Question

It can take $12$ hours to fill a swimming pool using two pipes. If the pipe of larger diameter is used for four hours and the pipe of smaller diameter for nine hours, only half of the pool can be filled. How long would it take for each pipe to fill the pool separately?

Answer 1

Step 1: Framing the equation:

Let $A$ and $B$ be the two pipes and the diameter of $A$ is larger than $B$

Let,

Pipe $A$ takes $x$ hours and pipe $B$ takes $y$ hours to fill the pool separately.

Part of the pool, Pipe $A$ can fill the pool in $1$hour $=\frac{1}{x}$

Part of the pool, Pipe $B$ can fill the pool in $1$ hour $=\frac{1}{y}$

If both pipes are together then they take $12$ hours to fill the pool

In $1$ hour, Part of the pool they fill the together

$\Rightarrow \frac{1}{x}+\frac{1}{y}=\frac{1}{12}$ ………..…… $\left(i\right)$

Now, part of pool, Pipe $A$ can fill in $4$ hours $=\frac{4}{x}$

Part of pool Pipe $B$ can fill in $9$ hours $=\frac{9}{x}$

Then, according to the question, they fill half of the pool together,

$\Rightarrow \frac{4}{x}+\frac{9}{y}=\frac{1}{2}$ ………..…… $\left(ii\right)$

Step 2: Calculating the time taken by each pipe to fill the pool separately

Now, solving equations $\left(ii\right)-4\left(i\right)$, we get

$\Rightarrow \frac{5}{y}=\frac{1}{6}$

$\Rightarrow y=30$

Now, put $y=30$ in equation $\left(i\right)$, we get,

$$\begin{gathered} \Rightarrow \frac{1}{x}+\frac{1}{30}=\frac{1}{12} \\ \Rightarrow \frac{1}{x}=\frac{1}{12}-\frac{1}{30} \\ \Rightarrow x=20 \end{gathered}$$

Thus, $x=20$ and $y=30$

Therefore, pipe $A$ would take $20$ hours and the pipe $B$ would take $30$ hours separately to fill the pool.