Math 20 - Lesson 9

A Portland Community College mathematics telecourse.
A Course in Arithmetic Review.
Produced at Portland Community College.
This lesson will be very short and to the point.
It deals with averaging,
a concept that today's world is very familiar with
from having read newspapers and listened to television.
Let's consider this rather simple problem.
The ages of 6 people are 35, 28, 37, 24, 31 and a second 31.
What is the average age of this group?
Now just what do we mean by average?
I'd be willing to bet that most of you know already
so this lesson is probably confirming what you already know.
You simply add the numbers of all the group being considered,
in this case the six ages,
then divide that sum by the number of numbers in the group.
In this case, 6.
In this case here are our 6 ages.
If we were to add them we would find, that's the sum of 186.
Our rule was to take the sum of the numbers,
divided by the number of numbers being averaged, there are 6.
And we get not 36, but 31.
So we would report that the average age is 31.
Now, what do we mean by that?
An interesting property of average is this.
If I were to replace every one of these by 31, the average,
and add those up: [ 31 + 31 + 31 + 31 + 31 + 31 ]
I could get, and I would get the same number.
So the average is that one number
which could replace every one of these different numbers
and get the same sum.
Average is a very important concept in statistics,
which is one of the fastest growing fields of mathematics
in college curriculums today.
A second somewhat messier problem:
We have students who were weighed and were concerned
about computing the average weight of the class.
Many students at the beginning without thinking would add these,
then divide it by 1,
2,
3,
4,
5,
6,
7,
And if you did that, you would be forgetting this.
You would be adding this as though there were just one of them
but in fact there is two of those,
that is two people weighed 110,
three people weighed 128, so adding just once is not enough.
You've got to add three of them.
There's not one of these, there's five of them,
two of this weight and six people weighed 170.
So in fact there is one person weighing 106.
There is two people, each weighing 110, which is 220.
One person weighing 118, [ 1 x 118 = 118 ]
three people weighing 128, [ 3 x 128 ]
which is 384 and so on. [ 3 x 128 = 384 ]
So we take the number of subjects,
which has each measure which is being averaged,
so we take the number of subjects times the measure in each case.
If we did that we would,
so what we have here are a group of sub-sums,
which are to be added to give us
the grand total sum which is 2,860.
We are to divide the sum of all of our measures
by the total count of those things being measured.
There's one of these students,
two of these, one of these, and a grand total of 20 students.
So now we divide the total weight by the total number of students
and we'll get the average weight for the class.
Doing that, the total weight divided by
total numbers of students [ 2860/20 ]
give us an average weight of 143 pounds. [ 2860/20 = 143 ]
That average is a very useful number in many, many areas,
specifically the health fields and the business field.
We call that an 'average.'
In statistics it's called an 'arithmetic mean'
because as it will turn out in more advances courses
there are other kinds of averages other than this one.
But we'll leave that for a future course.
This is the average with which most people are familiar.
Dropping back and reviewing now,
frequently when more than one subject has the same measure,
we don't list it many times,
we just list the measure once and how many times it occurred.
So, if this kind of a pattern is what you're facing
remember you always multiply
the number of times something occurred
times the thing that occurred to get the sub total.
Then add those to get the grand total
and add all of these counts to get the grand count.
That will occur in several problems in your homework.
That's it. That's the topic of average.
But first, did you notice
that the number we call 'average' of a set of numbers
is always greater than the smallest number
in that set of given measures?
And at the same time, less than the greatest number in the set?
I trust that that's a fairly obvious fact.
But sometimes we sort of forget
to divide by the number of measures
after we've added up all of our data,
and a good cross-check is when you're done
always check that number you call the average
and see if it's larger than the smallest in the set
and less than the greatest number in the set.
And this way you can check any great and obvious error
without having to redo your work.
So file this away at the back of your mind
particularly when we get to the section
on decimals and fractions.
It will prove to be helpful to us there.
At the same time,
let's recall at this moment that this is a symbol.
That means that this [number] is 'less than' [<] this [average].
Or reading backwards this [number] is 'greater than' [>] that.
