Question

A drain pipe can drain a tank in 12 hours, and a fill pipe can fill the same tank in 6 hours. A total of n pipes, which include a few fill pipes and the remaining drain pipes can fill the entire tank in 2 hours. How many of the following values could n have in this case? a. 24 b. 16 c. 33 d. 13 e. 9 f. 8

• A. 12
• B. 13
• C. 5
• D. 3
• E. 2

Hint:

In pipe problems, express rates as fractions of tank per hour. For fill pipes, rate is positive; for drain pipes, rate is negative.

Answer 1

The Correct Option is D

Step 1: Let the number of fill pipes be $x$ and drain pipes be $y$. Then $n = x + y$.
Step 2: Fill pipe rate = $\frac{1}{6}$ tank per hour. Drain pipe rate = $\frac{1}{12}$ tank per hour (negative since it drains).
Step 3: Net rate with $x$ fill and $y$ drain = $\frac{x}{6} - \frac{y}{12}$.
Step 4: This net rate fills the tank in 2 hours, so $\frac{x}{6} - \frac{y}{12} = \frac{1}{2}$.
Step 5: Multiply by 12: $2x - y = 6 \implies y = 2x - 6$.
Step 6: Since $y \ge 0$, we have $2x - 6 \ge 0 \implies x \ge 3$.
Step 7: Also $x$ and $y$ are integers. Then $n = x + y = x + (2x - 6) = 3x - 6$.
Step 8: $n = 3x - 6$, where $x \ge 3$ integer. So $n$ can be $3(3)-6=3$, $3(4)-6=6$, $3(5)-6=9$, $3(6)-6=12$, $3(7)-6=15$, $3(8)-6=18$, $3(9)-6=21$, $3(10)-6=24$, etc.
Step 9: From the given values: a. 24 (possible when $x=10$), b. 16 (not of the form $3x-6$), c. 33 (possible when $x=13$), d. 13 (not of the form $3x-6$), e. 9 (possible when $x=5$), f. 8 (not of the form $3x-6$).
Step 10: Values that work: 24, 33, 9. That's 3 values.
Step 11: Final Answer: 3 of the given values could be n.