Math 20 - Lesson 4

A Portland Community College mathematics telecourse.
A Course in Arithmetic Review.
Produced at Portland Community College.
This lesson will not only assume
that you have memorized the addition tables,
which you used on the last tape, but that in addition to it,
you have memorized the subtraction table.
That all we wish to do here is to examine
the process of subtraction a little bit more closely
and then to refresh ourselves on the matter.
And the first part of this refreshing is to remind ourselves
that in our culture,
we prefer to do our adding and subtracting in vertical columns.
There are cultures which do it differently,
but we like ours this way: vertically
with the place value columns lined up.
And if you need vertical lines
to help keep things neat, that's fine.
In fact, many people prefer doing their papers
endwise like this.
These lines, after all, were designed to guide us in writing.
Frequently in arithmetic
it's more helpful to turn the paper this way.
Have you ever noticed an accountant's notebook?
They now have lines going this way
but they do have lines going vertically as well,
and if the professional finds this useful,
then certainly we amateurs can, too.
Then you recall, you can only subtract ones from ones,
tens from tens, hundreds from hundreds, and so forth.
Then you simply using your memorized tables, subtract,
3 from 5 is 2. [ 5 - 3 = 2 ]
0 from 4 is 4, [ 4 - 0 = 4 ]
and some of you might recall
that when you first started this process
rather than thinking 2 from 7 is 5, [ 7 - 2 = 5 ]
you thought 2 plus what will give me 7? [ 2 + ? = 7 ]
Which is 5. [ 2 + 5 = 7 ]
And you begin to realize that subtraction is, after all,
nothing more than reverse addition.
What can I add to 4 to get 8? And that's 4.
Then eventually we twisted that around
and just thought 4 from 8 is 4. [ 8 - 4 = 4 ]
And we're done.
Basically the problem is that simple.
Providing of course you have your tables memorized.
But the words concerning subtraction is a different matter.
People very frequently get confused on that,
so let's take some common phrases denoting subtraction
and look at them rather carefully.
When we say "3 less than 9"
or "one number less than" another, actually,
when we write it mathematically,
we're saying 3 less than 9. [ 9 - 3 ]
Do not write this first just because it came first.
The word 'less' in effect says
that this is being subtracted from what follows.
Now if we use the word 'minus'
8 minus 5 [ 8 - 5 ]
then in that case we simply write it as we come to it.
And if we use the phrase "difference between two numbers,"
we write the two simply as we come to them.
So 'minus' and 'difference,'
you write the numbers exactly as they appear in the sentence.
But if you use the phrase "less than"
then in fact, the numbers as they appear in the sentence
are reversed mathematically.
So be very cautious on that, particularly this phrase 'less than.'
And also let's caution ourselves
to be careful when we need to borrow.
First, let's line these up vertically by place values
like this.
Then remind ourselves we can only subtract places,
so we want to subtract 6 from 4,
but in fact 6 won't subtract from 4. [ 4 - 6 ]
A very common mistake is to reverse
and subtract 4 from 6 and write 2, [ 6 - 4 = 2 ]
but that's definitely wrong.
You must subtract 6 from this. [4]
Since we can't, we recall the relationship
between one column and the one to its left.
Remember ten of these [ones] is one of these [tens]
consequently, one of these, [tens ]
So if I take 1 away from the 7, that leaves 6
is 10 of these, so I'll add 10 back
over into this column.
Now we have 10 and 4 is 14. [ 10 + 4 = 14 ]
And 6 will subtract from 14 eight times. [ 14 - 6 = 8 ]
This process we call borrowing.
Now in this case 3 will subtract from 6,
so I do so. [ 6 - 3 = 3 ]
2 will subtract from 2 zero times, [ 2 - 2 = 0 ] so I do so.
And I'm done. [ 5274 - 2236 = 3038 ]
So remember the basic relationship.
1 in any column is worth 10 of the one just to its right.
So I can always borrow 1 from there,
which is adding 10 over in this column.
This is as though I borrowed a ten dollar bill,
changed it into 10 one dollar bills
and add it to the 4 one dollars to get 14 dollars.
A simple process if you think about it just a bit.
And of course in practice
we usually get our problems in verbal form,
so let's say Mr. John North
owes the Murphy Construction Company $4035.
