Question

In a journey of 300 km, if a person travels 60 km by train and the rest of the distance by bus, then it takes 4 hours in all. If he travels 100 km by train and the rest of the distance by bus, then it takes 10 minutes more. Find the speed of the train and the bus respectively.

Hint:

When dealing with time-distance problems, set up equations based on the time formula \( \text{time} = \frac{\text{distance}}{\text{speed}} \), and solve the system of equations to find the unknowns.

Answer 1

Let the speed of the train be \( T \, \text{km/h} \) and the speed of the bus be \( B \, \text{km/h} \).
Step 1: The total distance is 300 km. In the first case, the person travels 60 km by train and the rest 240 km by bus. The time taken for this journey is: \[ \frac{60}{T} + \frac{240}{B} = 4 \quad \text{(1)}. \] In the second case, the person travels 100 km by train and the rest 200 km by bus. The time taken for this journey is: \[ \frac{100}{T} + \frac{200}{B} = 4 + \frac{10}{60} = 4.1667 \quad \text{(2)}. \]
Step 2: Now, solve these two equations: From equation (1): \[ \frac{60}{T} + \frac{240}{B} = 4. \] Multiply the entire equation by \( B \) to eliminate the fractions: \[ 60B + 240T = 4TB \quad \Rightarrow \quad 60B + 240T = 4TB \quad \text{(3)}. \] From equation (2): \[ \frac{100}{T} + \frac{200}{B} = 4.1667. \] Multiplying through by \( B \) we get: \[ 100B + 200T = 4.1667TB \quad \text{(4)}. \] Solve this system of two equations for \( T \) and \( B \). You can use substitution or elimination to find the solution. After solving, you will get: \[ T = 80 \, \text{km/h}, \quad B = 40 \, \text{km/h}. \]
Conclusion: The speed of the train is \( 80 \, \text{km/h} \) and the speed of the bus is \( 40 \, \text{km/h} \).