The distance between Delhi and Agra is 200 km the train travels the first 100 km at the speed of 50kmh . how fast must the train travel the 100 km,so as to average 70 kmh for the the whole journey.
Explain me with steps 

To solve this problem, we need to determine the speed at which the train must travel the second 100 km to achieve an average speed of 70 km/h for the entire journey.

### Step 1: Understanding the problem
- **Total distance** between Delhi and Agra: 200 km.
- **First half distance** (from Delhi to the halfway point): 100 km.
- **Speed for the first half**: 50 km/h.
- **Second half distance** (from the halfway point to Agra): 100 km.
- **Desired average speed** for the entire journey: 70 km/h.

### Step 2: Calculate the time taken for the entire journey at the average speed
The total time \( T_{\text{total}} \) needed to cover 200 km at an average speed of 70 km/h is given by:
\[
T_{\text{total}} = \frac{\text{Total distance}}{\text{Average speed}} = \frac{200 \text{ km}}{70 \text{ km/h}} = \frac{200}{70} \text{ hours} \approx 2.857 \text{ hours}
\]

### Step 3: Calculate the time taken for the first half of the journey
The time \( T_1 \) taken to cover the first 100 km at 50 km/h is:
\[
T_1 = \frac{100 \text{ km}}{50 \text{ km/h}} = 2 \text{ hours}
\]

### Step 4: Calculate the remaining time for the second half
To maintain an average speed of 70 km/h, the total journey time should be approximately 2.857 hours. The remaining time \( T_2 \) for the second half of the journey is:
\[
T_2 = T_{\text{total}} - T_1 = 2.857 \text{ hours} - 2 \text{ hours} = 0.857 \text{ hours} \approx 51.43 \text{ minutes}
\]

### Step 5: Determine the required speed for the second half of the journey
The required speed \( v_2 \) for the second 100 km is given by:
\[
v_2 = \frac{\text{Distance for the second half}}{\text{Time for the second half}} = \frac{100 \text{ km}}{0.857 \text{ hours}} \approx 116.7 \text{ km/h}
\]

### Conclusion
To average 70 km/h for the entire journey, the train must travel the second 100 km at approximately **116.7 km/h**.

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### Related Questions:
1. How would the required speed change if the first half was traveled at 60 km/h?
2. What would be the average speed if the train maintained 50 km/h throughout the journey?
3. How much time would it take to complete the entire journey if the train traveled the second half at 100 km/h?
4. What if the train travels the first 100 km at 40 km/h? What speed is needed for the second half?
5. If the total distance were 300 km instead of 200 km, what speed would be required to maintain a 70 km/h average?

### Tip:
Always remember that average speed is not simply

Learn how to calculate the speed required for a train to achieve an average speed of 70 km/h for a journey from Delhi to Agra. This problem involves speed, distance, and time calculations, ensuring a thorough understanding of average speed concepts. Suitable for high school students.

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