Question : A water tap fills a tub in '$p$' hours and a sink at the bottom empties it in '$q$' hours. If $p < q$, both tap and sink are open, and the tank is filled in '$r$' hours, then:
Option 1: $\frac{1}{r}$ = $\frac{1}{p}+\frac{1}{q}$
Option 2: $\frac{1}{r}$ = $\frac{1}{p}-\frac{1}{q}$
Option 3: $r = p + q$
Option 4: $r = p - q$
Correct Answer: $\frac{1}{r}$ = $\frac{1}{p}-\frac{1}{q}$
Solution :
Given: A water tap fills a tub in '$p$' hours and a sink at the bottom empties it in '$q$' hours where $p < q$.
So, in 1 hour the tap will fill $\frac{1}{p}$ part of the tub.
Also, in 1 hour the sink will empty $\frac{1}{q}$ part of the tub.
Now, part of the tub that will fill in 1 hour with both of them opened = ($\frac{1}{p}-\frac{1}{q}$)
Time taken to fill the whole tub = $\frac{1}{(\frac{1}{p}–\frac{1}{q})}$ hours, which is equal to $r$.
Therefore, $\frac{1}{(\frac{1}{p}–\frac{1}{q})}$ = $r$
⇒ $\frac{1}{r}$ = $\frac{1}{p}-\frac{1}{q}$
Hence, the correct answer is $\frac{1}{r}$ = $\frac{1}{p}-\frac{1}{q}$.
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