And reading this way, average is 'less than' [<] this
or reading backwards, this is 'greater than' [>] this.
So keep in mind as we close this chapter
that we did introduce these two symbols
which were called 'order symbols.'
If you plan on going on into algebra
you'll have great occasion to use those two symbols
many, many times.
So once again, 'average' is always a number
between the smallest and the largest number.
If have you a number that is not,
then you know that you've made an error
and go back and double-check
Well, we have this opportunity now of an easy lesson.
Let's spend a few minutes
to see if above the review that we're doing
that we are beginning to think mathematically.
Let's see what I mean by that.
During the last lesson we became acquainted with certain words
and we tightened our grips on the meanings of them
so that we can begin to move
from the written word into mathematical statements.
And we learned that 'quotient' meant
the results of having divided etc.,
for product, sum, and difference.
And we learned that this word 'result' frequently
means 'parentheses' or 'all of,' the entirety of something.
Then from these words you were introduced
into certain language phrases such as this,
to see if could you go from words into an arithmetic statement.
So we would start to read and as soon as we found enough
to make a mark on our paper, we stopped and did so.
So the 'product' means the results of having multiplied.
So I'm going to have to multiply
two numbers or expressions together
simply because this word was presented to me.
So now I'm thinking what are these two expressions
I'm going to be asked to multiply?
Then it says okay the 'product of the sum.'
So it's telling me that one of my multipliers
is the 'results,' 'parentheses,' of having 'added.'
So I would say okay what do I add first
in order to get one of my multipliers?
It says: the sum of 28 and 12. [ (28 + 12)x( ) ]
Okay. But I still don't have the expression
that will be my other multiplier or factor.
So it says I want and the 'difference.'
The 'difference' means the 'results,' 'parentheses,'
of having 'subtracted,' [ (28 + 12)x( - ) ]
so I want the 'results' of having 'subtracted'
the difference between 15 and 5. [ ( 28 + 12 )x( 15 - 5 ) ]
Now, the word 'product,' 'sum,' and 'difference,'
primarily is what told me to put parentheses in here.
Now that I have a number statement, I can forget the words
and simply do this by the order of operation
which says I do what's in parentheses first,
what's in the next parentheses next, [ ( 15 - 5 ) = 10 ]
and now, the only operation that's left [ 40 x 10 ]
[ 40 x 10 = 400 ] and that's 400.
So this is giving me instructions on what to do to four numbers
when and how in order to get the final evaluation of 400.
So we want you to practice
and get used to going from words to arithmetic statements.
But at the same time during the last few lessons,
we have occasionally introduced you to simple math sentences,
not a phrase which what is this is,
but an entire math sentence called in algebra,
of course, an 'equation.' [ N + 350 = 486 ]
And we hope that you realize strongly now that a 'variable,'
a letter if you would, [ N ]
is simply a temporary mark to stand for 'some number.'
So you see, algebra in formulas is not about letters and numbers.
This [N] is a number, it's just that I don't know it.
So if I read this sentence it says:
'some number' added to 350 is 486. [ N + 350 = 486 ]
What's that number? [ N = ? ]
And we showed you that in playing the algebra game,
particularly when things become more complicated,
that we want to work on this equation by certain acceptable laws
so that ultimately we isolate the variable
all by itself on one side
so it just tells me my number is [ N = ] whatever is over here,
and we'll learn that if this unknown number [N] is being altered
by adding 350 [ N + 350 ] to undo adding 350,
we use the obvious arithmetic fact
that we subtract 350, [ N + 350 - 350 ]
and we're right back to where we started from
with that unknown number. [ N = ]
But then the Subtraction Law of Equality says
if you subtract 350 from the expression,
on one side of the equation [ N + 350 - 350 ]
you must subtract the same amount
from the expression representing the other side of the equation,
which we call in this case the 'right member'
as opposed to the 'left member' of the equation.
In short, if I subtract 350 from this side of the equal
I have to subtract 350 from this side. [ = 486 - 350 ]
And in so doing, we get 6,
3,
1.
Now this statement simply says [ N = 136 ]
your number, which you named N, is 136. [ N = 136 ]
So we learn that if you have a variable
being added by some constant,
you undo adding by subtracting, and you do it to both sides.
