On a `120 km` track , a train travels the first `30 km` at a uniform speed of ` 30 km//h`. How fast must the train travel the next `90 km` so as to average `60 km//h` for the entire trip?

Options

A. 60 km/hr
B. 70 km/hr
C. 80 km/hr
D. 90 km /hr

Explanation

To solve this problem, we need to determine the speed at which the train must travel the next 90 km to ensure that the average speed for the entire trip is 60 km/h.

### Step-by-Step Solution:

1. **Calculate the total distance and required average speed:**
- Total distance, \( D_{\text{total}} = 120 \) km
- Required average speed, \( V_{\text{avg}} = 60 \) km/h

2. **Calculate the total time required to achieve the average speed:**
\[ \text{Total time} = \frac{\text{Total distance}}{\text{Average speed}} = \frac{120 \text{ km}}{60 \text{ km/h}} = 2 \text{ hours} \]

3. **Determine the time taken for the first part of the journey:**
- Distance for the first part, \( D_1 = 30 \) km
- Speed for the first part, \( V_1 = 30 \) km/h
\[ \text{Time for the first part} = \frac{D_1}{V_1} = \frac{30 \text{ km}}{30 \text{ km/h}} = 1 \text{ hour} \]

4. **Calculate the remaining time for the second part of the journey:**
- Total time required, \( T_{\text{total}} = 2 \) hours
- Time taken for the first part, \( T_1 = 1 \) hour
\[ \text{Time for the second part} = T_{\text{total}} - T_1 = 2 \text{ hours} - 1 \text{ hour} = 1 \text{ hour} \]

5. **Determine the speed required for the second part of the journey:**
- Distance for the second part, \( D_2 = 90 \) km
- Time for the second part, \( T_2 = 1 \) hour
\[ \text{Speed for the second part} = \frac{D_2}{T_2} = \frac{90 \text{ km}}{1 \text{ hour}} = 90 \text{ km/h} \]

### Final Answer:
The train must travel the next 90 km at a speed of 90 km/h to average 60 km/h for the entire trip.