Question

Two pipes running together can fill a cistern in $6$ minutes. If one pipe takes $5$ minutes more than the other to fill the cistern, find the time in which each pipe would fill the cistern.

Answer 1

Let the first pipe fill the cistern in $x$ minutes, then the second pipe requires $x+5$ minutes to fill it.
Applying the concept of the Unitary Method,
In one minute, both pipes will fill the part of cistern as below;
$\frac{1}{x}+\frac{1}{x+5}=\frac{1}{6}$
$\frac{x+5+x}{x^2+5x}=\frac{1}{6}$
$\frac{(2x+5)}{x^2+5x}=\frac{1}{6}$
$12x+30=x^2+5x$
$x^2-7x-30=0$
On factorising the same.
$x^2-10x+3x-30=0$
$x(x-10)+3(x-10)=0$
$(x-10)(x+3)=0$
$\therefore x=-3orx=10$
Then the first pipe will fill the cistern in $x$ minutes i.e., $10$ minutes and the second pipe will fill the cistern in $x+5$ minutes i.e., $15$ minutes.