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.
At what times Bus A and Bus B reach their destinations respectively? (2 Marks)

Options

A. 10 PM, 11:20 PM
B. 11 PM, 10:20 PM
C. 11:20 PM, 10 PM
D. 10:20 PM, 11 PM

Answer

To solve the problem step by step, we will calculate the time taken by both buses A and B to reach their respective destinations.

### Step 1: Calculate the distance each bus travels
The total distance between cities M and N is 600 km. Each bus travels one-third of this distance at different speeds.

- Distance for each one-third = \( \frac{600 \text{ km}}{3} = 200 \text{ km} \)

### Step 2: Calculate the time taken by Bus A
Bus A travels the first 200 km at 40 km/h, the second 200 km at 60 km/h, and the last 200 km at 40 km/h.

1. **First 200 km at 40 km/h:**
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{200 \text{ km}}{40 \text{ km/h}} = 5 \text{ hours} \]

2. **Second 200 km at 60 km/h:**
\[ \text{Time} = \frac{200 \text{ km}}{60 \text{ km/h}} = \frac{10}{3} \text{ hours} \approx 3.33 \text{ hours} \]

3. **Last 200 km at 40 km/h:**
\[ \text{Time} = \frac{200 \text{ km}}{40 \text{ km/h}} = 5 \text{ hours} \]

### Step 3: Total time taken by Bus A
Now, we sum up the times calculated:
\[ \text{Total time} = 5 + \frac{10}{3} + 5 = 10 + \frac{10}{3} = \frac{30}{3} + \frac{10}{3} = \frac{40}{3} \text{ hours} \]
Converting \( \frac{40}{3} \) hours to hours and minutes:
\[ \frac{40}{3} \text{ hours} = 13 \text{ hours and } 20 \text{ minutes} \]

### Step 4: Calculate the arrival time of Bus A
Bus A starts at 9 AM:
\[ 9:00 \text{ AM} + 13 \text{ hours } 20 \text{ minutes} = 10:20 \text{ PM} \]

### Step 5: Calculate the time taken by Bus B
Bus B travels the first 200 km at 50 km/h, the second 200 km at 60 km/h, and the last 200 km at 30 km/h.

1. **First 200 km at 50 km/h:**
\[ \text{Time} = \frac{200 \text{ km}}{50 \text{ km/h}} = 4 \text{ hours} \]

2. **Second 200 km at 60 km/h:**
\[ \text{Time} = \frac{200 \text{ km}}{60 \text{ km/h}} = \frac{10}{3} \text{ hours} \approx 3.33 \text{ hours} \]

3. **Last 200 km at 30 km/h:**
\[ \text{Time} = \frac{200 \text{ km}}{30 \text{ km/h}} = \frac{20}{3} \text{ hours} \approx 6.67 \text{ hours} \]

### Step 6: Total time taken by Bus B
Now, we sum up the times calculated:
\[ \text{Total time} = 4 + \frac{10}{3} + \frac{20}{3} = 4 + \frac{30}{3} = 4 + 10 = 14 \text{ hours} \]

### Step 7: Calculate the arrival time of Bus B
Bus B starts at 9 AM:
\[ 9:00 \text{ AM} + 14 \text{ hours} = 11:00 \text{ PM} \]

### Final Answer
- **Bus A reaches City N at:** 10:20 PM
- **Bus B reaches City M at:** 11:00 PM

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