Question

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.

Answer 1

(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?

Answer 2

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

Answer 3

yes I have

Answer 4

For
traveled downstream

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

Answer 5

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

Answer 6

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 ?

Answer 7

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

Answer 8

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

Answer 9

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

Answer 10

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 ?

Answer 11

hmm y+ x+y?

Answer 12

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?

Answer 13

(x+y)*1=d

Answer 14

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

Answer 15

(x+y)*3.6h=d

Answer 16

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

Answer 17

(x+y)*3h=d

Answer 18

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.

Answer 19

now you need to the speed going upstream
any ideas ?

Answer 20

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

Answer 21

(x+y) is how fast you go downstream

when you go upstream, you go slower

Answer 22

would it be x-y

Answer 23

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

Answer 24

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

Answer 25

yes, exactly.

now you have part a)
3(x+y)= d
3.6(x-y) = d

Answer 26

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 ?

Answer 27

It does make since d is the outcome in both equations

Answer 28

expressions i mean

Answer 29

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

Answer 30

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

Answer 31

yes,

Answer 32

3x+3y=3.6x-3.6y

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

Answer 33

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

Answer 34

6.6y=-0.833

Answer 35

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

Answer 36

0.6x

Answer 37

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 ?

Answer 38

11

Answer 39

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

Answer 40

Ok I get it

Answer 41

**(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 ?

Answer 42

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

Answer 43

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

Answer 44

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
\[ \frac{1 \cdot x }{11 \cdot x} = \frac{1}{11} \cdot \frac{x}{x}\]

Answer 45

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
\[ \frac{1}{11} \cdot 1 = \frac{1}{11} \]

Answer 46

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

Answer 47

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

Answer 48

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.

Answer 49

ok, 0.11

Answer 50

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

Answer 51

9.09 after multiplying 100

Answer 52

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

Answer 53

9.09% got it

Answer 54

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