Question 122110
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Start with distance = rate times time, or {{{d=rt}}}.  We can safely presume
that the distance up the hill is the same as the distance back down the hill.
Let's say that {{{d}}} is the distance up or down the hill, {{{r[u]}}} is the
rate up the hill (given as 3 km/h), {{{r[d]}}} is the rate down the hill (given
as 4 km/h), and {{{t[u]}}} and {{{t[d]}}} 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 {{{14/3}}} hours.
Since {{{t[u]+t[d]=14/3}}}, we can say that {{{t[d]=14/3-t[u]}}}


Using {{{d=rt}}}, we can say that the distance up the hill is {{{d=3t[u]}}},
and the distance downhill is {{{d=4t[d]}}}, but since we have established that
{{{t[d]=14/3-t[u]}}}, we can substitute and write {{{d=4(14/3-t[u])}}}.


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


{{{3t[u]=4(14/3-t[u])}}}


Next, solve for {{{t[u]}}}


{{{3t[u]=56/3-4t[u]}}}
{{{7t[u]=56/3}}}
{{{t[u]=8/3}}}


So the time to go uphill is {{{8/3}}} hour or 2 and 2/3 hour.


Since she travelled at 3 km/hr uphill, {{{d=3(8/3)=8}}} km.


<b>Check the answer</b>


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.


<b>Answer checks.</b>
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