Arjun travels half of his journey by train at the speed of 120 km/hr and rest half by car at 80 km/hr. What is his average speed ?

Question

Accepted Answer

To determine Arjun's average speed for his entire journey, we need to consider the speeds at which he travels different segments of his journey and the distances covered.

1. **Journey Details:**
   - Arjun travels half of his journey by train at a speed of 120 km/hr.
   - The other half of his journey is by car at a speed of 80 km/hr.

2. **Understanding Average Speed:**
   - Average speed is defined as the total distance traveled divided by the total time taken.

3. **Calculation:**
   - Let's assume the total distance of the journey is $2D$. Therefore, each half of the journey is $D$.
   - **Time taken by train:** 
     $$
     \text{Time by train} = \frac{D}{120} \text{ hours}
     $$
   - **Time taken by car:** 
     $$
     \text{Time by car} = \frac{D}{80} \text{ hours}
     $$
   - **Total time taken for the journey:**
     $$
     \text{Total time} = \frac{D}{120} + \frac{D}{80}
     $$
   - To add these fractions, find a common denominator:
     $$
     \text{Total time} = \frac{2D}{240} + \frac{3D}{240} = \frac{5D}{240} = \frac{D}{48} \text{ hours}
     $$
   - **Average speed:**
     $$
     \text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} = \frac{2D}{\frac{D}{48}} = 96 \text{ km/hr}
     $$

4. **Conclusion:**
   - The average speed of Arjun's journey is 96 km/hr, which corresponds to Option 3.

5. **Explanation of Incorrect Options:**
   - **Option 1 (88 km/hr):** This option is incorrect because the calculated average speed using the given speeds and distances does not match 88 km/hr.
   - **Option 2 (92 km/hr):** This option is incorrect for the same reason; the calculation does not yield 92 km/hr.
   - **Option 4 (100 km/hr):** This option is incorrect because the calculated average speed is not 100 km/hr.

Therefore, based on the calculations and logical reasoning, Option 3 (96 km/hr) is indeed the correct answer.