If a man starts at 7 a.m. at $$\frac{5}{4}$$ of his usual speed, he reaches the office 30 minutes earlier than usual. At this new speed, at what time does he reach his office?

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Accepted Answer

1. **Understand the relationship between speed and time**:  
   Time is inversely proportional to speed. If the man's speed increases to $$\frac{5}{4}$$ of his usual speed, his time decreases proportionally.

2. **Define variables**:  
   Let the usual time taken to reach the office be $$4x$$ minutes.  
   At the new speed, his time becomes $$5x$$ minutes (in proportion to the ratio of speeds).

3. **Use the given condition**:  
   The man reaches 30 minutes earlier than usual:  
   $$
   4x - 5x = 30
   $$  
   Solve for $$x$$:  
   $$
   x = 30
   $$

4. **Calculate the usual time**:  
   Usual time taken = $$4x = 120$$ minutes (or 2 hours).  
   If he starts at 7 a.m., he would usually reach by 9 a.m.

5. **Adjust for new speed**:  
   With the increased speed, he reaches 30 minutes earlier than usual, which is **9:00 a.m**.

Thus, the correct answer is **(c) 9:00 a.m**.

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