Then we also learned that you can undo subtraction, by adding,
and again to both sides or members of the equation.
So adding 482 undoes subtracting 482 [ R - 482 + 482 = R - 0 ]
leaving me with 'R' is [ R = ]
this [482] added to this [952] [ R = 482 + 952 ]
which is 1,434. [ R = 482 + 952 = 1,434 ]
And we reminded ourselves
that the reason we use different letters for our variable
is that in algebra, science, technology, and business,
it helps us to remember what this number really stands for.
Perhaps in this case 'R' stood for a 'revenue,'
which is something subtract 482 will leave me with 952.
Maybe this [R] is my monthly revenue,
this [ - 482] is my tax deductions,
leaving me with this [952] amount of money,
which tells me that I have $1,434 to begin with.
So by making a statement
about something we know regarding something,
which we might not know, we can play the algebra game
and find out that thing which we temporarily don't know.
Recall that we also learned
that we can undo multiplying by anything
by dividing by the same amount.
Let's say this is a problem that is saying [ 56 · W = 4032 ]
56 times some unknown weight is 4,032. [ 56 · W = 4032 ]
What was that weight? [ W = ? ]
It might be that we have 56 boxes all weighing the same thing,
which we've called 'W,' [ 56 · W = 4032 ]
and the total weight is 4032 pounds [ = 4032 ]
how much does each box weigh?
So you see again, the convenience of selecting a variable [W]
to help stand for, what we're talking about. [ W = Box weight ]
So if we can isolate that W, [ W = ] on one side, legally,
then the other side will present us with a solution.
And we learn that you undo multiplying by dividing both sides,
the number that you were multiplying the variable by [/56]
So, dividing by 56 undoes multiplying by 56
leaving one times W is, [ 1 · W = ]
and now of course you can see we get lots of review here.
We can either divide this [ 4032/56 ] by longhand,
if that's the kind of review we need.
Or, we can go to a calculator
and key in 4,032 then divide by, [4][0][3][2][÷]
[÷] which what this dash [/] is telling us,
56, equals, [4][0][3][2][÷][5][6][=]
and we get the weight of a single package is 72.
We advise you that at this phase
since the main purpose of this course is to practice arithmetic,
that it's well worth your while
to perform the division longhand yourself first.
And then when you have finished use the calculator
to do it once again to check your work
and in that manner, you are practicing hand arithmetic,
which you might need badly,
and calculator arithmetic at the same time.
We also covered the kind of equation
where your variable is being divided by a constant
and the results are equal to a known amount.
Perhaps here we're using the letter 'a' to stand for area
and someone knows if I were to divide an area by 48,
which might be a width, [ a/48 = 59 ]
I will get the length which is 59 of say, a rectangle.
The question might be:
What was the original area that made that happen?
So in order to isolate the variable 'a,'
we'd like to not get rid of the 48 so much as undo the division,
and we learn you undo division by multiplying.
Specifically, I want to undo division of 48,
so I undo it by multiplying 48,
but if I multiply the left member of the equation by 48
I must multiply the right member also by 48.
So, multiplication by 48 undoes division by 48,
isolating my variable,
which is what we mean in a sense by 'solving' an equation.
And this side now says look, your area
is the results of 59 times 48. [ a = 59 x 48 ]
or using our more technical vocabulary,
a is the product of 59 and 48. [ a = 59 · 48 ]
So now multiplying this, by hand, we begin to get
2,832. Two thousand eight hundred thirty two.
So a is 2,832. [ a = 2,832 ]
And if we check on a calculator
59 times 48 is 2,832. [5][9][x][4][8][=] 2,832
So our hand arithmetic checks our calculator and vice versa.
Next we introduced equations
and combinations of division and addition
or division and subtraction
or multiplication and addition or subtraction.
The question then comes up: Which of these two do I undo first?
And a simple rule you can use is simply to think
what is the order of operation between these two
if you knew that number and you had to do it.
Well I would take this number and divide it by 8 first,
and then to that quotient add 11.
