A person walks from point P to Q at 5 km/hr and returns from Q to P along the same route jogging at 10 km/hr. If the total time t…

Question

A person walks from point P to Q at 5 km/hr and returns from Q to P along the same route jogging at 10 km/hr. If the total time taken is 3 hours, what is the distance between P and Q?

1. A. 10 km (Correct answer)
2. B. 5 km
3. C. 15 km
4. D. 20 km

Correct Answer

Option A — 10 km

Detailed Solution & Explanation

The correct answer is 10 km.

Shortcut Trick

• For a there-and-back journey at $u$ and $v$, the average speed is the harmonic mean $\dfrac{2uv}{u+v} = \dfrac{2(5)(10)}{15} = \dfrac{100}{15} = \dfrac{20}{3}$ km/h.
• Total distance $= \dfrac{20}{3} \times 3 = 20$ km, which is the round trip.
• One way $= \dfrac{20}{2} = \textbf{10}$ km.

Alternate Method

• Let the one-way distance be $d$.
  • $\dfrac{d}{5} + \dfrac{d}{10} = 3$
  • $\dfrac{2d + d}{10} = 3 \Rightarrow \dfrac{3d}{10} = 3$
  • $d = \textbf{10}$ km

Additional Information

• Average speed for equal distances is the harmonic mean, never the arithmetic mean. Here it is $6.67$ km/h, not $\dfrac{5+10}{2} = 7.5$ — because more time is spent at the slower speed.
• The arithmetic mean applies only when equal times are spent at each speed, which is a different question entirely.
• The final halving is the step most often missed: the 20 km covers both legs, and the question asks for the distance between P and Q.