A widget company has a daily expense of 1500 dollars and each item they make costs them
a $1.45 to make.
If they sell each widget for $3, how many...
I don't know what kind of bad accent that is... but how many widgets must they sell
a day to break even?
Ok.
Well we got something in here about expense.
Then we have got something in here about selling an item and making some money.
So we have here two equations.
There are a couple of ways to set this up.
I want to identify this as two separate equations.
Let's talk about the expense.
How much is it costing this company to run everyday?
Well everyday and expense...
A widget company has a daily expense of $1500.00 and each item the make is costing them...
each item that they are making costs them $1.45.
Now I am using the same color for a different part of the question.
I really meant to highlight that in blue.
If you can tell the difference between the blue and the purple.
I hope so.
So talking about the expense, relating the expense of running the company to making items
each of them having a cost to make of $1.45.
So we are comparing the number of widgets and the reason we are comparing the number
of widgets to... well the other thing that we are comparing it to is expense... expense
and number of widgets.
We are talking about that because we have a cost dealing with something per day and
that cost to make each one of those widgets.
So we have our cost is going to be...
Let's see here.
I was about to write the equation.
But a discussion about independent and dependent, domain and range, explanatory and response,
and going back to that label and independent and dependent variable...
You have an expense.
You are making widgets.
What is helping to determine, you know, one variable or the other.
It seems like as a company you would have some control over how many widgets you are
making per day.
That value, the number of those widgets that you are going to make is going to help explain...
besides the daily cost... explain how much your expenses are to run the company.
So the independent variable therefor would be the number of widgets and the dependent
variable will be the expense.
The domain is going to be the number of widgets and the range, different word but same type
of idea, the answer or the range is going to be the expense.
So our cost based on a number of widgets is going to be that $1500.00 per day, and that
is just a one time fee per day.
Maybe it is a combination of your employees, your insurance, your electrical cost, renting
the building.
Whatever is making up the expense of your company.
Now plus maybe the materials it takes to make each one of those widgets which is a $1.45
times, that $1.45 is going to every time you make a new widget it is another $1.45, and
another $1.45, and another $1.45, and another $1.45... and when you do that repeated addition...
that new additional cost for every widget..., repeated addition is showing up as multiplication.
So we have $1.45 again times that number of widgets.
I like writing my equations, especially in early algebra, using words.
Even though widgets is just a generic term.
Maybe I am making watches, I am making ear rings, I am making something.
A widget is just a generic for something that is being made.
At least it is in words and you can see that the final answer is going to be the cost for
all of this production.
Well now we are going to be making money.
You know you are selling this stuff.
We are not making products, no one makes products to just give them away.
I mean that I know that there are give aways, but that is to drum up more business to ultimately
make it back and increase sales of something else maybe.
So they are selling this product at $3 a piece and they are trying to create an income.
What they...
Let's see here.
Let's call it income.
Their income is going to be again based on the number of widgets these people are selling.
We are doing that for $3.00 a piece.
Every addition item we sell is another $3 coming in.
3 plus 3 plus 3 plus3 for everyone we sell.
That is going to show up as $3 again times, because repeated addition can be short handed
into multiplication... a notation of multiplication.
We have our income is $3 times the number of widgets.
Now when you talk about revenue and income.
There is a gross income and a net income or gross revenue and net revenue, whatever language
you might be using.
When you sell something for $3, you did not really make $3 in profit.
That is called your gross income.
The net income is saying...
ok I sold something for $3 but there was a cost behind that.
If I only sell one item, then while I may have brought in $3 I have actually spent $1500.00
and $1.45 to create that one widget.
So I don't want to sell one widget and just bring in $3 because I have lost a lot of money
that day.
How many would I have to sell just to break even?
What would my gross income, my gross revenue, need to be so that my net income is equal
to zero?
You know, when will this break even point occur?
Then if we were solving inequalities, when you sold more than that then you would start
to incrementally make an actual net income of something.
