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Math 20 Lesson 37
A Portland Community College mathematics telecourse. A course in arithmetic review. Produced at
Portland Community College. If you did well on the addition of decimals, this lesson on
the subtraction of decimals will be quite an easy and short lesson. Recall that subtraction
of decimals works just like addition. We write these two numbers above each other vertically,
lining up the decimals. Decimal points are kept in a line but more important than that,
the like places are in a line, tens with tens, ones with ones, tenths with tenths, hundredths
with hundredths, thousandths, and there are no thousandths here. If it will help you,
insert a zero on the end as a place holder. Then at that point simply remember that the
decimal point of the answer stays in that line and we forget that itís there and simply
subtract as thought itís a whole number, 7 from 10 is 3, I borrowed 1, 8 from 13 is
5, I borrowed 1, 5 from 12 is 7, I borrowed 1, 18 from 17 is 9, I borrowed 1, and 4 from
11 is 7, just that simple. Keep the decimal points in a line, right down through the answer.
Keep the like places in a line and give yourself placeholder zeros, if necessary. Letís summarize
that for our thinking and notes. First you write in vertical columns, line up decimal
points, line up like place value positions, we inserted our zeros wherever itís useful,
now subtract just like whole numbers. So you see the bulk of your work is done not in the
actual arithmetic but simply in setting up the problem and more particularly in setting
up the decimal places. So on this one we would have 24.86. This 5 here is a one so it must
go under the ones position, decimal points are lined up, 2 in the tenths position, 1
in the hundredths position, 4 in the thousandths position and then give yourself inserted placeholder
zeros, if it will help your thinking. The same thing down here, simply write down your
first number, line up the decimal points, ones in the ones position, tenths, hundredths,
and, if it will make you feel more comfortable, insert your placeholder zero. Keep the decimal
point lined up even for the answer and now simply subtract as though that were a whole
number and youíre done. The same thing down here, once the decimal points are lined up,
youíve accounted for every place properly, simply subtract as though itís a whole number
and you get your results. Now in this case, we had to insert a placeholder zero on this
one but down here on this one. With practice youíll know that thereís a zero there and
you wonít bother writing it. How about when decimals are mixed with whole numbers? Well
all we need to recall here is that a whole number really does have a decimal, it is written
to the right. Itís that by itself itís usually not written. But your mind must still know
that itís there. So writing down our first number, lining up the decimal points, you
can see in this case we need two either mental or actually written placeholder zeros. Keep
the answer decimal point in the same line and then simply subtract, 4 from 10 is 6,
I had to borrow from up here, then 3 from 9 is 6, 7 from 7 is zero. And again, by convention not by necessity,
usually the zero in front of the decimal point will be written; in some cases even two zeros,
notably in billings from department stores youíll find quite frequently there are two
zeros here. Now once again, write down the first number, line up the decimal points,
so this 5 goes here. A person who is a little bit careless will sometimes want to write
it on the end but always remember to keep those decimal points in a line. And then subtract,
keeping it in a line. Now see, in this case, we know there is a zero there, zero from 4
is 4, without writing it down. Zero from 3 is 3, again without bother of writing it down,
and 5 from 8 is 3. Now while where at it, do you recall how to check these. You simply
take your results, add it back to the number you were subtracting. Just add upwards and
see if you get back where you started from, of course, 4 and zero is 4, 3 and zero is
3, 3 and 5 is 8, were done. Up here, 6 and 4 is 10, carry the 1, okay, thatís 7 and
3 is 10, carry the 1, thatís 7 which is 8, and that checks. So our check was to simply
take our answer, add the one above it, see if itís the one above that, so very easy
process to check. Now one more quickly. Line this up vertically, lining up the decimal
points, and we would get this as our working problem, keeping the decimal point in a line,
now subtracting. Notice in this case we get quite a stream of zeros and we did this to
remind ourselves, which of these zeros do I really have to keep to protect the value.
