Two cars start at the same time from A and B. If the two cars travels in opposite directions they meet each other in the mid of A and B. After meeting they take 3 hours and 1 hours 20 minutes to cover their distance completely. The ratio of the time taken by the trains to cover their distances is

Options

A. `3 : 2`
B. `16 : 24`
C. `13 : 16`
D. `16 : 13`

Answer

To solve the problem, we need to determine the ratio of the time taken by the two cars (C1 and C2) to cover their respective distances after they meet at point P, which is the midpoint between points A and B.

### Step-by-Step Solution:

1. **Understanding the Meeting Point**:
- Two cars start from points A and B and meet at point P, which is the midpoint between A and B.

2. **Time Taken After Meeting**:
- After meeting, Car C1 takes 3 hours to reach point A.
- Car C2 takes 1 hour and 20 minutes (which is equivalent to 1 + 20/60 = 1.33 hours) to reach point B.

3. **Calculating Total Time for Each Car**:
- Since they meet at the midpoint, the time taken to reach point P from A for Car C1 is the same as the time taken to reach point P from B for Car C2.
- Let’s denote the time taken by C1 to reach P from A as T1 and the time taken by C2 to reach P from B as T2.

4. **Finding the Total Time for Each Car**:
- For Car C1:
- Time taken to reach P = T1
- Time taken from P to A = 3 hours
- Total time for C1 = T1 + 3 hours
- For Car C2:
- Time taken to reach P = T2
- Time taken from P to B = 1 hour 20 minutes = 1.33 hours
- Total time for C2 = T2 + 1.33 hours

5. **Using the Midpoint Information**:
- Since they meet at the midpoint, the distances covered by both cars to reach P are equal, which means:
- T1 (time taken by C1 to reach P) = T2 (time taken by C2 to reach P)

6. **Setting Up the Equation**:
- Let T1 = T2 = T (the time taken to meet at point P).
- Therefore, the total time for C1 = T + 3 hours.
- The total time for C2 = T + 1.33 hours.

7. **Finding the Ratio**:
- The ratio of the total time taken by C1 to the total time taken by C2 is:
\[ \text{Ratio} = \frac{T + 3}{T + 1.33} \]
- To find the ratio in a simpler form, we can assume T = 0 (for simplicity):
\[ \text{Ratio} = \frac{3}{1.33} \approx \frac{300}{133} \approx \frac{18}{13} \]

8. **Conclusion**:
- Therefore, the ratio of the time taken by the two cars to cover their distances is **18:13**.