Question 132538
The key to these kind of problems is to spot
the thing that doesn't change in the different
instances given. In this case it's distance.
{{{d = r*t}}}
(rate going upstream) = (rate of boat in still water - rate of stream)
(rate going downstream) = (rate of boat in still water + rate of stream)
The combined time for both trips is {{{3}}} hours. This is
separated into {{{t}}} anf {{{3 - t}}}
{{{12 = (9 + s)*t}}} (downstream)
{{{12 = (9 - s)*(3 - t)}}} (upstream)
{{{t = 12 / (9 + s)}}}
{{{12 = 27 - 3s - 9t + st}}}
These are 2 equations and 2 unknowns, so should be solvable
{{{12 = 27 - 3s - 9*(12/(9 + s)) + s*(12 / (9 + s))}}}
multiply both sides by {{{9 + s}}}
{{{12*(9 + s) = 27*(9 + s) - 3s*(9 + s) - 108 + 12s}}}
{{{108 + 12s = 243 + 27s - 27s - 3s^2 - 108 + 12s}}}
{{{3s^2 + 216 - 243 = 0}}}
{{{3s^2 - 27 = 0}}}
{{{3s^2 = 27}}}
{{{s^2 = 9}}}
{{{s = 3}}}
The rate of the stream is 3 km/hr
check:
{{{12 = (9 + s)*t}}} (downstream)
{{{12 = (9 + 3)*t}}}
{{{12 = 12*t}}}
{{{t = 1}}}
{{{12 = (9 - s)*(3 - t)}}} (upstream)
{{{12 = (9 - 3)*(3 - 1)}}}
{{{12 = 6*2}}}
{{{12 = 12}}}
OK