And remember behind the decimal point on the end, zeros may be inserted or taken off without
changing the value. But this placeholder zero is necessary, if I didnít have it, I would
think that number is 14, without it I know itís 140. Now when we get to round off, frequently
we need to keep these zeros, even though they do not add or subtract to the value, just
to show how accurate our answer is. So recall that when we get there an a few lessons. Now
letís remind ourselves that a word use frequently to indicate that weíre going to subtract
is the word difference. Whenever the word difference is used, the second number written,
the second is the number being subtracted. So weíre going to subtract this from that,
so that gives me 18.6, weíre going to subtract from it, 9, but the 9 is in the ones place
so I must put in the ones under the ones place, keep the decimal point lined up, the tenths
in the tenths place, 4 in the hundredths place. You can see why we made such a large deal
about place value at the beginning of this in whole numbers because all of our instructions
are concerned with what to do with place value digits. Then, if it will help you, insert
a zero in those extra places not accounted for as one of your numbers, then subtract,
4 from 10 is 6 because I borrowed, 7 from 15 is 8 because I had borrowed, 9 from 17
is 8, and, of course, the decimal point stays in that line. Again, the word difference tells
me Iím going to subtract. In fact, technically the word difference stands for the results
of the subtraction. And with the word difference, the first number which is given you is written
first and the second number is the one that is being written or subtracted second. Now
letís contrast that with the wording in this next problem. What number is 4.68 less than
24.3? The key phrase here is the words is less than. When you see that combination or
slight variations on it, it always tells you that itís the first number that is being
subtracted from the second number. Again, keeping decimal points lined up. Itís exactly
the opposite of our use of the word difference in the previous problem. So this is a fact
you simply have to memorize. Less than tells me the first one is being subtracted, difference
tells me itís the second one being subtracted. Then, of course, in this case, simply follow
through with your placeholder zeros, so weíre borrowing, again, some more borrowing, 6 and
12 is 6, 4 from 13 is 9 because I have borrowed again. Then to check, add these two, 8 and
2 is 10, carry the 1, 7 and 6 is 13, carry the 1, 5 and 9 is 14, carry the 1, 2, and
if that sum is what I have started with, then our work was correct and this is answer is
okay. Okay, again, the phrase less than tells me this is being subtracted from that. The
word difference is just the opposite. Now, of course, if you were to do a subtraction
by calculator, you just remember that each mark on your paper is a button on the calculator
and when youíre all done, you want to know what itís equal to and that too is a button.
So on the calculator itís simply a matter of punching buttons, 24.86, subtract, 17.648,
equals, and youíre done, 7.212. The best way to check on a calculator is to simply
store that number or clear it and redo it, 24.86 minus 17.648 equals and see if you get
this same thing. The calculator is so fast that generally the best check is to simply
redo the problem. Again, since this lesson is such a simple review, letís use our second
half to at the same time use our decimals get ready for algebra. Letís recall what
we reviewed in our last lesson. Remember the variable just stands for some number so this
reads, some number minus 54.9. Or, if you wish, some number decreased by 54.9 is 17.36.
And to solve for the equation means I wish to isolate that variable all by itself on
one side of the equation. And the key to all of this was to recall that we concentrate
on the operations before the numbers. And we ask ourselves then, how do you undo subtraction
and, of course, itís by addition of the same amount. But our addition law of equality stated,
that whatever you choose to add to one side of the equation, and you may choose anything,
you must add the same amount to the other side. And this addition undoes this subtraction,
isolating the variable, which is what solving is all about. Then over here, we simply have
to add two decimals, which tells me to line the decimal points together. Keep the answer
in the same line, then add like decimal places, and weíre done. Now, with this lesson on
subtracting, see here weíre subtracting an equation, but, in fact, weíre solving it
by adding. But in this lesson, which is about subtraction of decimals, weíre going to ask,
how do I undo addition, not how do I get rid of 21.85 but how do I undo addition in order
to isolate the variable, which here is designated by the letter K, for whatever reason was chosen
by whoever developed the equation. And all we need to remember is to undo addition, you
subtract, the inverse of what we did last time and you subtract the same amount. But
now we have a subtraction law of the equation which says whatever I wish to subtract from
this side of the equation, thatís okay providing I subtract the same amount from the other
side. And, of course, over here weíre going to obey our rule for decimals points by being
very sure the decimal points stay in a line. And subtraction of anything undoes adding
the same amount, hence isolating the variable now subtracting but if it will help us to
insert a placeholder zero and being sure that we subtract like places, hundredths from hundredths,
tenths from tenths, ones from ones, tens from tens. So 5 from 10 is 5, I borrow 1, so 8
from 12 is 4, but I have borrowed 1, 1 from 1 is zero, 2 from 5 is 3. So between this
and the last lesson we recall that subtraction now undoes addition and I must subtract from
both sides, and from the previous problem, addition undoes subtraction. And we learned
that to get ready for story problems, itís a matter of learning what algebra symbols
stand for words. Well 15.3 is algebra as well as English in this case. Less than, from just
a few problems ago, we have to just learn means that the first number I gave is the
one being subtracted from the next number that Iím going to give. But in this case,
my next number is unknown. So if we donít know a number we can call it by a variable.
So I can read this as 15.5 less than some number or reading frontwards, some number
decreased by 15.3. Or I could even learn the difference between some number and 15.3. So
you see there are different ways of saying the same thing in algebra. So in English,
we just have to learn and memorize what words, combinations mean. In this youíll see that,
in fact, itís algebra that simpler and more consistent than the language. Do realize that.
