Question : Two taps P and Q can fill a tank alone in 10 hours and 12 hours respectively. If the two taps are opened at 9 a.m., then at what time should the tap P be closed to fill the tank at exactly 3 p.m.?
Option 1: 2 p.m.
Option 2: 1 p.m.
Option 3: 3 p.m.
Option 4: 12 p.m.
Correct Answer: 2 p.m.
Solution :
Let the capacity of the tank (Total Work) be L
Rate of work done by P
= $\frac{L}{10}$ litres
Rate of work done by Q
= $\frac{L}{12}$ litres
⇒ The combined rate = $\frac{L}{10}$ + $\frac{L}{12}$ = $\frac{11L}{60}$
The total time required to fill the tank is 6 hrs (9 am to 3 pm)
Let the time for which both worked together be $x$ hours and $(6-x)$ hours Q worked alone.
So, $\frac{11L}{60} × x + \frac{L}{12}× (6-x) = L$
⇒ $\frac{11x-5x}{60} + \frac{1}{2} = 1$
⇒ $\frac{6x}{60} = \frac{1}{2}$
⇒ $x = 5$
Tap P should be closed after 5 hours i.e. at 2 p.m.
Hence, the correct answer is 2 p.m.
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