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 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?
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(6\frac{1}{2}\) km
The man walks the same distance up and down, so if the distance is \(d\) km, the times for the two walks, each distance ÷ speed, add up to the total time.
The time for each walk (distance ÷ speed):
Uphill speed \(=2\frac{1}{2}=\frac{5}{2}\) km/hr
Time up \(\displaystyle =d\div\frac{5}{2}=\frac{2d}{5}\) hr
Downhill speed \(=3\frac{1}{4}=\frac{13}{4}\) km/hr
Time down \(\displaystyle =d\div\frac{13}{4}=\frac{4d}{13}\) hr
Finding \(d\) (the two times add up to the total time):
Total time \(=4\) hr 36 min \(\displaystyle =4\frac{36}{60}=\frac{23}{5}\) hr
\(\displaystyle \frac{2d}{5}+\frac{4d}{13}=\frac{26d+20d}{65}=\frac{46d}{65}\)
\(\displaystyle \frac{46d}{65}=\frac{23}{5}\)
\(\displaystyle d=\frac{23}{5}\times\frac{65}{46}=\frac{13}{2}=6\frac{1}{2}\) km