It takes eight hours for a 600 km journey, if 120 km is done by train and the rest by car. It takes 20 minutes more, if 200 km is done by train and the rest by car. The ratio of the speed of the train to that of the speed of the car is
A)4:3`
B)`3:4`
C)`3:2`
D)`2:3`

Options

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

Answer

To solve the problem, we need to find the ratio of the speed of the train (let's denote it as \( x \)) to the speed of the car (denote it as \( y \)). We will use the information given in the question to set up equations based on the distances traveled and the time taken.

### Step 1: Set up the equations based on the first scenario
In the first scenario, 120 km is traveled by train and the remaining 480 km is traveled by car. The total time taken for the journey is 8 hours.

The time taken by the train to cover 120 km is:
\[ \text{Time}_{\text{train}} = \frac{120}{x} \]

The time taken by the car to cover 480 km is:
\[ \text{Time}_{\text{car}} = \frac{480}{y} \]

According to the problem, the total time is:
\[ \frac{120}{x} + \frac{480}{y} = 8 \quad \text{(Equation 1)} \]

### Step 2: Set up the equations based on the second scenario
In the second scenario, 200 km is traveled by train and the remaining 400 km is traveled by car. The total time taken for this journey is 20 minutes more than the first journey, which is 8 hours + 20 minutes = 8 hours + \(\frac{1}{3}\) hours = \(\frac{25}{3}\) hours.

The time taken by the train to cover 200 km is:
\[ \text{Time}_{\text{train}} = \frac{200}{x} \]

The time taken by the car to cover 400 km is:
\[ \text{Time}_{\text{car}} = \frac{400}{y} \]

According to the problem, the total time is:
\[ \frac{200}{x} + \frac{400}{y} = \frac{25}{3} \quad \text{(Equation 2)} \]

### Step 3: Solve the equations
Now we have two equations:
1. \(\frac{120}{x} + \frac{480}{y} = 8\)
2. \(\frac{200}{x} + \frac{400}{y} = \frac{25}{3}\)

We can manipulate these equations to eliminate one variable. Let's multiply both equations by \(xy\) to clear the denominators.

From Equation 1:
\[ 120y + 480x = 8xy \quad \text{(Multiply by } xy\text{)} \]

From Equation 2:
\[ 200y + 400x = \frac{25}{3}xy \quad \text{(Multiply by } 3xy\text{)} \]

Now we have:
1. \(120y + 480x = 8xy\)
2. \(600y + 1200x = 25xy\)

### Step 4: Rearranging the equations
Rearranging both equations gives:
1. \(8xy - 120y - 480x = 0\)
2. \(25xy - 600y - 1200x = 0\)

### Step 5: Solve for the ratio
We can express \(y\) in terms of \(x\) using these equations. From the first equation, we can express \(y\):
\[ y = \frac{8xy - 480x}{120} \]

Substituting this expression for \(y\) into the second equation will allow us to find the ratio \( \frac{x}{y} \).

### Step 6: Finding the ratio
After substituting and simplifying, we will find that:
\[ \frac{x}{y} = \frac{3}{4} \]

Thus, the ratio of the speed of the train to the speed of the car is:
\[ \text{Ratio of } x \text{ to } y = 3:4 \]

### Final Answer
The correct option is **B) 3:4**.