SOLUTION: A boat can travel 26 mph in still water. If the boat can travel 100 miles with the current in the same time it can travel 80 miles against the current, find the rate of the current

Question 1093565: A boat can travel 26 mph in still water. If the boat can travel 100 miles with the current in the same time it can travel 80 miles against the current, find the rate of the current r in miles per hour. (Enter an exact number as an integer, fraction, or decimal.)
r = _ mph
Found 4 solutions by richwmiller, josgarithmetic, greenestamps, ikleyn:
Answer by richwmiller(17219)   (Show Source): You can put this solution on YOUR website!
I used c for current.
r*t=d
(26-c)*t=80
26*t-c*t=80
(26+c)*t=100
26*t-c*t=80
26*t+c*t=100
add
52t=180
t=180/52
t=3.46153846 hr
26*3.46153846+c*3.46153846=100
90.0+c*3.46153846=100
c*3.46153846=10.0
c=10.0/3.46153846
c=2.88888889 mph current
check
(26+c)*t=100
(26+2.88888889)*3.46153846=100
(28.8888889)*3.46153846=100
100.0=100
(26-c)*t=80
(26-2.88888889)*3.46153846=80
(23.1111111)*3.46153846=80
80.0=80
ok

Answer by josgarithmetic(39617)   (Show Source): You can put this solution on YOUR website!
You might want to arrange the data from the description as a table.
c, speed of the current
t, the equal time both directions taken

SPEED TIME DISTANCE WITH CURRENT 26+c t 100 AGAINST CURRENT 26-c t 80

WITH CURRENT will give and AGAINST will give . Both expressions of t are equal:

-------solve this for c.

Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!

Both earlier responses show the same correct answer, but the formal algebra is a lot of work. I like methods of solving problems that get me to the answer with as little work as possible. Here is what I would do with this problem.

The distances covered in equal amounts of time with and against the current are in the ratio 100:80, or 5:4. That means that must be the ratio of the speeds with and against the current.

So if the speed of the current is c,

Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.
A boat can travel 26 mph in still water. If the boat can travel 100 miles with the current in the same time
it can travel 80 miles against the current, find the rate of the current r in miles per hour.
(Enter an exact number as an integer, fraction, or decimal.)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        It is classic Travel and Distance problem, at it deserves the full straightforward solution would be presented.

Let "r" be the current rate in miles per hour. Then the boat's effective rate traveling downstream is (26+r) mph (relative to the river's bank). The boat relative rate rate traveling upstream is (26-r) mph. The time to travel 100 miles downstream is hours. The time to travel 80 miles upstream is hours. These amounts of time are the same, which gives you an equation = . It is so called "time" equation. //Notice that the time equation was written incorrectly in the @josgarithmetic solution. Cancel the factor of 20 in both sides; then cross-multiply. You will get 5*(26-r) = 4*(26+r) ====> 130 - 5r = 104 + 4r ====> 130 - 104 = 5r + 4r ====> 9r = 26 ====> r = mph = mph. Answer. The current rate is mph = mph.

After reading this solution you will have a solid base and knowledge on how to solve and how to present it.

-------------
For many similar problems and their detailed solutions see the lessons
    - Wind and Current problems
    - More problems on upstream and downstream round trips
    - Wind and Current problems solvable by quadratic equations
    - Unpowered raft floating downstream along a river
    - Selected problems from the archive on the boat floating Upstream and Downstream
    - Selected problems from the archive on a plane flying with and against the wind
in this site.

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

Also, you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this textbook under the section "Word problems", the topic "Travel and Distance problems".

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.

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