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The question in the image is as follows:

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**Q31.**  
The distance between Kolkata and Ranchi is 600 km. If Rakesh moves 120 km by train and the rest by car, she reaches in the same time as the entire journey by car. What would be the ratio between the speeds of the train and the car?

- (A) The required ratio is 4:5
- (B) The required ratio is 1:2
- (C) The required ratio is 2:3
- (D) The required ratio is 3:4

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### Solution:

Let the speed of the train be \( S_{\text{train}} \) and the speed of the car be \( S_{\text{car}} \).

- Distance traveled by train = 120 km
- Distance traveled by car = \( 600 - 120 = 480 \) km
- Total distance = 600 km

Time taken by train = \( \frac{120}{S_{\text{train}}} \)  
Time taken by car for the remaining distance = \( \frac{480}{S_{\text{car}}} \)

The total time when Rakesh splits the journey between the train and car is the same as the time it would take if the whole journey were done by car, i.e.,

\[
\frac{120}{S_{\text{train}}} + \frac{480}{S_{\text{car}}} = \frac{600}{S_{\text{car}}}
\]

Simplifying:

\[
\frac{120}{S_{\text{train}}} = \frac{600}{S_{\text{car}}} - \frac{480}{S_{\text{car}}}
\]
\[
\frac{120}{S_{\text{train}}} = \frac{120}{S_{\text{car}}}
\]

From this, we get:

\[
\frac{S_{\text{train}}}{S_{\text{car}}} = \frac{120}{120} = 1
\]

Thus, the ratio of the speed of the train to the speed of the car is 1:1, which means the ratio between speeds is equal.

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None of the answer choices directly match the result, so there may be a discrepancy in the setup or the options might be incomplete. However, based on this, we can explore further possibilities based on any clarifications.

Would you like a more detailed breakdown or have questions about this? Let me know!

Here are five related questions to expand your understanding:
1. What if the distance traveled by the train were different? How would that change the ratio?
2. How does the total time constraint impact the ratio of speeds in such problems?
3. Can this problem be solved using proportionality concepts instead of equations?
4. How would this change if the speed of the car varied during the journey?
5. What if the entire journey was completed by train? What would the result be?

**Tip:** When comparing speeds over fixed distances, always set up equations based on time relations (distance divided by speed) to explore the relationship.

The distance between Kolkata and Ranchi is 600 km. If Rakesh moves 120 km by train and the rest by car, she reaches in the same time as traveling the entire journey by car. What would be the ratio between the speeds of the train and the car?

Solve the problem of finding the speed ratio between a train and a car when Rakesh travels 120 km by train and the remaining 480 km by car, covering the same time as the total journey by car. Explore the time-speed-distance relation in this real-world problem.

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