Question : Pipes A and B together can fill an empty tank in $6 \frac{2}{3}$ minutes. If A takes 3 minutes more than B to fill the tank, then the time (in minutes) in which A alone would fill one-third part of the tank is:
Option 1: 6
Option 2: 5
Option 3: 5.5
Option 4: 4.5
Correct Answer: 5
Solution : Given: Pipes A and B together can fill an empty tank in $6 \frac{2}{3}=\frac{20}{3}$ minutes. A takes 3 minutes more than B to fill the tank. Let in $x$ minutes, pipe B by itself can empty the entire tank. So, the complete tank can be emptied in $(x + 3)$ minutes with just Pipe A. According to the question, $\frac{1}{x+3}+\frac{1}{x}=\frac{3}{20}$ ⇒ $\frac{x+x+3}{x^2+3x}=\frac{3}{20}$ ⇒ $\frac{2x+3}{x^2+3x}=\frac{3}{20}$ ⇒ $40x+60=3x^2+9x$ ⇒ $3x^2-31x-60=0$ ⇒ $3x^2-36x+5x-60=0$ ⇒ $3x(x-12)+5(x-12)=0$ ⇒ $(x-12)(3x+5)=0$ ⇒ $x-12=0$ [since time can not be negative] ⇒ $x=12$ Pipe B takes 12 minutes to empty the entire tank. The amount of time pipe A needs to empty a full tank is (12 + 3) = 15 minutes The time in which A alone would fill one-third part of the tank $=15\times \frac{1}{3}=5$ minutes Hence, the correct answer is 5.
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