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 ?

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 ? Correct Answer $$6\frac{1}{2}$$ km

Average speed :
$$\eqalign{ & = \frac{{\left( {2 \times \frac{5}{2} \times \frac{{13}}{4}} \right)}}{{\left( {\frac{5}{2} + \frac{{13}}{4}} \right)}}{\text{ km/hr}} \cr & {\text{ = }}\left( {\frac{{65}}{4} \times \frac{4}{{23}}} \right){\text{ km/hr}} \cr & = \left( {\frac{{65}}{{23}}} \right){\text{ km/hr}} \cr} $$
Total time taken :
= 4 hr 36 min
$$= 4\frac{36}{60}\,\, hr$$
$$= 4\frac{3}{5}\,\, hr$$
$$= \frac{23}{5}\,\, hr$$
Total distance covered uphill and downhill :
$$\eqalign{ & = \left( {\frac{{65}}{{23}} \times \frac{{23}}{5}} \right){\text{ km}} \cr & = 13{\text{ km}} \cr} $$
∴ Distance walked uphill :
$$\eqalign{ & = \left( {\frac{{13}}{2}} \right){\text{ km}} \cr & = 6\frac{1}{2}{\text{ km}} \cr} $$