A completes a journey of 540 km in 5 hours. If he travels a part of journey by train at 120 kmph and second part of journey by bus at speed of 100 kmph. Find the ratio between time taken to complete the first part and second part.

Options

A. A)`2:3`
B. B) `3:2`
C. C)`1:4`
D. D)`4:1`

Answer

To solve the problem, we need to find the ratio of the time taken by A to complete the journey by train and by bus. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the total journey
A completes a total journey of 540 km in 5 hours.

### Step 2: Calculate the overall speed
The overall speed can be calculated using the formula:
\[ \text{Overall Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{540 \text{ km}}{5 \text{ hours}} = 108 \text{ km/h} \]

### Step 3: Set up the speeds for the parts of the journey
- Speed of the train = 120 km/h
- Speed of the bus = 100 km/h

### Step 4: Use the concept of weighted averages
We can use the concept of weighted averages to find the ratio of the times taken for each part of the journey. The formula for the weighted average speed is:
\[ \text{Overall Speed} = \frac{(S_1 \cdot T_1 + S_2 \cdot T_2)}{(T_1 + T_2)} \]
Where \( S_1 \) and \( S_2 \) are the speeds for the two parts, and \( T_1 \) and \( T_2 \) are the times taken for each part.

### Step 5: Set the ratio of times
Let \( T_1 \) be the time taken by the train and \( T_2 \) be the time taken by the bus. We can express the distances covered by each mode of transport as:
\[ \text{Distance by Train} = S_1 \cdot T_1 = 120 \cdot T_1 \]
\[ \text{Distance by Bus} = S_2 \cdot T_2 = 100 \cdot T_2 \]

### Step 6: Total distance equation
The total distance is the sum of the distances covered by the train and the bus:
\[ 120 \cdot T_1 + 100 \cdot T_2 = 540 \]

### Step 7: Total time equation
The total time taken is:
\[ T_1 + T_2 = 5 \]

### Step 8: Solve the equations
From the total time equation, we can express \( T_2 \) in terms of \( T_1 \):
\[ T_2 = 5 - T_1 \]

Substituting \( T_2 \) into the total distance equation:
\[ 120 \cdot T_1 + 100 \cdot (5 - T_1) = 540 \]
Expanding this gives:
\[ 120 T_1 + 500 - 100 T_1 = 540 \]
Combining like terms:
\[ 20 T_1 + 500 = 540 \]
Subtracting 500 from both sides:
\[ 20 T_1 = 40 \]
Dividing by 20:
\[ T_1 = 2 \text{ hours} \]

Now substituting back to find \( T_2 \):
\[ T_2 = 5 - T_1 = 5 - 2 = 3 \text{ hours} \]

### Step 9: Find the ratio of times
The ratio of the time taken for the first part (by train) to the second part (by bus) is:
\[ \text{Ratio} = \frac{T_1}{T_2} = \frac{2}{3} \]

### Final Answer
The ratio between the time taken to complete the first part and the second part is \( 2:3 \).

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