More and more value.
But we are not solving inequalities, just equations.
So where is the break even point?
When does the cost equal the income?
I want to be able to subtract them and get a value of zero.
I don't want to loose money.
Making money would be great.
I will hopefully sell more widgets, but where is the break even point?
Where does the cost equal the income?
Well by saying that I am actually telling us how to set up the problem.
Where the cost is $1500.00 plus $1.45 times let's call it w for widgets.
When setting up word problems it is always extremely important to define the variable.
So we have $1500.00 plus $1.45w is equal to 3w. 3 times the number of widgets.
Ok, so now we have to solve for W. Really this is easier problem than pretty much any
of the ones we have solved for up till now in this lesson.
We need to look for parenthesis, we need to combine like terms and there is none of that.
We need to get the variables on the same side of the equation.
1.45 is less than 3.
I could subtract both sides by 3w if I liked to just keep my variable on the left hand
side of the equation.
But if I do that, there is not going to be anything on the right.
And as long as you write equals zero, then you are going to be ok.
A lot of students forget to do that though.
They just sort of move the equal sign between the two terms.
So I don't really encourage my students to move their terms around in the equation to
get one side to equal just zero at this level.
Not that it is wrong, but then they just let the equal sign float around and that is where
run into some troubles.
We are going to subtract both sides by 1.45w.
And we are going to get 1500 is equal to... now let's see here... 3 dollars minus 1 dollar
is 2 dollars.
2 dollars minus 40 cents is going to be $1.60.
Then $1.60 minus .05 is a $1.55.
Dollars makes sense for this problem because what we are talking about is money.
We are going to now, now that we have got that subtraction done and we have moved our
variable to the right hand side of the equation, we need to move everything to the left.
Well we are only one step away.
With our w we have a multiplication of 1.55, so we are going to divide that 1.55 from both
sides of the equation.
W is going to be equal to 1500 divided by 1.55 is 967.7.
Now if I want to check that answer, I need to take that and plug it back into the original
equation and make sure that the two sides are an exact match.
Now it will be a solution.
Sorry for the static.
It will be a solution assuming that is a correct setup for this word problem.
But at least you can show that it is a solution for the equation you came up with.
I am going to pause though and say that this is real life application problem.
You cannot make or sell I am assuming 967.7 widgets.
You need to make whole number of widgets and so I want to at least break even so we need
to say that W is equal to 968 widgets.
That will guarantee that we will break even.
Now if you sell any more than this you are going to then start making a net income that
is positive and actually create an income that is meaningful for your company.
I have one more word problem.
BAM!
Last example.
You travel to your parents home averaging 60 miles per hour.
Returning home you average 50 miles per hour and it takes you two more hours to get home.
How far do you live from your parents and what is the average of speed for the entire
trip?
Well anything in motion pretty much, especially in Algebra, is going to be dealing with distance
is equal to rate times time.
d=rt.
Distance equals Rate times Time.
And you are traveling to your parents home from your house.
So you are here at your home.
You travel to your parents.
You visit.
You come back.
So your home, parents, back home so the distance that you are traveling in each direction,
assuming that you have travelled the same path, are going to be equal.
So how far do you live from home?
You are going to take advantage of the fact that each of these distances are the same.
So we have a d1 and a d2.
d sub 1 and d sub 2 if you will.
But really you don't even need to write them with a different subscript because again they
are the same length.
So from home to your parents house your rate was 60 miles per hour and I don't know what
the time is because I am not given very much information about the time.
But going this way is equal to 60 times t.
Then when you head back you little bit slower, so your rate is going to be 50 and now here
you are being told something about the time it takes.
Coming home it is 2 more hours.
2 more hours than what?
It is 2 more hours than from your house to your parents.
So I do not know the time that each of these trips take, but I do know something about
the trip home.
Because you were going slower, whatever the time was to get to your parents plus two.
It was 2 more than.
It was two more hours to get home than it took to get there in the first place.
