Word Problems

[bmel37]

I am studying for a pre-calc assesment test. The last math class I took was trig almost 2 years ago, so I am pretty rusty!

My problem with this section of the study guide (Math Modeling - Word Problems) is that I have no idea where to begin!
If someone could just give me the basic formulas to use in order to solve these problems, I would REALLY appreciate it! I have the answers, and I don't mind working the problems out for myself, but I need to find out which formulas, etc. to use and memorize.
Please provide your input for any or all of the following questions:

1. A fast fish can swim the length of a river upstream in 5 hours and downstream in 3 hours. If the river is 40 miles long, how fast could the fish swim if there were no current in the river? What is the speed of the current?
Answer: The fish swims at 32mph. The current is 8mph.

2. Steve can write a quiz in 40 minutes. Steve and Ruth working together can write a quiz in 25 minutes. How long would it take Ruth working alone to write a quiz?
Answer:66 2/3 min

3. You have two saline solutions. One solution is 20% salt; the other is 50% salt. How much of the 20% solution would you mix with the 50% solution to make 5 liters of a 30.5% solution?
Answer:3.25 L of the 20% and 1.75 L of the 50%

Thanks in advance, everyone!

 

[Yogi]

1. If the fish swims downstream at 32+8 mph how does it take 3 hours going 40mph to go 40 miles? The given answer is incorrect.
Draw a detailed picture including all info given. You should know that mph(miles per hour) refers to velocity, so V=velocity=mph=miles per hour=miles/hour=distance/time=D/T or V=D/T What info do you know? What info don't you know? Make variables for the pieces you don't know...you do the rest.

2. Pick some arbitrary amount of time, say 40 minutes(or whatever you want). Figure out how many quizzes Steve can write in 40 minutes(already known). How many can they both write in 40 minutes? Now that it is a fixed unit of time you can figure out how many Ruth alone can do in 40 minutes. Then convert that to 1 quiz by doing the same thing to both sides of the equation.

3. Create 2 variables x and y where x is the percentage of the 20% solution used and y is the percentage of the 50% solution used so that x+y=1. Need 1 more equation with x and y, and solve 2 equations with 2 unknowns...you do the rest.

 

[soroban]

Hello, bmel37!

1. A fast fish can swim the length of a river upstream in 5 hours and downstream in 3 hours.
If the river is 40 miles long, how fast could the fish swim if there were no current in the river?
What is the speed of the current?
Answer: The fish swims at 32mph. The current is 8mph. .As Yogi said, these are wrong.

Formula: .\(\displaystyle \text{Distance} \;=\;\text{Speed} \times \text{Time}\)

Let \(\displaystyle f\) = speed of the fish (in still water).
Let \(\displaystyle c\) = speed of the current.

Swimming upstream, the current works against the fish.
Its speed is \(\displaystyle f\!-\!c\) mph.
In 5 hours, it swims 40 miles: .\(\displaystyle 5(f-c) \:=\:40 \quad\Rightarrow\quad f-c \:=\:8\) .[1]

Swimming downstream, the current works with the fish.
Its speed is \(\displaystyle f\!+\!c\) mph.
In 3 hours, it swims 40 miles. .\(\displaystyle 3(f+c) \:=\:40 \quad\Rightarrow\quad f+c \:=\:\frac{40}{3}\) .[2]

Add [1] and [2]: .\(\displaystyle 2f\:=\:8 + \frac{40}{3} \:=\:\frac{64}{3} \quad\Rightarrow\quad \boxed{f \:=\:\frac{32}{3}}\)

Substitute into [2]: .\(\displaystyle \frac{32}{3} + c \:=\:\frac{40}{3} \quad\Rightarrow\quad \boxed{c \:=\:\frac{8}{3}}\)

2. Steve can write a quiz in 40 minutes.
Steve and Ruth working together can write a quiz in 25 minutes.
How long would it take Ruth working alone to write a quiz?
Answer: 66 2/3 min

Steve does the job in 40 minutes.
In one minute, he does \(\displaystyle \frac{1}{40}\) of the job.
In 25 minutes, he does \(\displaystyle \frac{25}{40} \:=\:\frac{5}{8}\) of the job.

Ruth does the job in \(\displaystyle r\) minutes (working alone).
In one minute, she does \(\displaystyle \frac{1}{r}\) of the job.]
In 25 minutes, she does \(\displaystyle \frac{25}{r}\) of the job.

