Two boys Rishi and Vamsi start at the same time to ride from Lucknow to Kanpur that is 95 km away. Rishi travels 5 km/h slower than Vamsi. Vamsi reaches Kanpur and immediately start to travel back. On his return journey he meets Rishi who is 25 km away from Kanpur. Find Rishi's speed.
Rishi Speed Calculation: Lucknow Kanpur Journey
This problem requires calculating Rishi's speed using the given distance, relative speed difference, and information about where the travelers meet.
Defining Journey Variables
- Total distance (Lucknow to Kanpur): $D = 95$ km.
- Let Rishi's speed be $r$ km/h.
- Vamsi's speed is 5 km/h faster than Rishi's. So, Vamsi's speed is $v = r + 5$ km/h.
Calculating Meeting Distances
- Rishi travels from Lucknow towards Kanpur.
- Vamsi travels from Lucknow to Kanpur and starts returning.
- They meet when Rishi is 25 km from Kanpur.
- Distance Rishi traveled = $95 \text{ km} - 25 \text{ km} = 70$ km.
- Distance Vamsi traveled = $95 \text{ km}$ (to Kanpur) + $25 \text{ km}$ (returning) = $120$ km.
Equating Traveller Times
Since both started simultaneously, the time ($t$) taken to reach the meeting point is the same for both.
- Time = Distance / Speed
- Time for Rishi: $t = \frac{70}{r}$
- Time for Vamsi: $t = \frac{120}{v} = \frac{120}{r+5}$
- Equating the times: $ \frac{70}{r} = \frac{120}{r+5} $
Solving for Rishi's Speed
Solve the equation for $r$:
- Cross-multiply: $70(r+5) = 120r$
- Expand: $70r + 350 = 120r$
- Isolate $r$ terms: $350 = 120r - 70r$
- Simplify: $350 = 50r$
- Calculate $r$: $r = \frac{350}{50}$
- Result: $r = 7$ km/h.
Final Result: Rishi's Speed
Rishi's speed is 7 km/h.
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