Word problem just doesn't seem logical

[metoad2]

I have a word problem that does not seem to have enough information to make sense to me. Here it is: A boat traveled 210 miles down stream and back. The trip downstream took 10 hours, the trip back took 70 hours. What is the speed of the boat in still water, and what is the speed of the current?

 

[JeffM]

I have a word problem that does not seem to have enough information to make sense to me. Here it is: A boat traveled 210 miles down stream and back. The trip downstream took 10 hours, the trip back took 70 hours. What is the speed of the boat in still water, and what is the speed of the current?

The fundamental formula about distance, time, and speed is

\(\displaystyle s = \dfrac{d}{t} \implies d = s * t \implies t = \dfrac{d}{s}, where\)

\(\displaystyle where\ s = speed, d = distance, and\ t = time.\)

If you have any two variables, you can use one of the versions of that formula to find the third.

In a word problem, the first thing to do ALWAYS is to identify what the relevant variables are and assign letters to each in WRITING.

In distance/time/rate problems you already have a big clue about the relevant variables and what equations to use. You don't need a letter for distance because it is?

You don't need a letter for the times because you already have them. What are they?

Let w = the speed of the boat in still water.

Let x = the speed of the current.

Let y = the speed going downstream.

Let z = the speed going upstream.

This looks as though you have four unknowns, and you may think that you do not have enough information for a solution.

But, and here is the secret to this kind of problem:

y = w + x and z = w - x.

You have enough information to solve for y using the basic formula and enough information to solve for z using the basic formula.

What is y?

What is z?

Take a pause. You are half done. You now have just TWO linear equations in two unknowns.

What are they and how do you solve them?

Let us know exactly where this explanation loses you, but please answer all the questions up to that point.

 

[metoad2]

so X=4.5
w=6
y=10.5
z=1.5

are those correct?

 

[Deleted member 4993]

so X=4.5
w=6
y=10.5
z=1.5

are those correct?

No....

Let w = the speed of the boat in still water.

Let x = the speed of the current.

Let y = the speed going downstream. = 210/10 mph = 21

Let z = the speed going upstream. = 210/70 = 3

so:

y = 21 = w + x..................................(1)

Z = 3 = w - x....................................(2)

adding (1) and (2) → 2w = 24 → w = 12
and using (1) → x + 12 = 21 → x = 9

 

[soroban]

Hello, metoad2!

We will use: .\(\displaystyle \text{Time} \:=\:\dfrac{\text{Distance}}{\text{Speed}} \)

A boat traveled 210 miles down stream and back.
The trip downstream took 10 hours, the trip back took 70 hours.
What is the speed of the boat in still water, and what is the speed of the current?

Let \(\displaystyle b\) = speed of the boat (in still water).
Let \(\displaystyle c\) = speed of the current.

The boat went 105 miles downstream at \(\displaystyle b\!+\!c\) mph.
. . This took 10 hours: .\(\displaystyle \dfrac{105}{b+c} \:=\:10 \quad\Rightarrow\quad 105 \:=\:10(b+c)\;\;[1]\)

The boat went 105 miles upstream at \(\displaystyle b\!-\!c\) mph.
. . This took 70 hours: .\(\displaystyle \dfrac{105}{b-c} \:=\:70 \quad\Rightarrow\quad 105 \:=\:70(b-c)\;\;[2]\)

Equate [1] and [2]: .\(\displaystyle 10(b+c) \:=\:70(b-c) \quad\Rightarrow\quad 10b + 10c \:=\:70b - 70c\)

. . . . . . . . . . . . . . . \(\displaystyle 60b \:=\:80c \quad\Rightarrow\quad b \:=\:\frac{4}{3}c\;\;[3]\)

Substitute into [2]: .\(\displaystyle 105 \:=\:70\left(\frac{4}{3}c-c\right) \quad\Rightarrow\quad 105 \:=\:\frac{70}{3}c \)

Hence: .\(\displaystyle c \:=\:\frac{9}{2}\)

Substitute into [3]: .\(\displaystyle b \:=\:\frac{4}{3}\left(\frac{9}{2}\right) \:=\:6\)

The speeds are: .\(\displaystyle \begin{Bmatrix}\text{Boat:} & \text{6 mph} \\ \text{Current:} & \text{4.5 mph} \end{Bmatrix}\)

 

[lookagain]

Here it is: A boat traveled 210 miles down stream and back.

The OP wasn't clear with this sentence.

Until we know if the trip one-way was 210 miles, or if the trip one-way was 105 miles,
then Subhotosh Khan's fonal numbers and soroban's final numbers cannot be
determined to be the correct ones.

metoad2, you have to be clear on the meaning of the sentence in this quote box.

Which scenario is it, metoad2?

 

[metoad2]

That is exactly how the problem was written. We had to assume that it was 210 miles round trip, therefore 105 one way. That is part of the reason we couldn't figure out how to start to work the problem.

 

[JeffM]

That is exactly how the problem was written. We had to assume that it was 210 miles round trip, therefore 105 one way. That is part of the reason we couldn't figure out how to start to work the problem.

The question is ambiguously worded. But if trying to resove that ambiguity was your problem, why did you not say so? As it was, three people wasted their time dealing with the wrong problem. Actually you can solve the problem on either reading. If you assumed one reading and knew how to do the resulting math problem, then why did you waste everyone's time, your own included?