount of water pump C can fill $=\dfrac{L}{x}\text{ litres}...................\left( 3 \right)$
Now, it is given to us that when we use three pumps A, B and C together then it takes 2 hours to fill the tank.
Then,
As in 2 hours, all three pumps when used together fill the $L$ litres of water. So, in 1-hour all three pumps when used together can fill $\dfrac{L}{2}$ litres of water.
Then,
In 1-hour amount of water pump A, B and C can fill when used together $=\dfrac{L}{2}\text{ litres}...................\left( 4 \right)$

Now, as when the three pumps will be used together then the amount of water that they fill in 1-hour will be the sum of the individual capacity of each pump to fill the water in the tank in 1-hour.

Then,
In 1-hour amount of water pump A, B and C can fill when used together = (Amount of water pump A can fill in the tank in 1-hour) + (Amount of water pump B can fill in the tank in 1-hour) + (Amount of water pump C can fill in the tank in 1-hour).

Now, we will substitute value from (4), (1), (2) and (3). Then,
$\begin{align}
  & \dfrac{L}{2}=\dfrac{L}{6}+\dfrac{L}{8}+\dfrac{L}{x} \\
 & \Rightarrow \dfrac{L}{2}=\dfrac{7L}{24}+\dfrac{L}{x} \\
 & \Rightarrow \dfrac{L}{2}-\dfrac{7L}{24}=\dfrac{L}{x} \\
 & \Rightarrow \dfrac{5L}{24}=\dfrac{L}{x} \\
 & \Rightarrow x=\dfrac{24}{5L}\times L \\
 & \Rightarrow x=\dfrac{24}{5} \\
 & \Rightarrow x=4.8 \\
\end{align}$

Now, as the value of $x=4.8$ hours then, pump C will fill the tank alone in 4.8 hours. And in 1 hour pump C will fill $\dfrac{5L}{24}$ litres of water in the tank.
Thus, pump C will tank alone in 4.8 hours.

Note: Although the question is very easy to solve but the student must be careful while solving and avoid calculation mistakes. Moreover, one should not directly add the time taken by each pump to fill the tank when used alone directly to get the answer; it would be the wrong approach.