Aryan covers a certain distance in 1 hour 30 minutes. He covers two thirds of it at the rate of 4 kmph and remaining distance at the rate of 5 kmph. Find the total distance.

### Concept & Segmented Journey Time When a journey is split into segments, the total time is the sum of the time taken for each individual segment. We use the fundamental relationship: $$Total Time = \frac{Distance_1}{Speed_1} + \frac{Distance_2}{Speed_2}$$ ### Step-by-Step Solution 1. Let the total distance be $D$ km. 2. The journey is split into two parts: $\frac{2}{3}D$ and the remaining $\frac{1}{3}D$. 3. Time taken for the first part (at 4 kmph) = $\frac{\frac{2}{3}D}{4} = \frac{2D}{12} = \frac{D}{6}$ hours. 4. Time taken for the second part (at 5 kmph) = $\frac{\frac{1}{3}D}{5} = \frac{D}{15}$ hours. 5. Total time = 1 hour 30 minutes = 1.5 hours. 6. Set up the equation: $$\frac{D}{6} + \frac{D}{15} = 1.5$$ 7. Find a common denominator (30) to solve for $D$: $$\frac{5D + 2D}{30} = 1.5$$ $$\frac{7D}{30} = 1.5$$ $$7D = 45$$ $$D = \frac{45}{7} \approx 6.428 \text{ km}$$ 8. The closest approximate value from the options is 6.4 km. ### Exam Strategy & Shortcut Instead of writing out fractions with $D$, assume the total distance is a multiple of both the denominators (3, to easily take $\frac{1}{3}$ and $\frac{2}{3}$) and the speeds (4 and 5). Let distance be $3x$. Segments are $2x$ and $x$. Times are $\frac{2x}{4}$ and $\frac{x}{5}$. Total time = $\frac{x}{2} + \frac{x}{5} = \frac{7x}{10}$. Equate to 1.5 hours: $\frac{7x}{10} = 1.5 \Rightarrow x = \frac{15}{7}$. Total distance = $3x = \frac{45}{7} \approx 6.4$ km. ### Common Pitfall Misinterpreting "two thirds of it" as two-thirds of the total *time* rather than two-thirds of the total *distance*. ### Final Answer Therefore, the correct answer is **6.4 km**.