A can complete a journey in 10 hours. He travels first half of the journey at the rate of 21 km/hr and second half at the rate of 24 km/hr. Find the total journey in km.

### Concept & Formula When an object travels two equal distances at speeds $v_1$ and $v_2$, its average speed for the entire journey is the harmonic mean of the two speeds. $$ \text{Average Speed} = \frac{2 \cdot v_1 \cdot v_2}{v_1 + v_2} $$ Total Distance = Average Speed $\times$ Total Time. ### Step-by-Step Solution * **Given:** * Total time = $10$ hours * Speed for first half ($v_1$) = $21$ km/hr * Speed for second half ($v_2$) = $24$ km/hr * **Average Speed Calculation:** * Average Speed = $\frac{2 \times 21 \times 24}{21 + 24}$ * $= \frac{1008}{45}$ * Simplify by dividing numerator and denominator by 9: $\frac{112}{5}$ = $22.4$ km/hr. * **Total Distance Calculation:** * Total Distance = Average Speed $\times$ Total Time * Total Distance = $22.4 \times 10 = 224$ km. ### Exam Strategy & Shortcut Alternatively, let total distance be $2x$. The time equation is $\frac{x}{21} + \frac{x}{24} = 10$. $x \cdot \left(\frac{8+7}{168}\right) = 10 \Rightarrow x \cdot \left(\frac{15}{168}\right) = 10 \Rightarrow x = \frac{1680}{15} = 112$. Total distance is $2x = 224$. This algebra approach often avoids large multiplications if simplified early. ### Common Pitfall Taking the simple arithmetic average of the speeds ($\frac{21 + 24}{2} = 22.5$) instead of the harmonic mean, which would give an incorrect total distance of $225$ km. ### Final Answer Therefore, the correct answer is **224 km**.