A and B are two points 150 km apart on a highway. Two cars start with different speeds from A and B at same time. If they move in same direction, they meet in 15 hours. If they move in opposite direction, they meet in one hour. Find their speeds

Answer

To solve the problem, we will set up equations based on the information given about the two cars traveling towards each other and in the same direction.

### Step 1: Define Variables
Let:
- \( S_1 \) = speed of car A (in km/h)
- \( S_2 \) = speed of car B (in km/h)

### Step 2: Set Up Equations Based on the Given Information
1. **When the cars are moving in the same direction**:
- They meet after 15 hours.
- The distance covered by car A in 15 hours = \( 15S_1 \)
- The distance covered by car B in 15 hours = \( 15S_2 \)
- Since they are moving in the same direction, the distance covered by A minus the distance covered by B equals the distance between them (150 km):
\[ 15S_1 - 15S_2 = 150 \]
Dividing the entire equation by 15 gives:
\[ S_1 - S_2 = 10 \quad \text{(Equation 1)} \]

2. **When the cars are moving in opposite directions**:
- They meet after 1 hour.
- The distance covered by car A in 1 hour = \( S_1 \)
- The distance covered by car B in 1 hour = \( S_2 \)
- The total distance covered by both cars equals the distance between them (150 km):
\[ S_1 + S_2 = 150 \quad \text{(Equation 2)} \]

### Step 3: Solve the Equations
Now we have two equations:
1. \( S_1 - S_2 = 10 \)
2. \( S_1 + S_2 = 150 \)

We can solve these equations simultaneously.

#### Adding the two equations:
\[ (S_1 - S_2) + (S_1 + S_2) = 10 + 150 \]
\[ 2S_1 = 160 \]
\[ S_1 = 80 \quad \text{(speed of car A)} \]

#### Substituting \( S_1 \) back into Equation 1:
\[ 80 - S_2 = 10 \]
\[ S_2 = 80 - 10 = 70 \quad \text{(speed of car B)} \]

### Final Answer
The speeds of the cars are:
- Speed of car A (\( S_1 \)) = 80 km/h
- Speed of car B (\( S_2 \)) = 70 km/h

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