A man can walk uphill at the rate of $2 \frac{1}{2}$ km/hr and downhill at the rate of $3 \frac{1}{4}$ km/hr. If the total time required to walk a certain distance up the hill and return to the starting point was 4 hr 36 min, then what was the distance walked up the hill by the man?

### Concept & Time and Distance The core relationship is $Distance = Speed \times Time$. Since the distance traveled uphill and downhill is the same, we can equate the total time taken to the sum of the times taken for the individual legs of the journey. $$Total\ Time = \frac{Distance}{Speed_{uphill}} + \frac{Distance}{Speed_{downhill}}$$ ### Step-by-Step Solution * Convert speeds to improper fractions: * Uphill speed = $2 \frac{1}{2} = \frac{5}{2}$ km/hr. * Downhill speed = $3 \frac{1}{4} = \frac{13}{4}$ km/hr. * Convert total time to hours: * Total time = $4$ hr $36$ min = $4 + \frac{36}{60}$ hours = $4 + \frac{3}{5} = \frac{23}{5}$ hours. * Let the distance be $d$. * The time equation is: $\frac{d}{5/2} + \frac{d}{13/4} = \frac{23}{5}$ * Simplify the fractions: $\frac{2d}{5} + \frac{4d}{13} = \frac{23}{5}$ * Find a common denominator for the left side (65): * $\frac{26d + 20d}{65} = \frac{23}{5}$ * $\frac{46d}{65} = \frac{23}{5}$ * Solve for $d$: * $d = \frac{23 \times 65}{46 \times 5}$ * $d = \frac{23}{46} \times \frac{65}{5}$ * $d = \frac{1}{2} \times 13 = \frac{13}{2}$ * $d = 6.5$ km, or $6 \frac{1}{2}$ km. ### Exam Strategy & Shortcut You can also find average speed first: $\frac{2 \times (5/2) \times (13/4)}{(5/2) + (13/4)} = \frac{65/4}{33/4} = \frac{65}{33}$ km/hr. Then total distance (both ways) = Average speed $\times$ total time = $\frac{65}{33} \times \frac{23}{5}$. But the standard equation method is often less error-prone with mixed fractions. ### Common Pitfall Forgetting to convert the $36$ minutes into a proper fraction of an hour, or multiplying instead of dividing the time components. ### Final Answer Therefore, the correct answer is **$6 \frac{1}{2}$ km**.