We let the time taken when ....
the larger hose alone is used be x - 6
and the smaller hose as x.
This would make their rates [tex] \frac{1}{x-6} [/tex] and [tex] \frac{1}{x} [/tex] respectively.
Take note that:
1 / rate = time
so,
1 / rate of both hoses = 4
rate of both hoses = 1/4
[tex] \frac{1}{x-6} + \frac{1}{x} = \frac{1}{4}\\ \frac{2x-6}{x^2-6x} = \frac{1}{4} \\ 8x-24=x^2-6x \\ 0=x^2-14x+24 \\ 0=(x-12)(x-2)[/tex]
This would mean that x = 12 or 2 but x cannot be 2 since that would make the time taken for the larger hose negative.
Therefore it will take the smaller hose 12 hours to fill the swimming pool alone.
