Question 1111666
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<pre>
Let u = the shell rate in still water (in miles per hour) and
let v = the rate of the current.


Then you have TWO "time" equations


{{{8/(u+v)}}} + {{{8/(u-v)}}} = {{{3/2}}}    (1)    ("8 miles downstream and back in 1 1/2 hours")

{{{12/(u+v)}}} + {{{6/(u-v)}}} = {{{3/2}}}   (2)    ("12 miles downstream and halfway back in 1 1/2 hours)


To solve the syste, introduce new variables  x = {{{1/(u+v)}}},  y = {{{1/(u-v)}}}.

Then the system takes the form

 8x + 8y = {{{3/2}}}

12x + 6y = {{{3/2}}}.


Simplify it to

 x + y = {{{3/16}}}    (3)

2x + y = {{{3/12}}}    (4)


Subtract  eq(3)  from eq(4), eliminating "y".  You will get

 x = {{{3/12}}} - {{{3/16}}} = {{{12/48}}} - {{{9/48}}} = {{{3/48}}} = {{{1/16}}}.


Then from (3)  y = {{{3/16}}} - {{{1/16}}} = {{{2/16}}} = {{{1/8}}}.


It implies

u + v = {{{1/x}}} = 16,   (5)

u - v = {{{1/y}}} =  8.   (6)


Add equations (5) and (6), eliminating "v". You will get

2u = 16 + 8 = 24  ====>  u = {{{24/2}}} = 12.


Then from eq(5)  v = 16 - 12 = 4.


<U>Answer</U>.  The speed of the shell in still water is  12 mph.  The current rate is 4 mph.
</pre>

Solved.


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It is a typical and standard Upstream and Downstream round trip word problem.


You can find many similar fully solved problems on upstream and downstream round trips with detailed solutions and explanations in lessons 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/travel/Wind-and-Current-problems.lesson>Wind and Current problems</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/travel/More-problems-on-upstream-and-downstream-round-trips.lesson>More problems on upstream and downstream round trips</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/travel/Wind-and-Current-problems-solvable-by-quadratic-equations.lesson>Wind and Current problems solvable by quadratic equations</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Unpowered-raft-moving-downstream-along-a-river.lesson>Unpowered raft floating downstream along a river</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Selected-problems-from-the-archive-on-a-boat-floating-Upstream-and-Downstream.lesson>Selected problems from the archive on the boat floating Upstream and Downstream</A> 

in this site.


Read them attentively and learn how to solve this type of problems once and for all.


Also, &nbsp;you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this textbook under the section "<U>Word problems</U>", &nbsp;the topic "<U>Travel and Distance problems</U>".



Save the link to this online textbook together with its description


Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson


to your archive and use it when it is needed.