Cities M and N are 600 km apart. Bus A starts from City M towards N at 9 AM and bus B starts from city N towards M at the same time. Bus A travels the first `(1)/(3)`rd of the distance at a speed of 40 km/h, the second `(1)/(3)`rd at 60 km/h and the last `(1)/(3)`rd at 40 km/h. Bus B travels the first `(1)/(3)`rd of the total distance at a speed of 50 km/h, the second`(1)/(3)`rd at 60 km/h and the last `(1)/(3)`rd at 30 km/h.
If Bus A and Bus B will be completing their return journey at constant speeds of 40 km/h and 60 km/h respectively on the next day, then how far from the previous day's meeting point they will meet?

Options

A. 30 km
B. 90 km
C. 60 km
D. 120 km

Answer

To solve the problem step by step, we will first determine the meeting point of Bus A and Bus B on the first day, and then calculate how far they will meet from that point on the second day.

### Step 1: Calculate the distance each bus travels in the first segment.
- Total distance between cities M and N = 600 km.
- Each bus travels the first \( \frac{1}{3} \) of the distance, which is \( \frac{600}{3} = 200 \) km.

### Step 2: Calculate the time taken by each bus to travel the first segment.
- **Bus A** travels the first 200 km at a speed of 40 km/h:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{200 \text{ km}}{40 \text{ km/h}} = 5 \text{ hours} \]

- **Bus B** travels the first 200 km at a speed of 50 km/h:
\[ \text{Time} = \frac{200 \text{ km}}{50 \text{ km/h}} = 4 \text{ hours} \]

### Step 3: Determine the distance covered by each bus after 4 hours.
- After 4 hours, Bus B has traveled 200 km, and Bus A has traveled:
\[ \text{Distance by Bus A in 4 hours} = 40 \text{ km/h} \times 4 \text{ hours} = 160 \text{ km} \]
- Therefore, Bus A has traveled 160 km and Bus B has traveled 200 km.

### Step 4: Calculate the remaining distance to be covered by both buses.
- Distance covered by both buses = \( 160 + 200 = 360 \) km.
- Remaining distance = \( 600 - 360 = 240 \) km.

### Step 5: Calculate the time taken to cover the remaining distance.
- The remaining distance of 240 km will be covered by both buses moving towards each other.
- Combined speed = \( 40 \text{ km/h} + 50 \text{ km/h} = 90 \text{ km/h} \).
- Time to cover the remaining distance:
\[ \text{Time} = \frac{240 \text{ km}}{90 \text{ km/h}} = \frac{8}{3} \text{ hours} = 2 \text{ hours and } 40 \text{ minutes} \]

### Step 6: Determine the total time until they meet.
- Total time until they meet = \( 4 \text{ hours} + 2 \text{ hours and } 40 \text{ minutes} = 6 \text{ hours and } 40 \text{ minutes} \).
- Since both buses started at 9 AM, they will meet at:
\[ 9:00 \text{ AM} + 6 \text{ hours and } 40 \text{ minutes} = 3:40 \text{ PM} \]

### Step 7: Calculate the distance from the starting point to the meeting point.
- Distance covered by Bus A in 6 hours and 40 minutes:
\[ \text{Distance} = 40 \text{ km/h} \times 6.67 \text{ hours} \approx 267 \text{ km} \]
- Distance covered by Bus B in the same time:
\[ \text{Distance} = 50 \text{ km/h} \times 6.67 \text{ hours} \approx 333 \text{ km} \]

### Step 8: Calculate the distance from the meeting point to the return journey meeting point.
- On the second day, Bus A travels at 40 km/h and Bus B at 60 km/h.
- Combined speed = \( 40 + 60 = 100 \) km/h.
- Time taken to meet on the second day:
\[ \text{Time} = \frac{600 \text{ km}}{100 \text{ km/h}} = 6 \text{ hours} \]
- Distance covered by Bus A in 6 hours:
\[ \text{Distance} = 40 \text{ km/h} \times 6 \text{ hours} = 240 \text{ km} \]
- Distance covered by Bus B in 6 hours:
\[ \text{Distance} = 60 \text{ km/h} \times 6 \text{ hours} = 360 \text{ km} \]

### Step 9: Calculate the distance from the previous day's meeting point.
- The distance from the previous meeting point (270 km from M) to the new meeting point (360 km from N) is:
\[ \text{Distance} = 360 - 270 = 90 \text{ km} \]

### Final Answer:
The distance from the previous day's meeting point where they will meet again is **90 km**.

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