Now I have an expression for the distance between home and your parents house of 60t,
then getting home 50 the rate times times of t+2.
Now we are going to set these equations equal to each other, or these two expressions equal
to each other, because they are both equal to the same value of d... the same distance
between the two houses.
So we have got 50 times (t+2) is equal to 60t.
We are going to look for parenthesis that we can distribute through.
That means that we are going to distribute by the value of 50.
We get 50t plus 100.
I don't like the fact that my equal sign does not line up there.
Sorry about that.
50 times 2 is 100.
This is equal to 60t.
And I don't want to create any unnecessary negative coefficients.
I don't really want to move terms around to leave just a value of zero on one side of
the equation just because I see that mess up my students a little bit.
There is no reason why we cannot be comfortable with having variables on the right hand side
of the equation.
Just give yourself some practice.
So we divide both sides by 50 and we are going to get 100 equals 10t.
Now with the remaining variable that is almost isolated...
now that we have it not on both sides of the equation but just one, we are going to undo
that multiplication of 10 using the inverse math operation which is division.
And we get an answer which is that t is equal to 10.
We can assume that since we are talking about...
By the way, you might not understand this, mph is miles per hour or miles over hours.
And that means that our time period is 10 hours.
Now that t was 10.
It is a word problem.
It is applied to driving.
Now what the heck was t? t was the time it took you to go from home to your parents.
So it was 10 hours to your parents house.
And it took two more hours to get home.
That means that it was 12 hours to get home.
Now I am taking some time to write this, some values in context of the actual problem.
But I don't think I answered the question yet.
How far do you live from your parents home?
Well again the distance from home to parents is either d=60t or parents to home is d=50
times t+2.
I am actually going to check my answer of time is equal to 10 hours to parents and 12
to get home.
I am actually going to check that in the way that I am going to answer this question.
How far do you live from your parents.
I live from my parents 60 times the number of hours to get there which is 10.
So that comes out to be 600 miles.
If I really want to make sure that this is correct...
After I get the phone.
So again if I want to make absolutely sure I am correct, we said it was 12 hours to get
home so I can keep that value of t which of course is 10... or I have the 10+2 which where
I got the 12... and 50 miles per hour times 12 hours again that is going to come out to
be 600 miles.
So I have not only answered the question, yes in this particular problem with this scenario
you live 600 miles from your parents.
By doing it to both of these expressions and making sure I get the same answer I have checked
my solution.
What is the average rate of speed for the entire trip?
Well if distance is equal to rate times time, and I have a question about what is the average
rate of speed....
get out of the way here... then I can take this equation of d=rt and divide both sides
by t.
This is a little preview of what we are going to be doing in our next algebra lesson which
is solving literal equations.
That is an equation with more than one variable in it.
We want to isolate this r for rate.
We are going to do that by doing the inverse math operation of multiplication which is
division.
We get distance over time is equal to rate.
Now if we are talking about an average rate of speed, that is the total distance over
the total time.
The average rate of speed is not going to be an average of 50 and 60 because 50 plus
60 is 110 and 110 divided by 2 is 55 mph, and our average is not going to be right in
the middle or dead center because of the time difference.
We did not drive 50 and 60 miles per hour for the same period of time.
We have got to get that total distance over the total time.
Well if it is 600 miles to your parents house and 600 miles back, that is a total distance
of 1200 miles.
So our average rate is equal to 1200 over our total time for the average and our time
there was 10 and our time back was 12 so that is going to be 22 hours.
We have 1200 miles and a total of 22 hours.
We drove for 50 mph longer, so the average should not be right in the middle at 55.
The average is going to be closer to the value of 50 because we drove that speed for a longer
period of time.
1200 miles divided by 22 hours does come out to be 54.5 miles per hour.
There was a lot of examples in this video.
I hope that you got through all of them and understood everything.
This is the end of my lesson.
I am Mr. Tarrou.
BAM!
Go Do Your Homework.