Finding Speed of Train and Bus

Video Explanation

Question

Roohi travels 300 km partly by train and partly by bus. If she travels 60 km by train and the remaining by bus, she takes 4 hours. If she travels 100 km by train and the remaining by bus, she takes 10 minutes longer. Find the speed of the train and the bus.

Solution

Step 1: Concept

Time = Distance / Speed

Step 2: Let Variables

Let speed of train = \(x\) km/h

Let speed of bus = \(y\) km/h

Step 3: Convert Time

10 minutes = \( \frac{10}{60} = \frac{1}{6} \) hour

Second case time = \(4 + \frac{1}{6} = \frac{25}{6}\) hours

Step 4: Form Equations

First case:

\[ \frac{60}{x} + \frac{240}{y} = 4 \quad (1) \]

Second case:

\[ \frac{100}{x} + \frac{200}{y} = \frac{25}{6} \quad (2) \]

Step 5: Convert into Linear Form

Let:

\[ a = \frac{1}{x}, \quad b = \frac{1}{y} \]

Then:

\[ 60a + 240b = 4 \quad (3) \]

\[ 100a + 200b = \frac{25}{6} \quad (4) \]

Step 6: Solve Linear Equations

Multiply (3) by 5:

\[ 300a + 1200b = 20 \quad (5) \]

Multiply (4) by 6:

\[ 600a + 1200b = 25 \quad (6) \]

Subtract (5) from (6):

\[ 300a = 5 \]

\[ a = \frac{1}{60} \]

Substitute into (3):

\[ 60\left(\frac{1}{60}\right) + 240b = 4 \]

\[ 1 + 240b = 4 \]

\[ 240b = 3 \]

\[ b = \frac{1}{80} \]

Step 7: Back Substitute

\[ x = \frac{1}{a} = 60 \]

\[ y = \frac{1}{b} = 80 \]

Conclusion

\[ \text{Train speed} = 60 \text{ km/h}, \quad \text{Bus speed} = 80 \text{ km/h} \]

Verification

Case 1: \(60/60 + 240/80 = 1 + 3 = 4\) ✔

Case 2: \(100/60 + 200/80 = 1.67 + 2.5 = 4.17 = \frac{25}{6}\) ✔