Akshara and Siddharth travel from point P to Q, a distance of 72 km at rates of 8 km/h and 10 km/h, respectively. Siddharth reaches Q first and returns immediately and meets Akshara at point R. Find the distance from P to R.

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Accepted Answer

Step 1: Calculate the time taken by Siddharth to reach Q.
- Distance PQ = 72 km
- Siddharth's speed = 10 km/h
- Time taken by Siddharth = Distance / Speed = 72 / 10 = 7.2 hours

Step 2: Calculate the time taken by Akshara to reach Q.
- Akshara's speed = 8 km/h
- Time taken by Akshara = 72 / 8 = 9 hours

Step 3: Siddharth reaches Q first and immediately returns towards P.
- Siddharth starts returning after 7.2 hours.
- Akshara is still traveling towards Q and has traveled for 7.2 hours at 8 km/h.
- Distance Akshara has covered in 7.2 hours = 8 * 7.2 = 57.6 km

Step 4: Let the meeting point R be at a distance x km from P.
- Akshara's distance from P to R = x km
- Siddharth's distance from Q to R = 72 - x km

Step 5: Time taken by Akshara to reach R = x / 8
- Time taken by Siddharth to reach Q and then return to R = 7.2 + (72 - x) / 10

Step 6: Since they meet at R at the same time,
- x / 8 = 7.2 + (72 - x) / 10

Step 7: Solve for x:
- Multiply both sides by 40 (LCM of 8 and 10):
  5x = 288 + 4(72 - x)
  5x = 288 + 288 - 4x
  5x + 4x = 576
  9x = 576
  x = 576 / 9 = 64 km

Therefore, the distance from P to R is 64 km.

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