Finding Speed of Train and Taxi

Video Explanation

Question

Abdul travelled 300 km by train and 200 km by taxi in 5 hours 30 minutes. If he travels 260 km by train and 240 km by taxi, he takes 6 minutes longer. Find the speed of the train and the taxi.

Solution

Step 1: Concept

Time = Distance / Speed

Step 2: Let Variables

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

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

Step 3: Convert Time

5 hours 30 minutes = \(5.5 = \frac{11}{2}\) hours

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

Second case time:

\[ \frac{11}{2} + \frac{1}{10} = \frac{56}{10} = \frac{28}{5} \]

Step 4: Form Equations

\[ \frac{300}{x} + \frac{200}{y} = \frac{11}{2} \quad (1) \]

\[ \frac{260}{x} + \frac{240}{y} = \frac{28}{5} \quad (2) \]

Step 5: Convert into Linear Form

Let:

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

Then:

\[ 300a + 200b = \frac{11}{2} \quad (3) \]

\[ 260a + 240b = \frac{28}{5} \quad (4) \]

Step 6: Solve Linear Equations

Multiply (3) by 2:

\[ 600a + 400b = 11 \quad (5) \]

Multiply (4) by 5:

\[ 1300a + 1200b = 28 \quad (6) \]

Multiply (5) by 3:

\[ 1800a + 1200b = 33 \quad (7) \]

Subtract (6) from (7):

\[ 500a = 5 \]

\[ a = \frac{1}{100} \]

Substitute into (3):

\[ 300\left(\frac{1}{100}\right) + 200b = \frac{11}{2} \]

\[ 3 + 200b = \frac{11}{2} \]

\[ 200b = \frac{11}{2} – 3 = \frac{5}{2} \]

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

Step 7: Back Substitute

\[ x = \frac{1}{a} = 100 \]

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

Conclusion

\[ \text{Train speed} = 100 \text{ km/h}, \quad \text{Taxi speed} = 80 \text{ km/h} \]

Verification

Case 1: \(300/100 + 200/80 = 3 + 2.5 = 5.5\) ✔

Case 2: \(260/100 + 240/80 = 2.6 + 3 = 5.6 = \frac{28}{5}\) ✔