Points A and B are 120 km apart on a highway. A car starts from A and another car starts from B at the same time. If they travel in the same direction they meet in 9 h but if they travel towards each other they meet in 2 h. What are the speeds of the cars?

Options

A. 40 km/h, 40 km/h
B. 30 km/h, 30 km/h
C. 40 km/h, 20 km/h
D. None of these

Answer

To solve the problem, we need to find the speeds of two cars starting from points A and B, which are 120 km apart. Let's denote the speed of the car starting from A as \( x \) km/h and the speed of the car starting from B as \( y \) km/h.

### Step 1: Set up the equations based on the given information

1. **When the cars travel in the same direction**:
- They meet in 9 hours.
- The distance covered by the faster car (car A) minus the distance covered by the slower car (car B) equals the distance between A and B.
- Therefore, we can write the equation:
\[ 120 = (x - y) \times 9 \]
Simplifying this gives:
\[ x - y = \frac{120}{9} = \frac{40}{3} \quad \text{(Equation 1)} \]

2. **When the cars travel towards each other**:
- They meet in 2 hours.
- The total distance covered by both cars equals the distance between A and B.
- Therefore, we can write the equation:
\[ 120 = (x + y) \times 2 \]
Simplifying this gives:
\[ x + y = \frac{120}{2} = 60 \quad \text{(Equation 2)} \]

### Step 2: Solve the system of equations

Now we have the following two equations:
1. \( x - y = \frac{40}{3} \)
2. \( x + y = 60 \)

We can solve these equations simultaneously.

**Add Equation 1 and Equation 2**:
\[ (x - y) + (x + y) = \frac{40}{3} + 60 \]
This simplifies to:
\[ 2x = \frac{40}{3} + \frac{180}{3} = \frac{220}{3} \]
Thus, we find:
\[ x = \frac{220}{6} = \frac{110}{3} \quad \text{(Speed of car A)} \]

**Now substitute \( x \) back into Equation 2 to find \( y \)**:
\[ \frac{110}{3} + y = 60 \]
Rearranging gives:
\[ y = 60 - \frac{110}{3} = \frac{180}{3} - \frac{110}{3} = \frac{70}{3} \quad \text{(Speed of car B)} \]

### Final Speeds:
- Speed of car A (from A to B): \( \frac{110}{3} \) km/h
- Speed of car B (from B to A): \( \frac{70}{3} \) km/h

### Summary of Results:
- Speed of car A: \( \frac{110}{3} \approx 36.67 \) km/h
- Speed of car B: \( \frac{70}{3} \approx 23.33 \) km/h