A tourist covered a journey partly by foot and partly by bus. He walked for 90 km and rode the bus for 10 km. He spent 4 h less on the bus than on walking. If the tourist had reversed the time he travelled by foot and by bus the distances travelled on each part of the journey would be equal. How long did he ride the bus?

Answer

To solve the problem step by step, we need to set up the equations based on the information given in the question.

### Step 1: Define Variables
Let:
- \( T_w \) = time spent walking (in hours)
- \( T_b \) = time spent on the bus (in hours)
- \( x \) = speed of walking (in km/h)

### Step 2: Write Equations
From the problem, we know:
1. The tourist walked 90 km and rode the bus for 10 km.
2. The time spent on the bus is 4 hours less than the time spent walking:
\[ T_b = T_w - 4 \]

3. The time taken to walk 90 km at speed \( x \) is:
\[ T_w = \frac{90}{x} \]

4. The time taken to ride the bus for 10 km at speed \( y \) (where \( y \) is the speed of the bus) is:
\[ T_b = \frac{10}{y} \]

### Step 3: Substitute the Time Equation
Substituting \( T_b \) in terms of \( T_w \):
\[ \frac{10}{y} = \frac{90}{x} - 4 \]

### Step 4: Reverse the Time Condition
If the tourist had reversed the time spent on foot and by bus, the distances would be equal. This means:
\[ \text{Distance covered by foot} = \text{Distance covered by bus} \]
Using the times:
\[ y \cdot T_w = x \cdot T_b \]
Substituting \( T_b \):
\[ y \cdot \frac{90}{x} = x \cdot \left(\frac{90}{x} - 4\right) \]

### Step 5: Solve the Equations
Now we can solve the equations. From the first equation:
\[ \frac{10}{y} = \frac{90}{x} - 4 \]
Rearranging gives:
\[ \frac{10}{y} + 4 = \frac{90}{x} \]
Multiplying through by \( xy \):
\[ 10x + 4y = 90y \]
Thus:
\[ 10x = 86y \implies x = \frac{86y}{10} = 8.6y \]

### Step 6: Substitute Back
Now substitute \( x \) back into the equation for \( T_b \):
\[ T_b = \frac{10}{y} = \frac{90}{8.6y} - 4 \]
Cross-multiplying gives:
\[ 10 \cdot 8.6y = 90 - 4y \]
\[ 86y = 90 - 4y \]
\[ 90 = 90y \implies y = 1 \]

### Step 7: Calculate Time on Bus
Now substituting \( y \) back to find \( T_b \):
\[ T_b = \frac{10}{1} = 10 \text{ hours} \]

### Conclusion
The tourist rode the bus for **10 hours**.