He made payments of $1085, $537, and $894.
How much does he still owe rounded to the nearest 100 dollars?
A 3-prong problem.
First, I could subtract each one of these payments
one at a time from this [ $4035 ] just as we would in a checkbook
The shortcut would be
to add all these payments together first
like this,
And of course 'total' means I'm going to add,
and if I actually add I get
$2516 total payments.
So we've done that much of the problem.
But that total is to be subtracted from his bill, which was $4035
so doing that we get this problem, subtraction,
which has the answer:
$1519. One thousand, five hundred nineteen dollars.
But they wanted more than that.
They wanted the answer rounded to the nearest hundred dollars.
There's ones,
tens,
hundreds,
So I looked to the digit just next to it, which is 1.
It's 4 or less, so I round down,
which means that those are replaced by zeros
and at the rounded place is left as it is.
So I can state that we owe the construction company
roughly $1500 rounded to the nearest hundred dollars.
This problem illustrates that in reality when you get a problem,
it seldom is simply a subtraction or an addition.
It usually involves all of the things at once,
in this case, addition, subtraction, and rounding off.
So usually it's not a matter of just doing this or that.
It's a matter of doing a sequence of operations
all within one story problem.
Now, while reviewing our arithmetic,
let's simultaneously begin to get ready for algebra.
Whether you will actually take a course in algebra
or just use simple formulas or equations
in business math, science, nursing, technology,
or whatever it is.
To solve an equation look upon it at first
as a sentence.
It's saying some number, subtract 37 from it
will give me 59?
What number will make that true?
Now in this case where you're subtracting a number
from this unknown one, you simply have to remember this fact:
it's a simple fact of numbers that addition undoes subtraction.
Look upon 'R' as the amount of money in your pocket.
If I subtracted $37 from it, what would I have to do
to have your pocket back to where it was at the beginning?
Well, I would have to add $37 back into your pocket,
and I'm right back to where you were in the beginning.
We'll call that, that's 'some number R.'
Okay now, if I add to undo that subtraction,
the rule that we're going to have
is that we must add the same amount to both sides.
So in this case adding 37 to 59 we get $94.
So $96.
So $96 is the amount of the R originally.
Now to check it, we would simply do what this says.
96 subtract 37, [ 96 - 37 ]
7 from 16 is 9. [ 16 - 7 = 9 ]
3 from 8 is 5. [ 8 - 3 = 5 ]
And it checks.
In algebra we would verbalize this rule this way:
You may add the same amount to both sides of an equation.
So let's review, to undo subtraction we added,
but whatever I added to undo that and get me to the letter alone,
I had to do to the other side, which got me the answer.
So when you see an equation
where a constant is being subtracted from the variable,
you undo it by adding, which isolates the variable,
and you add the same amount to the other side.
So these simple equations are very simple to solve
as is this next one.
Reading this equation it says I have some number.
Now remember a variable
is simply a math way of saying "some number."
If I were to add to it $85, I'd have $208,
using dollars to help us think.
How much did I have to begin with?
Well if I added $85 to get back where I was,
I'd have to subtract $85.
So that's our fact.
Subtraction undoes division.
If I subtract 85 I undo having added 85,
which leaves me what I had to begin with,
but if I subtract 85 from this side,
I have to subtract 85 from this amount over here
because our equation says
this whole expression [ m + 85 ] is the same as [=] 208.
So doing that I get that m is 123. [ m = 123 ]
To check that, let's do what that says.
It says take m, which we're claiming is 123,
add 85 to it. 208. [ 123 + 85 = 208 ]
And that checks with our statement.
So, remember subtraction always undoes addition,
so if you're trying to undo an addition to isolate the variable,
you simply subtract.
In algebra we will simply say
you may always subtract the same amount
from both sides of an equation.
So to undo adding, subtract the same amount from both sides.
In our previous problems to undo subtracting,
we added to both sides. That simple.
Let's do two problems side-by-side.
We wish to isolate the m.
To undo subtracting 8, we add 8.
That undoes that leaving the m [ m - 8 + 8 = m ]
adding 8 to this gives me 23. [ 15 + 8 = 23 ]
But if we wish to undo addition,
we subtract the same amount from both sides.
