Question

Two friends can paddle a canoe at 4 miles per hour in still water. One day, they went 3 miles upstream against the current. They then traveled
downstream with the help of the current for the same distance. If the round trip took 2 hours, how fast was the current?

Will give a medal and become a fan! Please help & please show all steps!

Answer 1

@RadEn @phi

Answer 2

what formula do you know about speed , time and distance ?

Answer 3

Time= Distance/Rate ?

Answer 4

yes,
another way to write it is rate * time is distance

Not sure which version we will need yet.

Answer 5

they tell you
rate is 4 miles per hour in still water

Answer 6

I think you need common sense now... if there is a current, it will make you go faster if you go downstream and slower if you go upstream, so the rate will not be just 4 mph

Answer 7

The question asks

how fast was the current?

so the speed of the current is the unknown. call the speed of the current "x".

Answer 8

Now let's try to make an equation with this info

One day, they went 3 miles upstream against the current.

you know rate of the stream is x mph
rate of the boat in still water is 4 mph

any idea what the rate will be going upstream ?

Answer 9

4-x?

Answer 10

Or would it be 3/4-x instead..

Answer 11

yes. It should make sense. if the current was 4 mph, and they went up at 4 mph, they would be standing still.

or, going down stream, if you just floated (not paddling) you would still go down stream at x mph and if you go 4 mph in still water, going downstream you would go 4+x

** Or would it be 3/4-x instead.. ***
I am not sure what reasoning got you that.

Answer 12

4-x is the speed going upstream against the current

Answer 13

I got the 3 because it says that they travel 3 miles upstream, I thought that had to be added into the equation somehow

Answer 14

ok that makes sense. But I hope you agree the rate going up stream is 4-x
(where x is the rate of the stream)

now let's tackle

they went 3 miles upstream against the current.

list what you know: distance is 3, rate (upstream) is 4-x
time (no idea!)

and the formula
Time= Distance/Rate
T = D/R (easier to type)

replace the letters we know with the numbers:
T = 3/(4-x)

ok with that ?

Answer 15

can you do the same thing for this part

downstream with the help of the current for the same distance.

Answer 16

So for downstream it would just be 3/4+x then right?

Answer 17

you should write down the whole equation:
Time (downstream) = 3/(4+x) (and we should use parens otherwise it looks like ¾ +x which is different)

so far we have
Time (upstream)= 3/(4-x)
Time(downstream) = 3/(4+x)

The times will be different (I assume that is obvious... we are going slower upstream than downstream.

Answer 18

now focus on this info

the round trip took 2 hours

What can you say about the sum of the time upstream plus the time downstream?

Answer 19

The sum of both would be the total time it took, right? So 3/(4-x) + 3/(4+x) =4

Answer 20

pretty close. But what does
the round trip took 2 hours
mean?
in other words, how did you get a 4 ?

Answer 21

Oh yeah I guess it would be =2 instead of =4. I guess I thought that it would be 2 hours both ways, but it says 2 round trip. Oops..

Answer 22

2 hours for a "round trip" means 2 hours for there and back.

Answer 23

ok, now the interpreting part of the work is done. we now switch gears into
algebra mode, and solve for x in
3/(4-x) + 3/(4+x) =2

Answer 24

I distribute first right?

Answer 25

there is nothing to distribute. those parens are the denominators of fractions

if you had a*(x+4) you could distribute to ax+4a
but with a/(x+4) , you can't distribute.

one way to simplify is multiply the whole equation (both sides and *all* terms)
by (4-x)

(as a first step)

can you do that ?

Answer 26

to multiply each term by (4-x) you write (4-x) next to each term

Answer 27

On the top and bottom of both sides?

Answer 28

just the tops

Answer 29

3(4-x)/(4-x) + 3(4-x)/(4+x)

Answer 30

Then the first numerator would be 12-3x right?

Answer 31

also, both sides (we always do the same thing to both sides)

3(4-x)/(4-x) + 3(4-x)/(4+x) = 2(4-x)

now you could distribute the 3 in the first equation, but we will do something else.
there is a rule: a thing divided by itself is 1
we have (4-x) up top and a (4-x) in the bottom. that means (4-x)/(4-x) = 1

we now have
3 + 3(4-x)/(4+x) = 2(4-x)

(because the (4-x)/(4-x) is 1, and 3 times 1 is just 3)

next, multiply both sides (and *all terms) by (4+x)
that means write (4+x) next to each term

Answer 32

are you lost ?

you should have gotten
3(4+x) +3(4-x) = 2(4-x)(4+x)

Answer 33

now you can distribute (on the left side)
and multiply out (4-x)(4+x) on the right side.

Answer 34

Sorry @phi I had to do something. :/ I'm a little confused but I might be able to get it.

Answer 35

Ugh math stresses me out :/

Answer 36

So is it 12+3x+12+3x=2? then I combine like terms??

Answer 37

Do you know if thats right @whpalmer4?

Answer 38

Let me see:

3 + 3(4-x)/(4+x) = 2(4-x)

I would multiply both sides by (4+x), as suggested:

3*(4+x) + 3(4-x)/(4+x) * (4+x) = 2(4-x)*(4+x)
now we cancel out the (4+x)/(4+x) in the left side, because that equals 1:
3(4+x) + 3(4-x) = 2(4-x)(4+x)
Distribute the left side:
12 + 3x + 12 - 3x = 2(4-x)(4+x)
24 = 2(4-x)(4+x)
divide both sides by 2
12 = (4-x)(4+x)
expand the right side
12 = 16 + 4x - 4x - x^2
12 = 16 - x^2

can you finish solving that for me?

Answer 39

you'll get two answers, but only one of them makes sense in the context of this problem (no negative times need apply!)

Answer 40

we're within sight of the goal, don't up now! :-)

Answer 41

I got to -28=-x^2 but I'm not sure where to go from there

Answer 42

Is that right @whpalmer4 ?

Answer 43

12 = 16 - x^2
let's subtract 12 from both sides

Answer 44

14-x^2

Answer 45

no...

12 = 16 - x^2

subtract 12 from both sides

12 - 12 = 16 - x^2 - 12

can you simplify that for me?

Answer 46

12=16-x^2
12=-x^2+16
(2, -2)

Answer 47

right, x = 2 and x = -2 are the two solutions to that equation. which one makes sense for this problem?

Answer 48

Um x=2 I think

Answer 49

Well yeah it would have to be positive 2.

Answer 50

right. as a review of the early pages of this discussion shows, x was the speed of the stream in miles per hour. so, going with the current, we go 2+4=6mph, and against we go 4-2 = 2 mph.

now, does that answer satisfy the problem?

Answer 51

So this means that the current was going 2 mph?

Answer 52

let's check.

3 miles with the current, and 3 miles against the current, and the total time was 2 hours.

with the current, to cover 3 miles @ 6 mph = 3 miles / 6 (miles/hr) = 1/2 hour
against the current, to cover 3 miles @ 2 mph = 3 miles / 2 (miles/hr) = 1 1/2 hours

total time = 1/2 hour + 1 1/2 hour = 2 hours

our answer checks out!

Answer 53

yes, x was the speed of the current.

Answer 54

Thanks so much for your help :))

Answer 55

you're welcome! I like these problems :-)

@phi did a good job setting up the problem with you, too.

Answer 56

Yes, thank you @phi!! He's helped me with a few before :) You are both great!