Discussion
Two pipes M and N are opened together to fill a tank. Both the pipes fill the tank in time X minutes. When M alone is filling the tank, it takes 9 minutes more time than X to fill the tank. In a similar manner N takes 16 minutes more time than X to fill the tank. What is the value of X?
Answer: A
Work in rates: a pipe that fills a tank in \( n \) units of time fills \( \tfrac{1}{n} \) of it per unit, and rates add (an emptying pipe counting negative).
\[ \frac{1}{X + 9} + \frac{1}{X + 16} = \frac{1}{X} \]
Clearing denominators:
\[ X(X + 16) + X(X + 9) = (X+9)(X+16) \]
\[ 2X^{2} + 25X = X^{2} + 25X + 144 \]
\[ X^{2} = 144 \quad\Longrightarrow\quad X = 12 \text{ minutes} \]
The result is the geometric mean of the two extra times: \( X = \sqrt{9 \times 16} = 12 \), a useful shortcut for this standard form.
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