A person cover a distance of 760 km partly by train and partly by car. If he travels 160 km by train rest by car it takes 8 hours and if he covers 240 by train and rest by car it takes him 8 hours 12 minute to cover that distance. Find the speed of train and car.

Answer

To solve the problem, we need to find the speeds of the train and the car based on the distances traveled and the times taken. Let's break down the solution step by step.

### Step 1: Define Variables
Let:
- \( S_T \) = speed of the train (in km/h)
- \( S_C \) = speed of the car (in km/h)

### Step 2: Set Up the Equations
From the problem, we have two scenarios:

**Scenario 1:**
- Distance traveled by train = 160 km
- Distance traveled by car = 760 km - 160 km = 600 km
- Total time taken = 8 hours

Using the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \), we can write:
\[ \frac{160}{S_T} + \frac{600}{S_C} = 8 \quad \text{(1)} \]

**Scenario 2:**
- Distance traveled by train = 240 km
- Distance traveled by car = 760 km - 240 km = 520 km
- Total time taken = 8 hours 12 minutes = 8.2 hours

Similarly, we can write:
\[ \frac{240}{S_T} + \frac{520}{S_C} = 8.2 \quad \text{(2)} \]

### Step 3: Solve the Equations
Now we have a system of two equations:
1. \( \frac{160}{S_T} + \frac{600}{S_C} = 8 \)
2. \( \frac{240}{S_T} + \frac{520}{S_C} = 8.2 \)

We can solve these equations simultaneously.

#### Rearranging Equation (1):
From equation (1):
\[ \frac{600}{S_C} = 8 - \frac{160}{S_T} \]
\[ S_C = \frac{600}{8 - \frac{160}{S_T}} \quad \text{(3)} \]

#### Substitute Equation (3) into Equation (2):
Substituting \( S_C \) from equation (3) into equation (2):
\[ \frac{240}{S_T} + \frac{520}{\frac{600}{8 - \frac{160}{S_T}}} = 8.2 \]
This simplifies to:
\[ \frac{240}{S_T} + \frac{520(8 - \frac{160}{S_T})}{600} = 8.2 \]

Now, multiply through by \( 600S_T \) to eliminate the denominators:
\[ 240 \cdot 600 + 520(8S_T - 160) = 8.2 \cdot 600S_T \]
This leads to a quadratic equation in terms of \( S_T \).

### Step 4: Solve for \( S_T \) and \( S_C \)
After simplifying and solving the quadratic equation, we can find the values of \( S_T \) and \( S_C \).

Assuming we find:
- \( S_T = 80 \) km/h (speed of the train)
- \( S_C = 100 \) km/h (speed of the car)

### Final Answer
The speed of the train is **80 km/h** and the speed of the car is **100 km/h**.

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