A car travels 140 km partly at a speed of 6 km/h and the remaining at a speed of 10 km/h. If the speeds, are reversed then it travels 8 km more in the same time. Then find the time takes by car to travel 140 km and also find what was the average speed of the car?

Answer

To solve the problem step by step, we will break it down into manageable parts.

### Step 1: Define Variables
Let:
- \( x \) = time (in hours) traveled at 6 km/h
- \( y \) = time (in hours) traveled at 10 km/h

### Step 2: Set Up the First Equation
From the problem, we know that the total distance traveled is 140 km. The distance traveled at 6 km/h is \( 6x \) and the distance traveled at 10 km/h is \( 10y \). Therefore, we can write the first equation as:
\[ 6x + 10y = 140 \]
This is our Equation 1.

### Step 3: Set Up the Second Equation
If the speeds are reversed, the car travels 8 km more in the same time. This means at 10 km/h it travels for \( x \) hours and at 6 km/h for \( y \) hours, resulting in a total distance of 148 km. Thus, we can write the second equation as:
\[ 10x + 6y = 148 \]
This is our Equation 2.

### Step 4: Simplify the Equations
We can simplify both equations to make them easier to work with.

From Equation 1:
\[ 3x + 5y = 70 \quad \text{(divide the entire equation by 2)} \]
From Equation 2:
\[ 5x + 3y = 74 \quad \text{(divide the entire equation by 2)} \]

### Step 5: Solve the System of Equations
Now we have a system of equations:
1. \( 3x + 5y = 70 \)
2. \( 5x + 3y = 74 \)

We can use the method of elimination or substitution. Here, we will use elimination.

Multiply Equation 1 by 5:
\[ 15x + 25y = 350 \]

Multiply Equation 2 by 3:
\[ 15x + 9y = 222 \]

Now, subtract the second equation from the first:
\[ (15x + 25y) - (15x + 9y) = 350 - 222 \]
This simplifies to:
\[ 16y = 128 \]
So,
\[ y = 8 \]

### Step 6: Substitute to Find \( x \)
Now, substitute \( y = 8 \) back into Equation 1:
\[ 3x + 5(8) = 70 \]
\[ 3x + 40 = 70 \]
\[ 3x = 30 \]
\[ x = 10 \]

### Step 7: Calculate Total Time
The total time taken by the car to travel 140 km is:
\[ x + y = 10 + 8 = 18 \text{ hours} \]

### Step 8: Calculate Average Speed
The average speed is given by the formula:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]
Substituting the values we have:
\[ \text{Average Speed} = \frac{140 \text{ km}}{18 \text{ hours}} \]
This simplifies to:
\[ \text{Average Speed} = \frac{70}{9} \text{ km/h} \]

### Final Answers
- The time taken by the car to travel 140 km is **18 hours**.
- The average speed of the car is **\( \frac{70}{9} \) km/h**.