Two pipes $A$ and $B$ can fill a water irrigation tank in $12\text{ hours}$ and $18\text{ hours}$ respectively. A drain pipe $C$ can empty the full tank in $15\text{ hours}$. If all three pipes are opened simultaneously, calculate the time in $\text{hours}$ required to fill the empty tank completely.
Answer Key:
13.85
Step-by-Step Solution & Explanation
Net filling rate per hour:
$\text{Rate} = \frac{1}{12} + \frac{1}{18} - \frac{1}{15}$
Taking LCM of $12, 18, 15 = 180$:
$\text{Rate} = \frac{15 + 10 - 12}{180} = \frac{13}{180}\text{ tank/h}$
Time required to fill tank:
$T = \frac{180}{13} \approx 13.846\text{ hours}$