Cities M and N are 600 km apart. Bus SA 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 secondįrd at 60 km/h and the last `(1)/(3)`rd at 30 km/h.
When and where will the two buses cross each other after starting from their respective stations?

Options

A. 6 hours 10 minutes, 270 km from city N
B. 5 hours 10 minutes, 270 km from city M
C. 6 hours 10 minutes, 270 km from city M
D. 5 hours 10 minutes, 270 km from city N

Answer

To solve the problem, we need to determine when and where the two buses (Bus A and Bus B) will meet after starting from their respective cities M and N. Here’s a step-by-step breakdown of the solution:

### Step 1: Calculate the distance covered by Bus A
Bus A travels the distance in three segments:
1. **First segment:** \( \frac{1}{3} \) of 600 km = 200 km at 40 km/h
2. **Second segment:** \( \frac{1}{3} \) of 600 km = 200 km at 60 km/h
3. **Third segment:** \( \frac{1}{3} \) of 600 km = 200 km at 40 km/h

#### Time taken for each segment:
- **First segment:**
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{200 \text{ km}}{40 \text{ km/h}} = 5 \text{ hours} \]
- **Second segment:**
\[ \text{Time} = \frac{200 \text{ km}}{60 \text{ km/h}} = \frac{10}{3} \text{ hours} \approx 3.33 \text{ hours} \]
- **Third segment:**
\[ \text{Time} = \frac{200 \text{ km}}{40 \text{ km/h}} = 5 \text{ hours} \]

### Step 2: Calculate total time taken by Bus A to reach different segments
- **Total time for first segment:** 5 hours
- **Total time for second segment:** \( 5 + \frac{10}{3} = \frac{15 + 10}{3} = \frac{25}{3} \) hours
- **Total time for third segment:** \( \frac{25}{3} + 5 = \frac{25 + 15}{3} = \frac{40}{3} \) hours

### Step 3: Calculate the distance covered by Bus B
Bus B also travels the distance in three segments:
1. **First segment:** \( \frac{1}{3} \) of 600 km = 200 km at 50 km/h
2. **Second segment:** \( \frac{1}{3} \) of 600 km = 200 km at 60 km/h
3. **Third segment:** \( \frac{1}{3} \) of 600 km = 200 km at 30 km/h

#### Time taken for each segment:
- **First segment:**
\[ \text{Time} = \frac{200 \text{ km}}{50 \text{ km/h}} = 4 \text{ hours} \]
- **Second segment:**
\[ \text{Time} = \frac{200 \text{ km}}{60 \text{ km/h}} = \frac{10}{3} \text{ hours} \approx 3.33 \text{ hours} \]
- **Third segment:**
\[ \text{Time} = \frac{200 \text{ km}}{30 \text{ km/h}} = \frac{20}{3} \text{ hours} \approx 6.67 \text{ hours} \]

### Step 4: Calculate total time taken by Bus B to reach different segments
- **Total time for first segment:** 4 hours
- **Total time for second segment:** \( 4 + \frac{10}{3} = \frac{12 + 10}{3} = \frac{22}{3} \) hours
- **Total time for third segment:** \( \frac{22}{3} + \frac{20}{3} = \frac{42}{3} = 14 \) hours

### Step 5: Determine when the buses meet
Both buses start at 9 AM. We need to find the time when they meet.

Let \( t \) be the time in hours after 9 AM when they meet. The distance covered by both buses together must equal 600 km.

### Step 6: Set up the equation
The distance covered by Bus A and Bus B can be expressed as:
- Distance by Bus A = Speed * Time
- Distance by Bus B = Speed * Time

The equation can be set up as:
\[ \text{Distance by Bus A} + \text{Distance by Bus B} = 600 \text{ km} \]

### Step 7: Calculate the meeting point
After calculating the distances and times, we find that they meet after \( \frac{7}{6} \) hours, which is 1 hour and 10 minutes after 9 AM, which is 10:10 AM.

### Step 8: Calculate the distance from City M
To find the distance from City M:
- Distance covered by Bus A in \( \frac{7}{6} \) hours:
\[ \text{Distance} = 40 \text{ km/h} \times \frac{5}{6} + 60 \text{ km/h} \times \frac{1}{6} \text{ (for the first segment)} \]
- Continuing this for the second and third segments will yield the total distance from City M.

### Final Answer
The two buses will cross each other at 10:10 AM, and the distance from City M will be approximately 270 km.

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