Question 120264
James paddled his boat upstream for one mile. He continued for
another 15 minutes, then turned around and paddled downstream,
arriving at his starting point in exactly 1 hour.How fast is the
current of the stream?
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I am not real happy with what I came up with, but here it is:
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Going over what you had:
Solution:This is what i did,I defined 2 variables
B = speed of boat in still water, in miles per hour
C = speed of current, in miles per hour

Distance Rate Time
Upstream 1 mile + 1/4hr(B-C) B-C 1/4hr<<<<
The problem here is we don't know total time of the upstream trip at this point
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The distance both ways can be written as (1+.25(B-C))
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On the return trip, this distance was covered in 1 hr so we can also say:
 (1+.25(B-C)) mph,  the speed can also be written (B+C)mph
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A Speed equation: 
(1+.25(B-C)) = (B+C)
1 + .25B - .25C = B + C
1 +.25B - B - .25C - C = 0
-.75B - 1.25C = -1
or mult by -1
.75B + 1.25C = +1
Get rid of the decimals, multiply by 4
3B + 5C = 4
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But we have only 1 equation and two unknowns. I have been unable to generate
another equation from the information given.
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Taking the above equation and graphing it, where y is the speed in still water,
 and x is the speed of the current:
3y + 5x = 4
3y = -5x + 4
y = {{{-5/3}}}x + {{{4/3}}}; plotting this we have:
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{{{ graph( 300, 200, -1, 2, -2, 3, -(5/3)x + (4/3)) }}}
You can see the positive values are very limited,
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If you had a still water speed of 1 mph, the current would be .2 mph. 
this would work in our problem, B=1; C=.2
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Distance 1 + .25(1-.2) = 1 + .25(.8) = 1.2 mi is the distance
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The downstream speed 1 + .2 = 1.2 mph and of course to cover 1.2 mi would be 1 hr
This would be the only still water speed that is an integer
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As I said I am not real happy with it.