SOLUTION: Two kayakers paddle 18 km downstream with the current in the same time it takes them to go 8 km upstream against the current. The rate of the current is 3 km/hr. What is the rate o

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Question 1189181: Two kayakers paddle 18 km downstream with the current in the same time it takes them to go 8 km upstream against the current. The rate of the current is 3 km/hr. What is the rate of the kayakers in still water?
Fill in the details:
Downstream: Distance (km)? Rate (km/hr)? Time (hr)
Upstream: Distance (km)? Rate (km/hr)? Time (hr)
Found 4 solutions by josgarithmetic, ikleyn, Alan3354, greenestamps:
Answer by josgarithmetic(39618) About Me  (Show Source):
You can put this solution on YOUR website!
                   SPEED     TIME                DISTANCE

downstream         r+3       18/(r+3)             18

upstream           r-3        8/(r-3)              8

Those times are given as equal.

18%2F%28r%2B3%29=8%2F%28r-3%29
Solve.
----

9%2F%28r%2B3%29=4%2F%28r-3%29

4%28r%2B3%29=9%28r-3%29

4r%2B12=9r-27

12%2B27=5r

r=%2812%2B27%29%2F5

r=39%2F5

r=7%264%2F5

Answer by ikleyn(52794) About Me  (Show Source):
You can put this solution on YOUR website!
.
Two kayakers paddle 18 km downstream with the current in the same time it takes them to go 8 km upstream against the current.
The rate of the current is 3 km/hr. What is the rate of the kayakers in still water?
Fill in the details:
Downstream: Distance (km)? Rate (km/hr)? Time (hr)
Upstream: Distance (km)? Rate (km/hr)? Time (hr)
~~~~~~~~~~~~~~~~~ 

Let x be the rate of the kayakers in still water.


Then the rate paddling downstream is (x+3) km/h; time paddling downstream is  18%2F%28x%2B3%29 hours.

     the rate paddling   upstream is (x-3) km/h; time paddling   upstream is  18%2F%28x-3%29 hours.


The times are the same, giving this time equation


    18%2F%28x%2B3%29 = 8%2F%28x-3%29.


Solve by cross-multiplying


    18*(x-3) = 8*(x+3)

    18x - 54 = 8x + 24

    18x - 8x = 24 + 54

       10x   =   78

         x   =  78/10 = 7.8 km/h


ANSWER.  The rate of kayakers in still water is  7.8 km/h.


CHECK.  The time paddling downstream is  18%2F%287.8%2B3%29 = 18%2F10.8 hours = 180%2F108 = 10%2F6 hours = 100 minutes.

        The time paddling   upstream is  8%2F%287.8-3%29 = 8%2F4.8 = 80%2F48 hours = 10%2F6  hours = 100 minutes.   ! Correct !

Solved and explained.



Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Two kayakers paddle 18 km downstream with the current in the same time it takes them to go 8 km upstream against the current. The rate of the current is 3 km/hr. What is the rate of the kayakers in still water?
----------------
r = rate
---
t = 18/(r+3) = 8/(r-3)
18(r-3) = 8(r+3)
18r-54 = 8r+24
10r = 78
r = 7.8 km/hr

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


As suggested by the way the problem is posted, all three of the responses you have received to this point use some form of the standard equation

time = distance/rate

knowing that the times upstream and downstream are the same.

rate of kayaker: x
upstream rate: x-3
downstream rate: x+3

The time for 18km downstream is the same as the time for 8km upstream:

18/(x+3)=8(x-3)
etc...

Here is a different approach that I personally find easier for problems like this.

The times are the same, so the ratio of distances is the same as the ratio of rates.

The ratio of distances is 18:8, or 9:4, so let the two rates be 9x and 4x.

The difference between those two rates is 6km/h:

9x-4x=6
5x=6
x=1.2

The two rates are 9x=10.8km/h and 4x=4.8km/h; the rate of the kayaker is halfway between those two rates, 7.8km/h (i.e., 3km/h faster than 4.8km/h, and 3km/h slower than 10.8km/h).

ANSWER: 7.8km/h