a motor boat can go 24 miles downstream in the same amount of time that it takes to go 12 miles upstream. the boats speed in the still water is 12 mph.

## what is the speed of the river current

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# 2 Answers

Hi Daniel;

*a motor boat can go 24 miles downstream in the same amount of time that it takes to go 12 miles upstream. the boats speed in the still water is 12 mph.*

distance=(rate)(time)

downstream...distance=(speed+current)(time)

upstream...distance=(speed-current)(time)

x=current

24=(12+x)(time)

12=(12-x)(time)

Let's take the second equation and multiply both sides by 2..

2(12)=2(12-x)(time)

24=2(12-x)(time)

24=24

(12+x)(time)=2(12-x)(time)

(time) on both sides cancels...

(12+x)(time)=2(12-x)(time)

12+x=2(12-x)

12+x=24-2x

Let's subtract 12 from both sides...

-12+12+x=24-2x-12

x=12-2x

Let's add 2x to both sides...

2x+x=12-2x+2x

3x=12

x=4

current is 4 miles/hour.

To work this problem, set up two equations for the time, T, which is the same for both directions.

For the downstream trip, the appropriate speed is sboat + sstream, so we have the equation:

T = 24 /(sboat + sstream) { time = distance / rate }

For the up stream trip, the appropriate speed is sboat - sstrearm, so we have the equation

T = 12 /(sboat - sstream)

{ Here sboat is the boat's velocity in still water = 12 mph, and sstream is the stream speed. }

Since the left hand sides are both equal to T, we have

24/(sboat + sstream) = 12/(sboat - sstream). To solve, we can equate the multiplicative inverses of both sides to get

(sboat + sstream)/24 = (sboat - sstream)/12 which implies

sboat + sstream = 2 sboat - 2 sstream So sboat = 3 sstream. Since sboat = 12,

sstrearm = 4 mph.