Finding Speeds of A and B

Video Explanation

Question

A takes 3 hours more than B to walk 30 km. If A doubles his speed, he reaches 1.5 hours earlier than B. Find their speeds.

Solution

Step 1: Concept

Time = Distance / Speed

Step 2: Let Variables

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

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

Step 3: Form Equations

\[ \frac{30}{x} = \frac{30}{y} + 3 \quad (1) \]

\[ \frac{30}{2x} = \frac{30}{y} – \frac{3}{2} \quad (2) \]

Step 4: Convert into Linear Form

Let:

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

Then:

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

\[ \frac{a}{2} = b – \frac{3}{2} \quad (4) \]

Step 5: Solve Linear Equations

Multiply (4) by 2:

\[ a = 2b – 3 \quad (5) \]

Now from (3):

\[ b + 3 = 2b – 3 \]

\[ b = 6 \]

Substitute into (3):

\[ a = 9 \]

Step 6: Back Substitute

\[ \frac{30}{x} = 9 \Rightarrow x = \frac{30}{9} = \frac{10}{3} \]

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

Conclusion

\[ \text{Speed of A} = \frac{10}{3} \text{ km/h}, \quad \text{Speed of B} = 5 \text{ km/h} \]

Verification

A time = \(30 / (10/3) = 9\) hrs

B time = \(30 / 5 = 6\) hrs → difference = 3 ✔

Double speed of A = \(20/3\) → time = 4.5 hrs

B = 6 hrs → 1.5 hrs earlier ✔