Solve the following problems using two variables :
Some part of a journey of 780 km was made by car with a speed of 60 km/h and the remaining journey was made by train with a speed of 100 km/h . If total required was 9 hours . Find the time taken by train and distance covered by train .

Answer

To solve the problem, we will use two variables to represent the distances and times involved in the journey. Let's denote:

- \( x \): the distance traveled by car (in km)
- \( y \): the distance traveled by train (in km)

Given:
- Total distance of the journey = 780 km
- Speed of the car = 60 km/h
- Speed of the train = 100 km/h
- Total time taken = 9 hours

### Step 1: Set up the equations
From the problem, we can set up the following equations:

1. The total distance equation:
\[ x + y = 780 \]

2. The total time equation:
The time taken by car is given by \( \frac{x}{60} \) and the time taken by train is given by \( \frac{y}{100} \). Therefore, the total time equation is:
\[ \frac{x}{60} + \frac{y}{100} = 9 \]

### Step 2: Express \( y \) in terms of \( x \)
From the first equation, we can express \( y \) in terms of \( x \):
\[ y = 780 - x \]

### Step 3: Substitute \( y \) in the time equation
Now, substitute \( y \) in the second equation:
\[ \frac{x}{60} + \frac{780 - x}{100} = 9 \]

### Step 4: Clear the fractions
To eliminate the fractions, we can multiply the entire equation by the least common multiple (LCM) of 60 and 100, which is 300:
\[ 300 \left(\frac{x}{60}\right) + 300 \left(\frac{780 - x}{100}\right) = 300 \cdot 9 \]
This simplifies to:
\[ 5x + 3(780 - x) = 2700 \]

### Step 5: Simplify the equation
Now, distribute and simplify:
\[ 5x + 2340 - 3x = 2700 \]
Combine like terms:
\[ 2x + 2340 = 2700 \]

### Step 6: Solve for \( x \)
Subtract 2340 from both sides:
\[ 2x = 2700 - 2340 \]
\[ 2x = 360 \]
Now divide by 2:
\[ x = 180 \]

### Step 7: Find \( y \)
Now, substitute \( x \) back into the equation for \( y \):
\[ y = 780 - x = 780 - 180 = 600 \]

### Step 8: Find the time taken by train
The time taken by the train can be calculated using the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{y}{100} = \frac{600}{100} = 6 \text{ hours} \]

### Final Answers
- Distance covered by train = 600 km
- Time taken by train = 6 hours