Question : When operated separately, pipe A takes 5 hours less than pipe B to fill a cistern, and when operated together, the cistern gets filled in 6 hours. In how much time (in hours) will pipe A fill the cistern, if operated separately?
Option 1: 15
Option 2: 18
Option 3: 10
Option 4: 9
Correct Answer: 10
Solution :
Let time to fill the whole tank by pipe B alone be $x$ hours.
The time taken to fill the whole tank by pipe A will be $(x - 5)$ hours.
Let the capacity of the total tank be $6x(x - 5)$ units.
Efficiency of pipe A $=\frac{6x(x - 5)}{(x - 5)}=6x$
Efficiency of pipe B $=\frac{6x(x - 5)}{x}=6(x - 5)$
Efficiency of A and B together $=\frac{6x(x - 5)}{6}=x(x - 5)$
Equating combined efficiency,
$6x + 6(x - 5) = x(x - 5)$
⇒ $6x + 6x - 30 = x^2- 5x$
⇒ $12x - 30 = x^2- 5x$
⇒ $x^2- 5x - 12x + 30 = 0$
⇒ $x^2- 17x + 30 = 0$
⇒ $(x- 15)(x -2) = 0$
$\therefore x=15$ (since $x$ can't be 2)
Pipe A can fill the tank in $x-5=15-5=10$ hours
Hence, the correct answer is 10.
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