# Two drainpipes working together can drain a pool in 12 hours. Working​ alone, the smaller pipe would take 18 hours longer than the larger pipe to drain the pool. How long would it take the smaller pipe alone to drain the​ pool?

Jul 21, 2018

36hr

#### Explanation:

Time of both working together ${t}_{b} = 12$hr

Time of small pipe ${t}_{s}$ = (time of large pipe) $+ 18$

Let time of large pipe $= x$

${t}_{s} = x + 18$

For pipes in parallel $\frac{1}{T} = \frac{1}{t} _ 1 + \frac{1}{t} _ 2$

$\frac{1}{12} = \frac{1}{x + 18} + \frac{1}{x}$

$\frac{x}{12} = \frac{x}{x + 18} + \frac{x}{x}$

$\frac{x \cdot \left(x + 18\right)}{12} = \frac{x}{1} + \left(x + 18\right)$

${x}^{2} + 18 x = 12 \cdot x + 12 \cdot x + 216$

${x}^{2} - 6 x - 216 = 0$

$\left(x + 12\right) \left(x - 18\right) = 0$

$x = - 12 \mathmr{and} 18$
but time can't be -ve so

$x = 18$hr

so ${t}_{s} = 18 + 18 = 36$hr

36\ \text{hrs

#### Explanation:

Let $x \setminus \setminus \textrm{h r s}$ be the time taken by the smaller pipe alone to drain a pool of volume say $V$ . Then the time taken by the larger pipe alone to drain the same pool of volume $V$ will be $\left(x - 18\right) \setminus \textrm{h r s}$

Now, draining rate of smaller pipe $= \frac{V}{x}$

Similarly, draining rate of larger pipe $= \frac{V}{x - 18}$

Given that both the pipes working together can drain the same pool of volume $V$ in $12$ hours hence we have

$\setminus \textrm{\to t a l v o l u m e \mathrm{dr} a \in e d b y \bot h \pi p e s \to \ge t h e r \in 12 h r s}$

$= \setminus \textrm{v o l u m e o f p \infty l}$

$\setminus \therefore 12 \left(\frac{V}{x} + \frac{V}{x - 18}\right) = V$

$12 \left(\frac{x - 18 + x}{x \left(x - 18\right)}\right) = 1$

$12 \left(2 x - 18\right) = x \left(x - 18\right)$

${x}^{2} - 42 x + 216 = 0$

${x}^{2} - 36 x - 6 x + 216 = 0$

$x \left(x - 36\right) - 6 \left(x - 36\right) = 0$

$\left(x - 36\right) \left(x - 6\right) = 0$

$x - 36 = 0 , x - 6 = 0$

$x = 36 , 6$

But $x > 18$ hence

$x = 36$

hence, the time taken by the smaller pipe alone to drain the pool is 36\ \text{hrs