Finding Speed of Train and Car

Video Explanation

Question

Ramesh travels 760 km partly by train and partly by car. He takes 8 hours if he travels 160 km by train and the rest by car. He takes 12 minutes more if he travels 240 km by train and the rest by car. 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: Form Equations

First case:

\[ \frac{160}{x} + \frac{600}{y} = 8 \quad (1) \]

Second case (12 minutes = \( \frac{12}{60} = \frac{1}{5} \) hour):

\[ \frac{240}{x} + \frac{520}{y} = 8 + \frac{1}{5} = \frac{41}{5} \quad (2) \]

Step 4: Convert to Linear Form

Let:

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

Then equations become:

\[ 160a + 600b = 8 \quad (3) \]

\[ 240a + 520b = \frac{41}{5} \quad (4) \]

Step 5: Solve Linear Equations

Multiply (3) by 5:

\[ 800a + 3000b = 40 \quad (5) \]

Multiply (4) by 5:

\[ 1200a + 2600b = 41 \quad (6) \]

Now eliminate: Multiply (5) by 3:

\[ 2400a + 9000b = 120 \quad (7) \]

Multiply (6) by 2:

\[ 2400a + 5200b = 82 \quad (8) \]

Subtract (8) from (7):

\[ 3800b = 38 \]

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

Substitute into (3):

\[ 160a + 600\left(\frac{1}{100}\right) = 8 \]

\[ 160a + 6 = 8 \]

\[ 160a = 2 \]

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

Step 6: Back Substitute

\[ x = \frac{1}{a} = 80 \]

\[ y = \frac{1}{b} = 100 \]

Conclusion

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

Verification

Case 1: \(160/80 + 600/100 = 2 + 6 = 8\) ✔

Case 2: \(240/80 + 520/100 = 3 + 5.2 = 8.2 = \frac{41}{5}\) ✔