A man on tour travels 160 km by car at 64 km/hr and another 160 km by bus at 80 km/hr. The average speed for the whole journey is (L.I.C.A.D.O., 2008)

### Concept & Harmonic Mean for Equal Distances When a journey consists of two equal distances traveled at different speeds ($x$ and $y$), the average speed for the entire journey is given by the harmonic mean of those two speeds. The actual distance value ($160$ km) is irrelevant to this specific formula. $$Average\ Speed = \frac{2xy}{x + y}$$ ### Step-by-Step Solution * **Step 1: Identify the speeds and verify distances are equal.** * Both segments are $160$ km, so distances are equal. * Speed $x = 64$ km/hr. * Speed $y = 80$ km/hr. * **Step 2: Apply the harmonic mean formula.** * Average Speed $= \frac{2 \times 64 \times 80}{64 + 80}$ * Average Speed $= \frac{10240}{144}$ * **Step 3: Simplify the fraction.** * Divide by 16: $\frac{10240 \div 16}{144 \div 16} = \frac{640}{9}$ * Calculate decimal: $640 \div 9 = 71.111...$ km/hr. ### Exam Strategy & Shortcut You can completely ignore the $160$ km figure and plug $64$ and $80$ directly into the $\frac{2xy}{x+y}$ shortcut. This saves you from calculating the individual times ($\frac{160}{64}$ and $\frac{160}{80}$), adding them, and dividing the total $320$ km by that sum. ### Common Pitfall Taking the simple arithmetic average of the two speeds: $\frac{64 + 80}{2} = 72$ km/hr. Average speed must account for the fact that more time is spent traveling at the slower speed. ### Final Answer Therefore, the correct answer is **71.11 km/hr**.