A, B and C start from Delhi at 10 am, 11 am, & 12.00 towards Goa & their speed are 3 km/hr, 4 km/hr & 5 km/hr. & After meeting on the way, B send back A to C with a message. At what time C will get the message ?

Options

A. `2 : 45`
B. `2 : 35`
C. `2 : 15`
D. `2 : 05`

Answer

To solve the problem step-by-step, we will analyze the movements of A, B, and C and calculate when C will receive the message from A.

### Step 1: Determine the starting times and speeds
- A starts at 10 AM with a speed of 3 km/hr.
- B starts at 11 AM with a speed of 4 km/hr.
- C starts at 12 PM with a speed of 5 km/hr.

### Step 2: Calculate the positions of A and B when C starts
At 12 PM:
- A has been traveling for 2 hours:
\[ \text{Distance covered by A} = \text{Speed} \times \text{Time} = 3 \text{ km/hr} \times 2 \text{ hrs} = 6 \text{ km} \]
- B has been traveling for 1 hour:
\[ \text{Distance covered by B} = 4 \text{ km/hr} \times 1 \text{ hr} = 4 \text{ km} \]

### Step 3: Calculate the distance between A and B at 12 PM
- The distance between A and B:
\[ \text{Distance between A and B} = 6 \text{ km} - 4 \text{ km} = 2 \text{ km} \]

### Step 4: Calculate the time taken for A and B to meet
- The relative speed of A and B:
\[ \text{Relative speed} = 4 \text{ km/hr} - 3 \text{ km/hr} = 1 \text{ km/hr} \]
- Time taken to cover the gap of 2 km:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{2 \text{ km}}{1 \text{ km/hr}} = 2 \text{ hours} \]
- They will meet at:
\[ 12 \text{ PM} + 2 \text{ hours} = 2 \text{ PM} \]

### Step 5: Determine C's position at 2 PM
- C has been traveling for 2 hours:
\[ \text{Distance covered by C} = 5 \text{ km/hr} \times 2 \text{ hrs} = 10 \text{ km} \]

### Step 6: Calculate the distance between A and C at 2 PM
- The distance between A and C at 2 PM:
\[ \text{Distance between A and C} = 6 \text{ km} + 10 \text{ km} = 16 \text{ km} \]

### Step 7: Calculate the time taken for A to return to C with the message
- The relative speed of A and C when A is returning to C:
\[ \text{Relative speed} = 5 \text{ km/hr} + 3 \text{ km/hr} = 8 \text{ km/hr} \]
- Time taken to cover the distance of 16 km:
\[ \text{Time} = \frac{16 \text{ km}}{8 \text{ km/hr}} = 2 \text{ hours} \]

### Step 8: Determine the time C receives the message
- Time when A starts returning to C:
\[ 2 \text{ PM} + 2 \text{ hours} = 4 \text{ PM} \]

### Final Answer
C will receive the message at **4 PM**.