A car departs from A to B and another car depart from B to A. Two cars meet at mid-point of A and B. After meeting they complete their journey in 2 hours and 1 hour 20 min. respectively, Ratio of time taken by then for whole journey-

Options

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

Answer

To solve the problem step by step, we will analyze the journey of both cars and calculate the total time taken by each car for the entire journey.

### Step 1: Understand the scenario
- Two cars, C1 and C2, start from points A and B respectively and meet at the midpoint X.
- After meeting, C1 takes 2 hours to reach B, and C2 takes 1 hour 20 minutes to reach A.

**Hint:** Visualize the journey and the meeting point to understand the distances involved.

### Step 2: Convert time into a consistent unit
- Convert the time taken by C2 from hours and minutes to minutes:
- 1 hour 20 minutes = 60 minutes + 20 minutes = 80 minutes.

**Hint:** Always convert time into the same unit (here, minutes) for easier calculations.

### Step 3: Determine the distances after meeting
- Let the distance from the meeting point X to B be Y.
- C1 takes 2 hours (which is 120 minutes) to cover this distance Y.
- C2 takes 1 hour 20 minutes (which is 80 minutes) to cover the same distance Y.

**Hint:** Define distances clearly to relate them to the time taken.

### Step 4: Calculate the speeds of the cars after meeting
- Speed of C1 after meeting = Distance / Time = Y / 120 minutes.
- Speed of C2 after meeting = Distance / Time = Y / 80 minutes.

**Hint:** Remember that speed is calculated as distance divided by time.

### Step 5: Calculate the time taken to reach the midpoint
- Since they meet at the midpoint, the distance from A to X (for C1) is also Y.
- The time taken by C1 to reach X from A is twice the time taken to cover Y after meeting:
- Time taken by C1 = 2 * 120 minutes = 240 minutes.

**Hint:** Use the relationship of distances and speeds to find the time taken before the meeting.

### Step 6: Calculate the total time for C2
- Similarly, the time taken by C2 to reach X from B is twice the time taken to cover Y after meeting:
- Time taken by C2 = 2 * 80 minutes = 160 minutes.

**Hint:** Apply the same logic for both cars to find their total journey times.

### Step 7: Calculate the total journey times
- Total time taken by C1 for the whole journey = Time to X + Time after meeting = 240 minutes + 120 minutes = 360 minutes.
- Total time taken by C2 for the whole journey = Time to X + Time after meeting = 160 minutes + 80 minutes = 240 minutes.

**Hint:** Ensure you sum the times correctly for the entire journey.

### Step 8: Find the ratio of the total times
- The ratio of time taken by C1 to C2 = 360 minutes : 240 minutes.
- Simplifying this ratio:
- Divide both sides by 120:
- 360/120 : 240/120 = 3 : 2.

**Hint:** Simplifying ratios helps in finding the final answer easily.

### Final Answer
The ratio of the time taken by the two cars for the whole journey is **3 : 2**.