A man travels by bus to work . For each trip, the man pays $x. He has an option of purchasing a montly pass for $P which would reduce the fare for each trip by $r. If n be the number of trip s he must make per months so that purchasing the pass is a profitable to him, which of the following is the correct inequality satisfying the above information?

Options

A. `n lt (P)/(r)`
B. `n gt (P)/(r)`
C. `n lt (P)/(x -r)`
D. `n gt (p)/(x -r)`

Answer

To determine the number of trips \( n \) that a man must make per month for purchasing a monthly pass to be profitable, we can set up the following inequality based on the information provided:

1. **Cost without the pass**: If the man makes \( n \) trips and pays \( x \) dollars per trip, the total cost without the pass is:
\[ \text{Total cost without pass} = n \cdot x \]

2. **Cost with the pass**: If he buys a monthly pass for \( P \) dollars, the fare for each trip is reduced by \( r \) dollars. Therefore, the effective cost per trip with the pass becomes \( x - r \). The total cost with the pass for \( n \) trips is:
\[ \text{Total cost with pass} = P + n \cdot (x - r) \]

3. **Setting up the inequality**: For the pass to be profitable, the cost with the pass must be less than the cost without the pass:
\[ P + n \cdot (x - r) < n \cdot x \]

4. **Simplifying the inequality**:
\[ P + n \cdot (x - r) < n \cdot x \]
Subtract \( n \cdot (x - r) \) from both sides:
\[ P < n \cdot (x - r) - n \cdot (x - r) \]
This simplifies to:
\[ P < n \cdot r \]

5. **Rearranging the inequality**: We can rearrange this to find \( n \):
\[ n > \frac{P}{r} \]

Thus, the correct inequality that satisfies the condition for the man to find the pass profitable is:
\[ n > \frac{P}{r} \]