A man can walk uphill at the rate of $2\frac{1}{2}$ km/hr and down hill at the rate of $3\frac{1}{4}$ km/hr. If the total time required to 22 walk a certain distance up the hill and return to the starting point is 4 hr. 36 min, then what is the distance walked up the hill by the man?

$6\frac{1}{2}$ km

The correct answer is Option (1) → $6\frac{1}{2}$ km

Let the uphill distance be $x$ km.

Uphill speed = $\frac{5}{2}$ km/h, Downhill speed = $\frac{13}{4}$ km/h

Total time = 4 hr 36 min = $\frac{23}{5}$ hr

Time taken uphill = $\frac{x}{5/2} = \frac{2x}{5}$

Time taken downhill = $\frac{x}{13/4} = \frac{4x}{13}$

Total time: $\frac{2x}{5} + \frac{4x}{13} = \frac{23}{5}$

LCM of 5 and 13 = 65

$\Rightarrow \frac{26x + 20x}{65} = \frac{23}{5}$

$\Rightarrow \frac{46x}{65} = \frac{23}{5}$

$\Rightarrow x = \frac{23}{5} \cdot \frac{65}{46} = \frac{13}{2} = 6.5$