Finding Speeds of Ajeet and Amit

Video Explanation

Question

While covering a distance of 30 km, Ajeet takes 2 hours more than Amit. If Ajeet doubles his speed, he would take 1 hour less than Amit. Find their speeds.

Solution

Step 1: Concept

Time = Distance / Speed

Step 2: Let Variables

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

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

Step 3: Form Equations

Time taken:

\[ \frac{30}{x} = \frac{30}{y} + 2 \quad (1) \]

Second condition:

\[ \frac{30}{2x} = \frac{30}{y} – 1 \quad (2) \]

Step 4: Convert into Linear Form

Let:

\[ a = \frac{30}{x}, \quad b = \frac{30}{y} \]

Then:

\[ a = b + 2 \quad (3) \]

\[ \frac{a}{2} = b – 1 \quad (4) \]

Step 5: Solve Linear Equations

Multiply (4) by 2:

\[ a = 2b – 2 \quad (5) \]

Now from (3):

\[ b + 2 = 2b – 2 \]

\[ b = 4 \]

Substitute into (3):

\[ a = 6 \]

Step 6: Back Substitute

\[ \frac{30}{x} = 6 \Rightarrow x = 5 \]

\[ \frac{30}{y} = 4 \Rightarrow y = 7.5 \]

Conclusion

\[ \text{Speed of Ajeet} = 5 \text{ km/h}, \quad \text{Speed of Amit} = 7.5 \text{ km/h} \]

Verification

Ajeet time = \(30/5 = 6\) hrs

Amit time = \(30/7.5 = 4\) hrs → difference = 2 ✔

Double speed of Ajeet = 10 km/h → time = 3 hrs

Amit = 4 hrs → 1 hour less ✔