All too often in story problems a student thinks that itís math thatís causing the
difficulty when, in fact, itís English thatís causing the difficulty. So 15.3 less than
some number, will be is a form of the verb is, so thereís will be 19.47. Now once we
have, hopefully, correctly gone from English to a math sentence called an equation, itís
not even important what that sentence was at that point. Now our problem is to simply
solve the equation and all we have to recall is to solve an equation means I wish to isolate
the variable and that I undo subtracting by adding the same amount and that I must add
the same amount to both sides and that to add or subtract, I line up the decimal points
and then just add the like places and Iím done. So I hope you begin to realize that, in fact, algebra, once you learn the
very simple button pushing rules, is really very, very simple. That is, our thinking and
our language that causes algebra to seem hard but bear that in mind and put your emphasis
of understanding where the difficulty is and thatís in being very clear about what the
language is trying to say. So one of the best ways to get ready for algebra, other than
reviewing your arithmetic itself, is learning what different English words mean in algebra.
So difference is simply the subtraction sign. Itís going to be between two numbers so now
our question is grammatically what in the sentence tells me to write first and what
to write second. Well with the word difference, the first number that I come to is written
first and the second number second. So the difference between an unknown number, so the
first number I come to in my sentence is the unknown number, so I use a letter to stand
for the known number, perhaps N for number. And 205 is the second number I come to. Now
the word is simply the equals sign and then 69.3. Now once I have from English to a math
statement, we simply play the game. And again, that game is to isolate the variable all by
itself on one side of the equation. So we look at whatís happening to it and whatís
happening to it is its being subtracted by another number. Now we undo subtracting by
adding and the addition law of equality says to add the same amount to both sides. And
the addition law of decimals told us to line the decimal points up, inserting placeholder
zeros if necessary, and to keep that decimal point lined up right for the answer, then
to add like places, so 3 in the tenths place, 4 in the units place, 7 in the tens place,
and 2 in the hundreds place. See how very, very important it is that weíre comfortable
now in that place values that we perhaps struggled with a bit in chapter one? The way you learn
any language from English to Spanish, from English to French, is to speak it many, many
times until you can go back and forth comfortably and so it is going from English to mathematics.
You must do many, many problems. So letís us here at least to a few together, hoping
that you will do even more by yourself. So beginning to translate, an unknown number,
so in algebra thatís simply some letter, your choice of what itís going to be. Increased
by, well in algebra that means youíre going to add, and it tells us how much weíre going
to add. Gives, in algebra very frequently like is means equal, sometimes it doesnít
so weíll have to be a little bit cautious here, 408.35. And once we have translated
correctly from English to math, itís simply a matter of playing the game. In this case,
the game is to undo addition by subtraction in order to isolate the variable. So subtracting
105.2 undoes adding 105.2 with a net effect of isolating the variable on this side of
the equation. But our subtraction law of equality says you had better subtract the same amount
on the other side. And the rules of decimal subtraction says to line up the decimal point,
use placeholder zeros if we want to and then subtract like place digits, so subtracting
the hundredths, subtracting the tenths, subtracting the units, subtracting the tens, subtracting
the hundreds. And now reading back to myself, it says, my number, which is what X stood
for, is 303.15. Are you beginning to get the feeling that this really could be very simple?
It really is. Even though sometimes it seems tricky, but the trickiness will always be
in your feeling with language rather than math. 25, after all, is 25. Thereís no trickiness
about that whatsoever. But we just need to flat learn that less than means algebraically
that youíre going in the reverse order. Less than a number, well a number is a variable.
So 25 less than a number, this stands for a number, results in, thatís equal, 48.6.
So we have to remember that less than means that weíre writing in reverse order in algebra.
Now once weíre in algebra, see there is no problem. To undo subtracting 25, you add 25
to both sides, being careful of decimal points. Addition undoes subtraction, weíve isolated
the variable, doing the addition on the other side, weíre done. So again, the algebra should
begin to be very, very simple; which tells us now where to begin to focus on where the
problem really is and that is language. And, of course, remember that the main point of
this lesson is that if weíre subtracting decimals, you line up the decimal places in
both the problem and answer and keep the place values lined up even if that means you insert
placeholder zeros. Then go ahead and subtract as though this were a whole number, in this
case 6 from 10 is, 8 from 8 is zero, 2 from 8 is 6, 4, and itís that simple. And thatís
the point of decimals; itís to allow us to treat fractions as though they were whole
numbers. This is your host Bob Finnell, until our next lesson. Good luck.