But in 25 minutes, she does the other \(\displaystyle \frac{3}{8}\) of the job.

We have: .\(\displaystyle \dfrac{25}{r} \:=\:\dfrac{3}{8} \quad\Rightarrow\quad r \:=\:\frac{200}{3}\)

Therefore: .\(\displaystyle r \:=\:66\frac{2}{3}\text{ minutes.}\)

3. You have two saline solutions: one solution is 20% salt; the other is 50% salt.
How much of each solution would you mix to make 5 liters of a 30.5% solution?
Answer: 3.25 L of the 20% and 1.75 L of the 50%

Consider the amout of salt at each stage.

Let \(\displaystyle x\) = liters of the 20% solution.
It contains: .\(\displaystyle 0.2x\) liters of salt.

Then \(\displaystyle 5-x\) = liters of the 50% solution.
It contains: .\(\displaystyle 0.5(5-x)\) liters of salt.

The mixture contains: .\(\displaystyle 0.2x + 0.5(5-x)\) liters of salt. [1]


But we know that the mixture will be 5 liters which is 30.5% salt.
Hence, it contains: .\(\displaystyle 0.305(5) \:=\:1.525\) liters of salt. [2]


We just described the final amount of salt in two ways.

There is our equation! . . . . \(\displaystyle 0.2x + 0.5(5-x) \:=\:1.525\)


Solve for \(\displaystyle x\!:\;0.2x + 2.5 - 0.5x \:=\:1.525 \quad\Rightarrow\quad -0.3x \:=\:-0.975 \)

. . . . . . . . . .\(\displaystyle x \:=\:\dfrac{-0.975}{-0.3} \:=\:3.25\)


Therefore: .\(\displaystyle \begin{Bmatrix} 3.25\text{ L of the 20% solution} \\ 1.75\text{ L of the 50% solution} \end{Bmatrix}\)

 

[lookagain]

;
. . . that I have no idea where to begin!

> > > If someone could just give me the basic formulas to use in order to solve these problems < < <


I would REALLY appreciate it!


I have the answers, and > > > I don't mind working the problems out for myself, ... < < <

...

 

[bmel37]

Thanks Yogi and Soroban. Although Yogi did give me the information I asked for, I ended up really needing the detailed walk-throughs that Soroban provided. After studying your responses I tried solving each problem again on my own, and everything came together with some trial and error.

-B

 

[lookagain]

Thanks Yogi and Soroban. Although Yogi did give me the information I asked for,


> > > I ended up really needing the detailed walk-throughs that Soroban provided. <<< [/b]

No, you did not need that detailed walk-through that he provided.
You are not here to be spoon-fed.

You needed to work it out for yourself now and in any future problems after you're given help, not solutions.

Please provide your input for any or all of the following questions:

...

"Input?" See your word there. You are not to be given full solutions despite any veteran
users who don't follow the spirit of the guidelines of this board.


An please read the posting guidelines:

http://www.freemathhelp.com/forum/threads/41536-Read-Before-Posting!!


As part of them,please post only one problem per thread,
along with your attempts.

We are akin to coaches, and you are likened to the athlete
being coached. You are to prepare by getting in shape
and practicing, and then we are here to guide you further
along after we see what you can do first.

 

[bmel37]

So you're going to ignore the emphasis I put on not being given the answers and also disregard that I explained that this is for studying purposes. Lecture the people who ask for answers and the "coaches" who give them out, not me. I feel I'm being treated more harshly than I deserve. And as I also explained, I had no idea how to even begin working the problems out! So how was I supposed to provide my "attempts" when I came to this site in order for someone to show me the direction I should even start in. Soroban knowingly did something wrong, but it helped me, so I thanked him anyway. So now I'm at fault for showing my gratitude for what Soroban did to help me? I make my first post here and suddenly there are people nit-picking over every little thing and blaming me for all of it. Way to make me feel welcome.

 

[lookagain]

picking over every little thing and blaming me for all of it. Way to make me feel welcome.

As you later mature in life, you will realize that you're not special, not get defensive,
not make statements with hyperbole, and stop making excuses for your lack of work.
Post numbers 4 and 5 spell out what you wanted. There's no point now trying to
go back and discount what you requested. Next time, thank users who are
putting you on the right track for the posting guidelines.
I will welcome you then.