SOLUTION: The speed of a stream is 5km/h. If a boat travels 76kms downstream in 2 hrs less time than it takes to travel 54km upstream, what is the speed of the boat in still water. I hav

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Question 139479: The speed of a stream is 5km/h. If a boat travels 76kms downstream in 2 hrs less time than it takes to travel 54km upstream, what is the speed of the boat in still water.
I have gotten this far but am not sure if this is correct.
Downstream Boat D= 76km Rate= r+5 Time - t-2. Equation: 76= (r+5) * (t-2)
Upstream Boat D = 54 kms. Rate = r-5 Time = t. Equation 54km= (r-5) *t
To solve Upstream Boat of 54=(r-5)*t = 54= tr-5. Then I divide 54 by 5 to get 10.8. But I am stuck from here and do not think I am on the right track to solving this answer on the speed of the boat. Help would be appreciated. Thanks
Found 2 solutions by frostusna, ankor@dixie-net.com:
Answer by frostusna(7) About Me  (Show Source):
You can put this solution on YOUR website!
You started correctly and the two equations you obtained are correct.
76=%28r%2B5%29%28t-2%29
54=%28r-5%29t
Now, to eliminate one of the unknowns, solve each equation for t.
76%2F%28r%2B5%29=t-2, and solving for t gives t=+%2876%2F%28r%2B5%29%29+%2B+2, and
54%2F%28r-5%29=t.
From this we can show that %2876%2F%28r%2B5%29%29+%2B+2+=+54%2F%28r-5%29
Rewrite this equation to 76%2F%28r%2B5%29+-+54%2F%28r-5%29+%2B+2+=+0
To subtract the terms they must have common denominators. To get this we multiply as follows:
%28%28r-5%29%2F%28r-5%29%29%2A%2876%2F%28r%2B5%29%29+-+%28%28r%2B5%29%2F%28r%2B5%29%29%2A%2854%2F%28r-5%29%29+%2B2+=+0
Now carry out the multiplication to get:
%2876%28r-5%29+-+54%28r%2B5%29%29%2F%28r%2B5%29%28r-5%29+%2B+2+=+0
Carry out the multiplication in the numerator and multiply each term by the denominator to get:
76r+-+380+-+54r+-+270+%2B+2%28r%2B5%29%28r-5%29+=+0
Continuing, you get: 22r+-+650+%2B+2r%5E2+-+50+=+0. Rewriting becomes:
2r%5E2+%2B+22r+-+700+=+0. Devide through by 2 to simplify. You get
r%5E2+%2B+11r+-+350+=+0. This is a basic quadratic equation, so using the quadratic equation solver we get:
r+=%28-11+%2B-+sqrt%2811%5E2+-+%284%29%281%29%28-350%29%29%29%2F%282%2A1%29
Solving for r you get two possible solutions:
r+=+%28-11+%2B-+39%29%2F2 or r=-25 or r=14. The correct answer being
+r=14km%2Fh+
I hope this helped.

Answer by ankor@dixie-net.com(12701) About Me  (Show Source):
You can put this solution on YOUR website!
The speed of a stream is 5km/h. If a boat travels 76kms downstream in 2 hrs less time than it takes to travel 54km upstream, what is the speed of the boat in still water.
:
Here is a better approach. If we write a time equation, we only have 1 unknown
Let r = speed of boat in still water
then
(r-5) = speed upstream
(r+5) = speed downstream
:
Time = dist/speed:
Time upstream - 2 hrs = time downstream
54%2F%28%28r-5%29%29 - 2 = 76%2F%28%28r%2B5%29%29
:
Get rid of the denominators, multiply equation by (r+5)(r-5):
(r+5)(r-5)*54%2F%28%28r-5%29%29 - 2(r+5)(r-5) = (r+5)(r-5)*76%2F%28%28r%2B5%29%29
:
Cancel out the denominators:
54(r+5) - 2(r^2-25) = 76(r-5)
:
54r + 270 - 2r^2 + 50 = 76r - 380
:
-2r^2 + 54r - 76r + 270 + 50 + 380 = 0
:
-2r^2 - 22r + 700 = 0
Simplify divide equation by -2 and you have:
r^2 + 11r - 350 = 0
Factor this to:
(r-14)(r+25) = 0
The positive solution:
r = +14 is the speed in still water
:
:
Check solution by finding the time for each:
speed upstream = 9 mph; speed downstream = 19 mph
54/9 = 6 hrs
76/19 = 4
----------
diff = 2 hrs