Start with distance = rate times time, or .  We can safely presume
that the distance up the hill is the same as the distance back down the hill.
Let's say that  is the distance up or down the hill,  is the
rate up the hill (given as 3 km/h),  is the rate down the hill (given
as 4 km/h), and  and  are the times she took going up and
coming down.


We know she rested for 30 minutes, so she spent 5 hrs 10 min minus 30 minutes
or 4 hrs 40 min actually hiking.  4 hr 40 min is 4 and 2/3 or  hours.
Since , we can say that 


Using , we can say that the distance up the hill is ,
and the distance downhill is , but since we have established that
, we can substitute and write .


Now we have two different expressions for the same distance, so we can set
them equal to each other:





Next, solve for 







So the time to go uphill is  hour or 2 and 2/3 hour.


Since she travelled at 3 km/hr uphill,  km.


Check the answer


If the time uphill was 8/3, and the total time was 14/3, the time downhill
must have been 6/3, or 2 hours.  2 hours times 4 km/hr equals 8 km.


Answer checks.