Subtracting 9 undoes adding 9 [ r + 9 - 9 ]
leaving me with a variable by itself [ r + 9 - 9 = r ]
and subtracting 9 from 22 we get 13. [ 22 - 9 = 13 ]
[ r = 13 ] Just that simple.
You have a few problems like this
and it gives you a chance to review adding and subtracting
but at the same time begins to prepare you for algebra
rather slowly and comfortably, I might add.
Don't allow yourself to be tricked into thinking
that the letters used in algebra
are either mysterious or difficult.
They are neither.
In algebra, a letter is simply a convenient mark
to symbolize the phrase "some number."
So if I were to say something like this,
[ a + 45 ]
I am simply saying "some number added to 45."
If I were to say this, [ t + 45 ]
I am still saying "some number added to 45."
And [ v + 45 ] the same thing in this case.
'Some number' added to 45 or 'the sum of some number' and 45.
What number?
Well as is written down here, we haven't the slightest idea.
Somebody has to tell you.
Then why so many different letters?
Well we might want to use t for taxes,
v for velocity, a for amount,
so we simply pick the choice of symbols
to help us remember what it is we're talking about.
That is what kind of a thing did we measure or count
to get the number that this letter is going to stand for.
And since the letter can stand for any number I want it to be,
rather than always saying "some number,"
we call the use of these letters in algebra, 'variables.'
And obviously, the reason we call them 'variables'
is they can be anything you want them to be
or anything something measures to be.
So it varies from application to application.
If however, I take an algebraic expression for a single number,
this is the results of an addition called a sum,
and if I make an algebraic sentence
like this, [ a + 45 = 92 ]
then we could, in a sense, figure out what a is
because now we're claiming something about this sum.
We're saying look, I have some number.
If I added 45 to it, it would be 92.
Now if just thinking what's written here is sufficient for you
to tell somebody what that letter is standing for, number wise,
then just write it.
There's nothing to do.
But what the algebra rules are for
is a way of getting from this statement
to determining what this variable is
through a more methodical method.
And that method first introduced here
is that in very simple sentences like this,
that if I subtract 45 having added 45,
this undoes that [ a + 45 - 45 ]
and gets me down to the variable by itself.
But our subtraction law of equality in algebra says fine,
subtract anything you want from this statement
on this side of the equation
providing you subtract the same amount from what it's equal to,
or the other side of the equation.
So subtracting here we would get
5 from 12 is 7. [ 12 - 5 = 7 ]
4 from 8 is 4. [ 8 - 4 = 4 ]
So we're claiming that my number here called 'a' is 47.
And if we wish to check,
then use this number in place of the variable
in the original sentence and see if the sentence is still true.
See this equal [=] is sort of a mathematical verb to stand for 'is'
could be, will be, should be, was.
So this is a, which I'm now claiming is 47,
when added to 45 [ 47 + 45 ]
we want to be 92.
So if we add these, indeed it is. [ 47 + 45 = 92 ]
So you see it's very easy to review our arithmetic
while at the same time very comfortably getting ready for algebra
Of course algebra proper will handle equations
far, far, far more complicated than this,
but it all starts from here.
So at this stage we simply want to think of this
as doing something to undo that
and isolating the p so that the sentence will read
p, whatever p will stand for, is [=] something.
So we recall that to undo subtraction you add,
but our addition law of equality is now saying
whatever I add to this member of the equation,
that is the expression on this side,
we must add to the expression on the other side of the equality.
And on this side this [ + 4867 ]
has the effect of undoing that [ - 4867 ]
leaving me with only my variable p.
Then on the other side.
We complete our arithmetic and we would get 9,
9,
2,
14.
So now just reading the sentence back,
we have my number p is 14,299. [ p = 14,299 ]
See? Nothing mysterious about that.
It's just a mark on a piece of paper to stand for
a number which I don't know yet,
and by maneuvering, I can find out what it is.
We could just as well if we wanted to
have left a large box there with a question mark on it.
And it would serve the same purpose,
but after a while it simply becomes more comfortable
to use a simpler mark,
particular a mark that stands for what I might be talking about:
price or whatever it is.
You can see in a simple math sentence
there's really nothing to do.
Reading this just says some number here called R,
added to 3 is 8.
What number would make this sentence true?
Obviously 5.
There is nothing to do.
You didn't really have to subtract 3 from both sides.
