SOLUTION: A paddle boat can move at a speed of 8 km/h in still water. The boat is paddled 4 km downstream in a river in the same time it takes to go 2 km upstream. What is the speed of the r

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Question 1068609: A paddle boat can move at a speed of 8 km/h in still water. The boat is paddled 4 km downstream in a river in the same time it takes to go 2 km upstream. What is the speed of the river?
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39616) About Me  (Show Source):
You can put this solution on YOUR website!
c   speed of stream current --- UNKNOWN in this example
r   speed of paddle boat if no current--- 8
t   time either direction -- UNKNOWN in this example
d   downstream distance ---  4
u   upstream distance  --- 2


                  SPEED        TIME       DISTANCE
DOWNSTREAM        r+c           t           d
UPSTREAM          r-c           t           u

Question asks for c, the river current speed.

Basic constant travel rates rule, RT=D to relate rate, time, distance.

d%2F%28r%2Bc%29=u%2F%28r-c%29---solve this for c in terms of the other variables.

-
d%28r-c%29=u%28r%2Bc%29

dr-dc=ur%2Buc
dr-ur=uc%2Bdc
dr-ur=c%28u%2Bd%29

highlight%28c=%28dr-ur%29%2F%28u%2Bd%29%29----substitute the given values into this formula.

Answer by ikleyn(52776) About Me  (Show Source):
You can put this solution on YOUR website!
.
A paddle boat can move at a speed of 8 km/h in still water. The boat is paddled 4 km downstream in a river in the same time
it takes to go 2 km upstream. What is the speed of the river?
~~~~~~~~~~~~~~~~~~~~~~ 

Let "v" be the speed of the river (or, better to say, current rate), in km/h.

Then the speed of the boat downstream is (8+v) km/h (relative to the river banks),

and  the speed of the boat downstream is (8-v) km/h.


Therefore, the time for traveling 4 km downstream is 4%2F%288%2Bv%29 hours,

and        the time for traveling 2 km   upstream is 2%2F%288-v%29 hours.


According to the condition, time 4%2F%288%2Bv%29 is the same as time 2%2F%288-v%29, which gives you an equation

4%2F%288%2Bv%29 = 2%2F%288-v%29.    (1)


It is your equation to find the unknown "v". To solve it, multiply both sides of (1) by (8+v)*(8-v). You will get

4*(8-v) = 2*(8+v).


Simplify and solve for v:

32 - 4v = 16 + 2v  --->  32 - 16 = 2v + 4v  --->  6v = 16  --->  v = 16%2F6 = 8%2F3 km/h.

Answer.  The current rate is 8%2F3 km/h = 22%2F3 km/h.

Check.  Speed downstream is 8%2B+8%2F3 = %2824%2B8%29%2F3 = 32%2F3 km/h;  Time downstream is 4%2F%28%2832%2F3%29%29 = %284%2A3%29%2F32 = 12%2F32 = 3%2F8 of an hour.

        Speed   upstream is 8-+8%2F3 = %2824-8%29%2F3 = 16%2F3 km/h;  Time   upstream is 2%2F%28%2816%2F3%29%29 = %282%2A3%29%2F16 = 6%2F16 = 3%2F8 of an hour.

Solved and checked.

It is a typical and standard Upstream and Downstream trip word problem.
You can find many similar fully solved problems on upstream and downstream trips with detailed solutions in lessons
    - Wind and Current problems
    - More problems on upstream and downstream round trips
    - Selected problems from the archive on the boat floating Upstream and Downstream
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".