A person cover a distance of 450 km partly by train and partly by car. If he travels 100 km by train rest by car it takes 8 hours and if he covers 170 by train and rest by car it takes him 8 hours 12 minute to cover that distance. Find the speed of train.

Answer

To solve the problem, we need to determine the speed of the train based on the information provided about the distances traveled and the time taken. Let's break it down 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:

1. **First Scenario:**
- Distance traveled by train = 100 km
- Distance traveled by car = 450 km - 100 km = 350 km
- Total time taken = 8 hours

The equation for the first scenario can be expressed as:
\[ \frac{100}{S_t} + \frac{350}{S_c} = 8 \quad \text{(1)} \]

2. **Second Scenario:**
- Distance traveled by train = 170 km
- Distance traveled by car = 450 km - 170 km = 280 km
- Total time taken = 8 hours 12 minutes = \( 8 + \frac{12}{60} = 8.2 \) hours

The equation for the second scenario can be expressed as:
\[ \frac{170}{S_t} + \frac{280}{S_c} = 8.2 \quad \text{(2)} \]

### Step 3: Solve the Equations
We have two equations:
1. \( \frac{100}{S_t} + \frac{350}{S_c} = 8 \)
2. \( \frac{170}{S_t} + \frac{280}{S_c} = 8.2 \)

We can solve these equations simultaneously.

#### Rearranging Equation (1):
From equation (1):
\[ \frac{350}{S_c} = 8 - \frac{100}{S_t} \]
\[ S_c = \frac{350}{8 - \frac{100}{S_t}} \quad \text{(3)} \]

#### Substituting into Equation (2):
Substituting equation (3) into equation (2):
\[ \frac{170}{S_t} + \frac{280}{\frac{350}{8 - \frac{100}{S_t}}} = 8.2 \]
This simplifies to:
\[ \frac{170}{S_t} + \frac{280(8 - \frac{100}{S_t})}{350} = 8.2 \]
\[ \frac{170}{S_t} + \frac{8(280) - 280 \cdot \frac{100}{S_t}}{350} = 8.2 \]
\[ \frac{170}{S_t} + \frac{2240 - 28000/S_t}{350} = 8.2 \]

#### Finding a common denominator:
Multiply through by \( 350S_t \) to eliminate the denominators:
\[ 170 \cdot 350 + (2240 - 28000/S_t) \cdot S_t = 8.2 \cdot 350S_t \]
This leads to a quadratic equation in terms of \( S_t \).

### Step 4: Solve for \( S_t \)
After simplifying and solving the quadratic equation, we find:
\[ S_t = 50 \text{ km/h} \]

### Step 5: Conclusion
The speed of the train is \( 50 \) km/h.

---