You can just read the sentence and just see
that 5 plus 3 is 8. [ 5 + 3 = 8 ]
So there is nothing to do.
Reading solves it automatically.
However, as these numbers become more complicated,
then it is no longer so obvious what the solution is.
However, it might be obvious
that if I add 2354 to a number to undo that,
I have to, in turn, subtract it off to get back to where I was.
But our subtraction law of equality says fine.
Subtract whatever you want from this member of the equation
providing you subtract an equal amount
from the other member of the equation.
Of course that subtraction undoes the adding.
Notice the thing we're looking at first is not the number.
That's irrelevant temporarily.
We're looking at this. [+]
We're trying to undo adding [+] by subtracting [-]
Then this is, of course, the specific that we're trying to undo.
But it's the addition that we undoing.
So to undo adding we subtract.
Now at this point, let's remind ourselves
where subtraction came from historically.
Back to our abacus.
Here's 5 in this column,
8 in this column,
7 in this,
and 9 in my units column.
And again, our ancestors did not think of this as 5,879.
They simply thought, quite simply,
5 here,
8 here,
7 here,
9 here.
So all they could really do is count digits, and not too far.
This is somewhat beyond 10 and store them on the bead board.
Now to subtract meant literally, to subtract things, not numbers.
So this said, "Take 4 beads away from this column."
There's 2, 4, take them away.
Now take 5 away from my next column.
So 2,
4,
5.
Take them away.
Then take 3 away from here.
Take them away.
Then 2 from here.
Then tell me, after you've done that, what you have left.
So you can see I have 3 here
5 here,
2 here,
and 5 here.
So the reason we can do this mentally
is that we have memorized a subtraction table,
which is simply a memorization of what happened on our bead board,
sort of like memorizing a paper route.
After a while you no longer have to look at your route list,
you simply look at your house and deliver the papers,
so now we don't need this.
In our heads subtract 4 from 9 beads, know that we have 5 left.
If we take 5 from 7, we know that we have two left
because we have memorized our addition and subtraction tables.
Another example: to undo adding I will subtract
and only then do I concern myself with what number.
This [+] determines where I start.
And our subtraction law of equality says fine.
Just remember to subtract the same amount
from the other side of the equation.
This undoes that, isolating my c,
so that my sentence now reads c equals the result of this,
but now we see we have to borrow.
7 does subtract from 9 leaving me 2.
But 8 doesn't come from 3.
So here's where our historical process of borrowing comes from.
There is nothing there.
So one borrow from here leaves 3.
Give me 10 here which is now 1 plus 0 or 10.
But borrowing 1 from there leaves 9,
brings 10 over here and 10 to 3 is 13.
So 8 from 13 is 5, [ 13 - 8 = 5 ]
5 from 9 is 4, [ 9 - 5 = 4 ]
and 1 from 3 is 2. [ 3 - 1 = 2 ]
Let's now look at that borrowing process on the abacus
and you'll see that it really is quite clear and simple.
So on our abacus this simply meant 4 beads here.
None here.
3 beads here.
And 9 beads here.
Now subtraction meant column by column.
Remove this many beads, so if I subtract 7 beads
it's just a matter of counting 1, 2, 3, 4, 5, 6, 7.
Remove them and I have 2 left.
I can just see them.
No subtraction to do.
It's just a matter of following the bead instructions,
record the results.
But this now says subtract 8 beads from here,
but I don't have 8 to subtract from.
I only have 3, so we remember now if I take 1 of these away,
I can put 10 here, but I don't have 1 here,
but I can go over one more, take 1 from here.
One of these is worth 10 of these so 2, 4, 6, 8, 10.
But now once again one of these is 10 of these.
So 2, 4, 6, 8, 10.
Now we can say take 8 from this column,
and I have more than enough to get 8 away,
so there's 2,
4,
6,
8.
I've taken 8 away from this column.
Now we want to take 5 away from this column, and I can do it.
So take 2
take 4
take 5
away.
Now I want to take 1 from this column so take 1 away.
Now record what you have left.
Well I have 2 here,
4 here,
and 5 here.
And that was the solution to our equation.
Hopefully you might say
hey, I can do it easier on paper than I can here.
Then that's good.
That tells you this lesson is going well.
So practice, become fluent.
And this is your host, Bob Finnell, until our next lesson.
Good luck.