Question

A water tank has $M$ inlet pipes and $N$ outlet pipes. An inlet pipe can fill the tank in $8$ hours while an outlet pipe can empty the full tank in $12$ hours. If all pipes are left open simultaneously, it takes $6$ hours to fill the empty tank. What is the relationship between $M$ and $N$?

• A. $M:N = 1:1$
• B. $M:N = 2:1$
• C. $M:N = 2:3$
• D. $M:N = 3:2$
• E. Cannot be determined.

Hint:

When mixing multiple identical-rate pipes, equate the {net} rate to $\frac{1}{\text{time}}$. If the resulting linear relation between counts admits many integer pairs, the ratio cannot be uniquely determined.

Answer 1

Answer: (E)

Step 1: Write individual rates.
Each inlet fills $\frac{1}{8}$ tank/hour. Each outlet empties $\frac{1}{12}$ tank/hour.
With $M$ inlets and $N$ outlets open together, the net rate is \[ \text{Net rate} \;=\; M\!\left(\frac{1}{8}\right)\;-\;N\!\left(\frac{1}{12}\right) =\frac{3M-2N}{24}\ \text{tank/hour}. \] Step 2: Use the given filling time.
It takes $6$ hours to fill the tank $\Rightarrow$ net rate $=\frac{1}{6}$ tank/hour. Hence \[ \frac{3M-2N}{24}=\frac{1}{6}\ \Rightarrow\ 3M-2N=4. \] Step 3: Interpret the relationship.
The diophantine equation $3M-2N=4$ has {many} positive integer solutions, e.g. $M=2,\ N=1$ (ratio $2:1$), \quad $M=4,\ N=4$ (ratio $1:1$), \quad $M=8,\ N=10$ (ratio $4:5$), \dots
Since multiple $(M,N)$ pairs satisfy the condition, the ratio $M:N$ is {not unique}. \[ \boxed{\text{Cannot be determined from the given information.}} \]