Two persons start from the opposite ends of a 90 km straight track and run to and fro between the two ends. The speed of first person is 30 m/s and the speed of other is 120/6 m/s. They continue their motion for 10 hours. How many times they pass each other?

Options

A. 10
B. 9
C. 12
D. none of these

Answer

To solve the problem of how many times the two persons pass each other while running on a 90 km track for 10 hours, we can follow these steps:

### Step 1: Convert the speeds from m/s to km/h
- The speed of the first person is given as 30 m/s.
- To convert this to km/h, we use the conversion factor \( \frac{18}{5} \):
\[ \text{Speed of first person} = 30 \times \frac{18}{5} = 108 \text{ km/h} \]

- The speed of the second person is given as \( \frac{120}{6} \) m/s, which simplifies to 20 m/s.
- Converting this to km/h:
\[ \text{Speed of second person} = 20 \times \frac{18}{5} = 72 \text{ km/h} \]

### Step 2: Calculate the relative speed
- Since they are moving towards each other, we can add their speeds to find the relative speed:
\[ \text{Relative speed} = 108 + 72 = 180 \text{ km/h} \]

### Step 3: Calculate the time duration in hours
- They run for a total time of 10 hours.

### Step 4: Calculate the distance covered in 10 hours
- The total distance they cover in 10 hours at the relative speed is:
\[ \text{Distance covered} = \text{Relative speed} \times \text{Time} = 180 \text{ km/h} \times 10 \text{ h} = 1800 \text{ km} \]

### Step 5: Determine how many times they pass each other
- To find out how many times they pass each other, we need to divide the total distance covered by the length of the track:
\[ \text{Number of passes} = \frac{\text{Distance covered}}{\text{Length of track}} = \frac{1800 \text{ km}}{90 \text{ km}} = 20 \]

### Conclusion
Thus, the two persons pass each other **20 times** in 10 hours.

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