Pipe A can fill a tank in 492 minutes. Pipe B can fill the same tank in 984 minutes. If both pipes are opened together, how many minutes will it take to fill the empty tank?

1. Problem Description

Pipe A can fill a tank in 492 minutes. Pipe B can fill the same tank in 984 minutes. If both pipes are opened together, how many minutes will it take to fill the empty tank?

2. Solution Steps

Let the capacity of the tank be $C$.

Pipe A's rate of filling is $R_A = \frac{C}{492}$ (capacity per minute).

Pipe B's rate of filling is $R_B = \frac{C}{984}$ (capacity per minute).

When both pipes are opened together, their combined rate is:

$R_{A+B} = R_A + R_B = \frac{C}{492} + \frac{C}{984}$

We can factor out $C$:

$R_{A+B} = C(\frac{1}{492} + \frac{1}{984})$

Since $984 = 2 \times 492$, we can rewrite the equation as:

$R_{A+B} = C(\frac{1}{492} + \frac{1}{2 \times 492}) = C(\frac{2}{2 \times 492} + \frac{1}{2 \times 492}) = C(\frac{2+1}{984}) = C(\frac{3}{984})$

$R_{A+B} = \frac{3C}{984} = \frac{C}{328}$

Let $t$ be the time it takes to fill the tank when both pipes are opened together. Then, $R_{A+B} \times t = C$.

Substituting $R_{A+B} = \frac{C}{328}$, we get:

$\frac{C}{328} \times t = C$

Divide both sides by $C$:

$\frac{t}{328} = 1$

$t = 328$ minutes.

3. Final Answer

The time it takes to fill the tank when both pipes are opened together is 328 minutes.

Answer: (C) 328