Question 1204016
.
A motorboat takes 4 hours to travel 128 kilometers going upstream. 
The return trip takes 2 hours going downstream. What is the rate of the boat in still water 
and what is the rate of the current?
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<pre>
Let  u  be the motorboat speed in still water (in kilometers per hour, km/h).

Let  v  be the rate of the current, in km/h.


Then  the boat's  effective speed downstream is  u+v  km/h,
while the boat's  effective speed upstream   is  u-v  km/h.


    // It is first major point you need understand and use in this sort of problems.


Now, "speed" equation for boat floating upstream is 

{{{128/4}}} = u - v    (1)    (speed upstream = the distance divided by time upstream)


Next, "speed" equation for boat floating downstream is 

{{{128/2}}} = u + v    (2)    (speed downstream = the distance divided by time  downstream)


    // It is the second major point in solving such problems: you must understand and write these equation automatically !


Simplify equations (1) and (2)


u - v = 32     (3)
u + v = 64     (4)


Now add equations (3) and (4) to eliminate "v". You will get


2u = 32 + 64 = 96  ====>  u = {{{96/2}}} = 48.


Thus you just found the boat' speed in still water. It is 48 kilometers per hour.


Now find the current rate from equation (4)  v = 64 - u = 64 - 48 = 16 km/h.


<U>Answer</U>.  The boat' speed in still water is 48 kilometers per hour.

         The current speed is 16 km/h.
</pre>

Solved.


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<H3>Post-solution note</H3>

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;I solved the problem formally, and this formal answer and the solution both are correct.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;But you should keep in your mind the following.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;In this problem, the speed of the motorboat in still water is very high: 48 km/h.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Even if imagine so high speed, it will mean that the river is lowland.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;But for lowland river, the current rate of 16 km/h is too high.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;So, the input data in this problem are not realistic.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;I wrote this post-solution note to explain this fact to you.



&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;When a professional Math composer creates his (or her) problems, usually he &nbsp;(or she) 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;knows a subject good enough to avoid &nbsp;(= to exclude) &nbsp;such inconsistencies.


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Because he &nbsp;(or she) &nbsp;does think on what he &nbsp;(or she) &nbsp;is doing.