chetak and Ashwa two horses atrat galloping from Patna to Rachi which are 80 km apart the speed of Chetak is 160km/h and that of Ashwa is 150km/h they start galloping simultaneously from Patna to Rachi Chetak reached to Rachi and return to Patna but Ashwa returned fro jamshedpur(which is somewhere between patna and Ranchi) tro Patna at the same time what is the ratio of distances between Patna and jamshedpur and jamshedpur and Ranchi?

Options

A. 0.62569444444444
B. 0.14236111111111
C. 0.62638888888889
D. none of these

Answer

To solve the problem step by step, we will analyze the situation involving the two horses, Chetak and Ashwa, and their respective distances and speeds.

### Step 1: Understand the Distances and Speeds
- The distance from Patna to Ranchi is given as 80 km.
- The speed of Chetak is 160 km/h.
- The speed of Ashwa is 150 km/h.

### Step 2: Calculate the Time Taken by Chetak
Chetak travels from Patna to Ranchi and then returns back to Patna. Therefore, the total distance covered by Chetak is:
\[ \text{Total Distance} = \text{Distance from Patna to Ranchi} + \text{Distance from Ranchi to Patna} = 80 \text{ km} + 80 \text{ km} = 160 \text{ km} \]
Now, we can calculate the time taken by Chetak to cover this distance:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{160 \text{ km}}{160 \text{ km/h}} = 1 \text{ hour} \]

### Step 3: Analyze Ashwa's Journey
Ashwa starts from Patna and travels towards Ranchi but returns from Jamshedpur. Let the distance from Patna to Jamshedpur be \( x \) km. Therefore, the distance from Jamshedpur to Ranchi will be:
\[ \text{Distance from Jamshedpur to Ranchi} = 80 \text{ km} - x \text{ km} \]
The total distance Ashwa travels is:
\[ \text{Total Distance} = x + (80 - x) = 80 \text{ km} \]
Since Ashwa also returns to Patna, the total distance covered by Ashwa is:
\[ \text{Total Distance} = x + (80 - x) + x = 80 + x \text{ km} \]

### Step 4: Calculate the Time Taken by Ashwa
The time taken by Ashwa to travel this distance can be calculated as:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{80 + x}{150} \]
Since both horses reach their respective destinations at the same time, we can set the time taken by Chetak equal to the time taken by Ashwa:
\[ 1 = \frac{80 + x}{150} \]

### Step 5: Solve for \( x \)
To find \( x \), we can rearrange the equation:
\[ 80 + x = 150 \]
\[ x = 150 - 80 = 70 \text{ km} \]
Thus, the distance from Patna to Jamshedpur is 70 km.

### Step 6: Calculate the Distance from Jamshedpur to Ranchi
Now, we can find the distance from Jamshedpur to Ranchi:
\[ \text{Distance from Jamshedpur to Ranchi} = 80 \text{ km} - 70 \text{ km} = 10 \text{ km} \]

### Step 7: Find the Ratio
Finally, we need to find the ratio of the distances:
\[ \text{Ratio} = \frac{\text{Distance from Patna to Jamshedpur}}{\text{Distance from Jamshedpur to Ranchi}} = \frac{70 \text{ km}}{10 \text{ km}} = 7:1 \]

### Conclusion
The ratio of distances between Patna and Jamshedpur to the distance between Jamshedpur and Ranchi is **7:1**.