SOLUTION: Meg rowed her boat upstream a distance of 65 mi and then rowed back to the starting point. The total time of the trip was 18 hours. If the rate of the current was 4 ​mph, find

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Question 1174765: Meg rowed her boat upstream a distance of 65 mi and then rowed back to the starting point. The total time of the trip was 18 hours. If the rate of the current was 4 ​mph, find the average speed of the boat relative to the water
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.

The "time" equation is


     +  = 18  hours.


The solution can be guessed MENTALLY in 4 seconds:  u = 9  miles per hour.


CHECK.    +  =  +  = 13 + 5 = 18 hours.   ! Correct !


Alternatively, you may reduce this time equation to quadratic equation and the solve it.


To do it, your first step is to multiply both sides of the equation by (u+4)*(u-4) = u^2 - 16.

Solved.

-------------

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 in 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
in this site, where you will find other similar solved problems with detailed explanations.

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

The closest to your problem is the lesson (*) of the list.

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.


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


Using formal algebra....

Let x be her speed in still water.

Then her upstream speed is x-4; her upstream time (distance divided by speed) is 65/(x-4).
And her downstream speed is x+4; her downstream time is 65/(x+4).

The total time is 18 hours:



Multiply through by (x-4)(x+4) to get a quadratic equation that can be solved to find the solution.

Knowing how to solve the problem using formal algebra is good. But you can get to the answer much faster -- and get a lot more good mental exercise -- by solving it using logical reasoning and simple arithmetic.

Since the total time is (exactly) 18 hours, the times for the two legs of the trip are probably whole numbers of hours.

The difference between the upstream and downstream speeds is 8 mph. So we need to find two numbers whose difference is 8 that both divide evenly into 65.

But 65 is 5*13; and the difference between 5 and 13 is 8.

So suppose the upstream speed is 5 mph and the downstream speed is 13 mph; that makes her speed in still water 9 mph. And her time upstream is 65/5 = 13 hours, and her time downstream is 65/13 = 5 hours -- making the total time 18 hours, which is what we needed.

ANSWER: Her speed in still water is 9 mph.
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