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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  \frac{1}{x-6} and  \frac{1}{x}  respectively.

Take note that:
1 / rate = time 
1 / rate of both hoses = 4
rate of both hoses = 1/4

 \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)

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.
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