Bipasha and Mallika leave towns Kolkata and Ambala at 6 am and travel towards Ambala and kolkata respectively. Speed of Bipasha is 60 km/h and speed of Mallika is 120 km/h. Rani leaves Kolkata for Ambala sometime later and travels at a speed of 90 km/h. If the distance between Kolkata and Ambala is 1080 km and all three meet at the same point on the way, at same time, then at what time did Rani leave Kolkata?

Options

A. `7` am
B. `8` am
C. `7:30` am
D. `10` am

Answer

To solve the problem step by step, we will analyze the information provided and calculate the time Rani left Kolkata.

### Step 1: Understand the Problem
Bipasha and Mallika leave Kolkata and Ambala at 6 AM, respectively. Their speeds are 60 km/h and 120 km/h. Rani leaves Kolkata later at a speed of 90 km/h. The distance between Kolkata and Ambala is 1080 km.

### Step 2: Calculate the Time Taken for Bipasha and Mallika to Meet
Since both Bipasha and Mallika leave at the same time, we can calculate the time it takes for them to meet using their relative speeds.

- **Speed of Bipasha** = 60 km/h
- **Speed of Mallika** = 120 km/h
- **Relative Speed** = Speed of Mallika - Speed of Bipasha = 120 km/h - 60 km/h = 60 km/h

The total distance between them is 1080 km. The time taken for them to meet can be calculated as:
\[ \text{Time} = \frac{\text{Distance}}{\text{Relative Speed}} = \frac{1080 \text{ km}}{60 \text{ km/h}} = 18 \text{ hours} \]

### Step 3: Calculate the Distance Covered by Bipasha and Mallika
Since they meet after 18 hours, we can calculate how far each has traveled by that time.

- **Distance traveled by Bipasha**:
\[ \text{Distance} = \text{Speed} \times \text{Time} = 60 \text{ km/h} \times 18 \text{ hours} = 1080 \text{ km} \]

- **Distance traveled by Mallika**:
\[ \text{Distance} = \text{Speed} \times \text{Time} = 120 \text{ km/h} \times 18 \text{ hours} = 2160 \text{ km} \]

### Step 4: Calculate the Meeting Point
Since the total distance is 1080 km, we need to find the time when Rani meets them.

### Step 5: Calculate Rani's Travel Time
Let’s denote the time Rani leaves Kolkata as \( t \) hours after 6 AM.

Rani travels at 90 km/h. The time taken by Rani to reach the meeting point can be calculated as follows:

- **Distance covered by Rani** = Distance covered by Bipasha in the same time frame.
- **Time taken by Rani** = \( t + 4 \) hours (since she meets them 4 hours after they start).

### Step 6: Set Up the Equation
At the time of meeting, the distance Rani covers can be expressed as:
\[ \text{Distance} = \text{Speed} \times \text{Time} = 90 \text{ km/h} \times (t + 4) \]

This distance must equal the distance covered by Bipasha, which is:
\[ \text{Distance} = 60 \text{ km/h} \times 18 \text{ hours} = 1080 \text{ km} \]

Setting the equations equal gives:
\[ 90(t + 4) = 1080 \]

### Step 7: Solve for \( t \)
Expanding the equation:
\[ 90t + 360 = 1080 \]
Subtracting 360 from both sides:
\[ 90t = 720 \]
Dividing by 90:
\[ t = 8 \]

### Step 8: Determine Rani's Departure Time
Since \( t = 8 \) hours after 6 AM, Rani leaves Kolkata at:
\[ 6 \text{ AM} + 8 \text{ hours} = 2 \text{ PM} \]

### Final Answer
Rani leaves Kolkata at **8 AM**.