Finding Speed of Train and Car

Video Explanation

Question

A man travels 600 km partly by train and partly by car. If he covers 400 km by train and the rest by car, it takes 6 hours 30 minutes. If he travels 200 km by train and the rest by car, he takes half an hour longer. Find the speed of the train and the car.

Solution

Step 1: Concept

Time = Distance / Speed

Step 2: Let Variables

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

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

Step 3: Convert Time

6 hours 30 minutes = \(6.5\) hours

Second case = \(6.5 + 0.5 = 7\) hours

Step 4: Form Equations

First case:

\[ \frac{400}{x} + \frac{200}{y} = 6.5 \quad (1) \]

Second case:

\[ \frac{200}{x} + \frac{400}{y} = 7 \quad (2) \]

Step 5: Convert into Linear Form

Let:

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

Then:

\[ 400a + 200b = 6.5 \quad (3) \]

\[ 200a + 400b = 7 \quad (4) \]

Step 6: Solve Linear Equations

Multiply (3) by 2:

\[ 800a + 400b = 13 \quad (5) \]

Subtract (4) from (5):

\[ 600a = 6 \]

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

Substitute into (3):

\[ 400\left(\frac{1}{100}\right) + 200b = 6.5 \]

\[ 4 + 200b = 6.5 \]

\[ 200b = 2.5 \]

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

Step 7: Back Substitute

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

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

Conclusion

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

Verification

Case 1: \(400/100 + 200/80 = 4 + 2.5 = 6.5\) ✔

Case 2: \(200/100 + 400/80 = 2 + 5 = 7\) ✔