Well as it turns out when you're trying to isolate the variable
once you established what the order of operation is,
this first and this second,
then in solving an equation surprisingly enough,
you will eliminate in the reverse order.
So in this case order of operation first
if I could do this arithmetic I'd do this first [k/8]
then this [ + 11 ], so this is the last thing I would do [ + 11 ]
So, it's the first thing that I will eliminate.
So I undo adding 11 to the results of this
by subtracting 11 and of course on both sides.
So this undoes that leaving me with the quotient
of k divided by 8 and over here, of course, this is 9.
Now there's only one operation, division.
To undo division by 8 I multiply by 8.
Then multiply both sides.
Then this undoes this telling me that k is 9 times 8 or 72.
So we're saying that 72 divided by 8 added to 11 will be 20.
So we've gone from words to solving simple,
very simple, equations and back to words again.
Soon you're going to find we're going to begin
to put these skills together going from words to arithmetic
and we will add to our vocabulary a few new words.
For instance, the verb 'to be' 'is' or any other form
like 'was,' 'will be,' 'could be,' frequently translates in math,
frequently, not always, to an equal sign [=]
And anything that designates 'what'
'How much' in math will simply be 'some number.'
It's just that I don't know it yet,
so we can simply use a variable,
any letter that is convenient to you.
Watch this following problem:
It's going to really give us a peek ahead
as to where this is all going.
Let's try to translate this into an arithmetic or math statement.
The 'sum,' that's a result of having added.
'9 and' [ 9 + ]
9 and the quotient. [ 9 + / ]
'Quotient' is the result of having divided
'some number,' let's call it 'N' for number, [ 9 + N/ ]
and 4. [ 9 + N/4 ]
See? The sum of 9
and the quotient of some number and 4 [ 9 + N/4 ]
is 21. [ 9 + N/4 = 21 ]
So see, after we have translated word by word
this into arithmetic and as it turns out algebraic symbols,
we are now left with an algebraic sentence
instead of a word statement. [ 9 + N/4 = 21 ]
And this we know how to do.
Order of operation, I would do this first [N/4], then this [9 +].
So we do the reverse order.
To undo adding 9, I will subtract 9. [ 9 - 9 + N/4 ]
In this case, from both sides. [ 9 - 9 + N/4 = 21 - 9 ]
And that leaves me
with my number divided by 4 is 12. [ N/4 = 12 ]
21 subtract 9. [ 21 - 9 = 12 ]
Then to undo dividing by 4 we multiply by 4.
Both sides. [ 4 · N/4 = 12 · 4 ]
And that tells me my number is 48. [ N = 48 ]
Isn't that nice?
That's exactly the purpose of mathematics.
Very frequently, we know what's happening to something
without knowing what that something is.
So if we can describe what's happening to that something,
in this case called 'some number,'
then translate our verbal description into a math statement,
then begin to play the math game called 'simplification,'
'evaluation,' 'order of operations,' 'solving' an equation,
we can begin to push these symbols around just like any game
until we have isolated the variable
and discovered what that number was
that we had described up here in,
by the statement of whatever is occurring to it.
But at this phase we want you just to get used to going
from numbers to arithmetic statements.
The sum.
The results of having added two things.
9 and the quotient.
The results of having divided two things.
Some number, that's the variable,
and 4 is 21. [ + N/4 = 21 ]
So it's not a matter of 'how do I do this problem?'
it's simply a matter of 'what are they saying?'
and can I say it, from this word language into a symbol language
because a symbol language will allow us manipulative power.
So through this example perhaps you can begin to get a feel
for where this course is trying to nudge you
while, at the same time, reviewing your basic arithmetic.
In this lesson of course, 'averaging' is our focal of review.
So recall that averaging means divide the sum of all of the data,
by the total number of data pieces data or data bits
or however we wish to say it,
but here again see how we're saying with words
what we're going to do with numbers and symbols?
Divide the sum.
The results of having added.
Added what?
All the data.
Well, what are we going to divide that sum of the data by?
The total number of those data pieces.
So this lesson is not only an arithmetic review,
but a very large part getting used to this verbalism
so we can communicate more exactly in future chapters.
This is your host, Bob Finnell.