3. When a boat traveled downstream from Town A to Town B, the trip took 3 h. When the same boat traveled upstream from Town B to Town A, the trip took 3.6 h. For each trip, the speed of the boat and the water current were unchanged. Let x represent the speed of the boat and let y represent the speed of the water.

- freemap

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- freemap

(a) Write an expression for the distance traveled downstream using 3 h for the time. Then write an expression for the distance traveled upstream using 3.6 h for the time.
(b) Set the expressions in Part (a) equal to each other. Then solve the equation for y. Show your work.
(c) What percent of the boat’s speed is the water current?

- phi

have you learned
rate * time = distance
or
speed * time = distance
or
velocity * time = distance
?

- freemap

yes I have

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## More answers

- phi

For
traveled downstream
first, what is the speed of the boat going with the current?

- freemap

I don't remember how to do this. I'm I suppose to multiply?

- phi

The only reason people study math is to give them an excuse to think (or puzzle out a problem)
Say you were in a canoe but did not paddle, and drifted downstream. According to the info in this problem, how fast will you be moving ?

- freemap

Umm I really don't mean to sound stupid, but 1 maybe. not very fast

- phi

you will be going at a speed of "y"
(they tell us let y represent the speed of the water.)

- phi

though y is a letter, it means "speed of the current" (it's short-hand)

- phi

If you start paddling you will go faster. You would add on the speed you can paddle
they say
Let x represent the speed of the boat (when paddling in still water)
do you know how to show x added to y ?

- freemap

hmm y+ x+y?

- phi

just x+y
that is how fast you go if you go downstream
we might want to put parens around it , so we remember it as one thing: speed of the boat: (x+y)
now we use
speed * time = distance
(x+y)* time = distance
they don't tell us the distance, we let's call the distance "d"
but we know the time.
can you fill in the time with a number and write the full equation?

- freemap

(x+y)*1=d

- phi

looks good, except why are you using 1 for the time ? what does the question say the time is for going downstream ?

- freemap

(x+y)*3.6h=d

- phi

Write an expression for the distance traveled downstream using 3 h for the time.

- freemap

(x+y)*3h=d

- phi

ok. I think the h mean hours. Probably we should leave it off.
so
(x+y)*3 = d
or
3(x+y)= d
is the answer to the first part.

- phi

now you need to the speed going upstream
any ideas ?

- freemap

(x+y)*3.6=d
or
3.6(x+y)=d

- phi

(x+y) is how fast you go downstream
when you go upstream, you go slower

- freemap

would it be x-y

- phi

the river is taking you backwards at a speed of y , as you paddle upriver at a speed of x
yes
(x-y) is the speed going upstream

- freemap

ok would we still multiply the time like (x-y)*3.6=d

- phi

yes, exactly.
now you have part a)
3(x+y)= d
3.6(x-y) = d

- phi

now
***Set the expressions in Part (a) equal to each other. Then solve the equation for y. Show your work. ***
we see that 3(x+y) = d
and 3.6(x-y) also equals d. if both are equal to d, we can say
3(x+y)= 3.6(x-y)
Does that make sense ?

- freemap

It does make since d is the outcome in both equations

- freemap

expressions i mean

- phi

to solve, distribute the 3 on the left side. that means multiply 3 times x and times y
ditto for the 3.6 on the other side

- freemap

ok 3x+3y=3.6x-3.6y do we then add like terms?

- phi

yes,

- phi

3x+3y=3.6x-3.6y
I would add 3.6y to both sides
3x + 3y + 3.6y = 3.6x -3.6y +3.6y

- phi

you get
3x+6.6y = 3.6x
now add -3x to both sides

- freemap

6.6y=-0.833

- phi

you lost the x?
what is 3.6x - 3x ?

- freemap

0.6x

- phi

yes, so you get
3x+6.6y = 3.6x
6.6y = 0.6 x
now divide both sides by 6.6
what do we get ?

- freemap

11

- phi

you get
\[ y= \frac{0.6}{6.6} x\]
that simplifies to
\[ y = \frac{1}{11} x \]

- freemap

Ok I get it

- phi

**(c) What percent of the boat’s speed is the water current? ***
I would find the ratio of y/x and change it to a percent.
of course, we can't use just "y", but we know y is the same as x/11
so use x/11 instead.
can you do that ?

- phi

you do
\[ \frac{y}{x}= y \cdot \frac{1}{x} \]
but y is x/11
so
\[ \frac{y}{x}= \frac{x}{11} \cdot \frac{1}{x} \]

- freemap

ok, so if i multiply it turns back into 1x and 11x so I'm really sure how to solve this

- phi

you should learn that if you have the same thing "up top" and "below" , they cancel
when you multiply fractions, you multiply top times top and bottom times bottom
\[ \frac{x \cdot 1 }{11 \cdot x}\]
as you know
we can change the order of the multiply (right ?)
so it's the same as
\[ \frac{1 \cdot x }{11 \cdot x}\]
but that is the same as multiplying the two fractions
\[ \frac{1 \cdot x }{11 \cdot x} = \frac{1}{11} \cdot \frac{x}{x}\]

- phi

the last step, we "undid" the multiply
the point is, we have x/x and
anything divided by itself is 1
in other words we get
\[ \frac{1}{11} \cdot 1 = \frac{1}{11} \]

- phi

that is the long way. the short way is to say
"x up top, x down below" cross off both
\[ \frac{y}{x}= \frac{\cancel{x}}{11} \cdot \frac{1}{\cancel{x}} = \frac{1}{11}\]

- freemap

Thats what I meant to say 1/11 because I did cross multiply

- phi

no, you should not cross multiply. Try to follow what I posted up above.
meanwhile, to answer the question, change 1/11 to a decimal, and then multiply by 100 to make it a percent.

- freemap

ok, 0.11

- phi

0.11 means 11/100
that is different from
1/11
to change it to a decimal, divide 11 into 1 (a calculator will do that)
or type
1/11=
into google

- freemap

9.09 after multiplying 100

- phi

and add a % sign (which is how we show we multiplied by 100)
9.09%

- freemap

9.09% got it

- freemap

Thank you so very much. I really appreciate your help.

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