A man performs 2/15 of the total journey by rail, 9/20 by bus and the remaining 10 km, on the cycle. His total journey is

### Concept & Fractional Parts of a Whole Journey To find the total distance, we add the given fractions of the journey and subtract the sum from 1 (the whole). This remaining fraction corresponds exactly to the absolute remaining distance provided in the problem. ### Step-by-Step Solution 1. Let the total distance of the journey be $D$. 2. The fractions of the journey covered by rail and bus are $\frac{2}{15}$ and $\frac{9}{20}$, respectively. 3. Add these fractions together. The least common multiple (LCM) of 15 and 20 is 60. $$\frac{2}{15} + \frac{9}{20} = \frac{2 \times 4}{60} + \frac{9 \times 3}{60} = \frac{8}{60} + \frac{27}{60} = \frac{35}{60}$$ 4. Simplify the combined fraction: $$\frac{35}{60} = \frac{7}{12}$$ 5. Determine the fraction representing the remaining journey on the cycle: $$1 - \frac{7}{12} = \frac{5}{12}$$ 6. We are given that this remaining fraction equals 10 km. Therefore: $$\frac{5}{12} \text{ of } D = 10 \text{ km}$$ 7. Solve for $D$: $$D = 10 \times \frac{12}{5} = 2 \times 12 = 24 \text{ km}$$ ### Exam Strategy & Shortcut Use the LCM of the denominators (15 and 20) to assume a hypothetical total distance. Let the total distance be 60 units. Rail journey = $\frac{2}{15}$ of 60 = 8 units. Bus journey = $\frac{9}{20}$ of 60 = 27 units. Total covered = $8 + 27 = 35$ units. Remaining distance = $60 - 35 = 25$ units. The problem states the remaining distance is 10 km. So, 25 units = 10 km. We need the value of 60 units. Total Distance = $\frac{10}{25} \times 60 = \frac{2}{5} \times 60 = 24$ km. ### Common Pitfall Incorrectly calculating the least common multiple of the fractions or making a basic arithmetic error during addition, leading to a wrong remaining fraction. ### Final Answer Therefore, the correct answer is **24 km**.