A, B, C travels a distance of 1200 km from Delhi to Mumbai B & C takes a Car & A starts with Tonga and after a certain distance C throws B from Car. B takes a cycle and C returns to takes A and finally they reach Mumbai at same time. If the speed of car is 50 km/hr and of Cycle of Tonga is 10 km/hr each. Find the total time of journey.

Options

A. 25 hours
B. 28 hours
C. 48 hours
D. 58 hours

Answer

To solve the problem, we need to analyze the situation step by step.

### Step 1: Understand the journey
- A starts from Delhi to Mumbai using a Tonga.
- B and C travel together in a car.
- After a certain distance, C throws B out of the car.
- B then takes a cycle.
- C returns to pick up A.
- Finally, both A and C reach Mumbai at the same time.

### Step 2: Define the speeds
- Speed of the car (B and C) = 50 km/hr
- Speed of the Tonga (A) = 10 km/hr
- Speed of the cycle (B) = 10 km/hr

### Step 3: Let’s denote the distances
- Let the distance traveled by car before B is thrown out be \( x \) km.
- The remaining distance for B to cycle to Mumbai will be \( 1200 - x \) km.
- The distance C travels back to pick up A and then both of them travel to Mumbai.

### Step 4: Calculate the time taken by each
1. **Time taken by B in the car until he is thrown out**:
\[ \text{Time}_{B, \text{car}} = \frac{x}{50} \]

2. **Time taken by B to cycle the remaining distance**:
\[ \text{Time}_{B, \text{cycle}} = \frac{1200 - x}{10} \]

3. **Time taken by C to return to A**:
- The distance C travels back to A is \( x \) km.
\[ \text{Time}_{C, \text{return}} = \frac{x}{50} \]

4. **Time taken by C and A to reach Mumbai together**:
- After picking up A, they have \( 1200 - x \) km to travel.
\[ \text{Time}_{C, A, \text{to Mumbai}} = \frac{1200 - x}{50} \]

### Step 5: Set up the equation
Since A and C reach Mumbai at the same time as B, we can set up the equation:
\[ \text{Time}_{B, \text{car}} + \text{Time}_{B, \text{cycle}} = \text{Time}_{C, \text{return}} + \text{Time}_{C, A, \text{to Mumbai}} \]

Substituting the times:
\[ \frac{x}{50} + \frac{1200 - x}{10} = \frac{x}{50} + \frac{1200 - x}{50} \]

### Step 6: Simplify the equation
1. The left side becomes:
\[ \frac{x}{50} + \frac{1200 - x}{10} = \frac{x}{50} + \frac{1200 - x}{10} = \frac{x + 1200 \cdot 5 - 5x}{50} = \frac{6000 - 4x}{50} \]

2. The right side simplifies to:
\[ \frac{x}{50} + \frac{1200 - x}{50} = \frac{1200}{50} = 24 \]

### Step 7: Solve for x
Setting the two sides equal:
\[ \frac{6000 - 4x}{50} = 24 \]
Multiplying both sides by 50:
\[ 6000 - 4x = 1200 \]
Rearranging gives:
\[ 4x = 6000 - 1200 = 4800 \implies x = 1200 \]

### Step 8: Calculate total time
Now, we can find the total time taken by A, B, and C:
- Time taken by B in the car:
\[ \text{Time}_{B, \text{car}} = \frac{1200}{50} = 24 \text{ hours} \]
- Time taken by B in the cycle:
\[ \text{Time}_{B, \text{cycle}} = \frac{0}{10} = 0 \text{ hours} \]
- Time taken by C to return to A:
\[ \text{Time}_{C, \text{return}} = \frac{1200}{50} = 24 \text{ hours} \]
- Time taken by C and A to reach Mumbai:
\[ \text{Time}_{C, A, \text{to Mumbai}} = \frac{1200}{50} = 24 \text{ hours} \]

### Total time of journey:
Thus, the total time of the journey is:
\[ \text{Total time} = 24 + 0 + 24 = 48 \text{ hours} \]

### Final Answer:
The total time of the journey is **48 hours**.