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.

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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.

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(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 ?
yes I have

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Other answers:

  • phi
For traveled downstream first, what is the speed of the boat going with the current?
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 ?
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 ?
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?
(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 ?
(x+y)*3.6h=d
  • phi
Write an expression for the distance traveled downstream using 3 h for the time.
(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 ?
(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
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
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 ?
It does make since d is the outcome in both equations
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
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
6.6y=-0.833
  • phi
you lost the x? what is 3.6x - 3x ?
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 ?
11
  • phi
you get \[ y= \frac{0.6}{6.6} x\] that simplifies to \[ y = \frac{1}{11} x \]
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 \[ \frac{y}{x}= \frac{x}{11} \cdot \frac{1}{x} \]
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 \[ \frac{y}{x}= \frac{\cancel{x}}{11} \cdot \frac{1}{\cancel{x}} = \frac{1}{11}\]
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.
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
9.09 after multiplying 100
  • phi
and add a % sign (which is how we show we multiplied by 100) 9.09%
9.09% got it
Thank you so very much. I really appreciate your help.

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