Distance between Kota and Jaipur is 240 km. Two persons A and B both start simultaneously from Kota to Kota to Jaipur. A travels half distance with 40 km/hr and remaining half distance with 60 km/hr. B travels with 40 km/hr for half time of journey and half time withy 60 km/hr. Find out
(i) Average speed of A and B
(ii) When and Who will reach the Jaipur first

Answer

To solve the problem step by step, we will calculate the average speeds of both persons A and B, and then determine who reaches Jaipur first.

### Step 1: Calculate Average Speed of A

1. **Distance covered by A**: The total distance from Kota to Jaipur is 240 km.
2. **Half distance**: A travels half of the distance (120 km) at 40 km/hr and the other half (120 km) at 60 km/hr.
3. **Time taken for the first half**:
\[ T_1 = \frac{120 \text{ km}}{40 \text{ km/hr}} = 3 \text{ hours} \]
4. **Time taken for the second half**:
\[ T_2 = \frac{120 \text{ km}}{60 \text{ km/hr}} = 2 \text{ hours} \]
5. **Total time taken by A**:
\[ T_A = T_1 + T_2 = 3 \text{ hours} + 2 \text{ hours} = 5 \text{ hours} \]
6. **Average speed of A**:
\[ \text{Average Speed}_A = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{240 \text{ km}}{5 \text{ hours}} = 48 \text{ km/hr} \]

### Step 2: Calculate Average Speed of B

1. **Total time of journey**: Let the total time taken by B be \(2T\).
2. **Time for first half**: B travels with 40 km/hr for half the time \(T\) and with 60 km/hr for the other half \(T\).
3. **Distance covered in the first half**:
\[ D_1 = 40 \text{ km/hr} \times T \]
4. **Distance covered in the second half**:
\[ D_2 = 60 \text{ km/hr} \times T \]
5. **Total distance equation**:
\[ D_1 + D_2 = 240 \text{ km} \implies 40T + 60T = 240 \implies 100T = 240 \implies T = 2.4 \text{ hours} \]
6. **Total time taken by B**:
\[ T_B = 2T = 2 \times 2.4 = 4.8 \text{ hours} \]
7. **Average speed of B**:
\[ \text{Average Speed}_B = \frac{240 \text{ km}}{4.8 \text{ hours}} = 50 \text{ km/hr} \]

### Step 3: Determine Who Reaches First

1. **Time taken by A**: 5 hours
2. **Time taken by B**: 4.8 hours

Since \(4.8 \text{ hours} < 5 \text{ hours}\), B reaches Jaipur first.

### Final Results

- **Average Speed of A**: 48 km/hr
- **Average Speed of B**: 50 km/hr
- **Who reaches first**: B reaches first after 4.8 hours.