Math 20 - Lesson 50


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Transcript:
A Portland Community College mathematics telecourse.
A Course in Arithmetic Review.
Produced at Portland Community College.
Having done the last lesson and faced with a problem like this,
you would immediately probably think Well, let's do the division,
change to it a decimal, and change a decimal to a percent.
That's what we've been doing in the last two lessons.
But for a change of pace,
let's approach this first from another angle.
Let's go back to basics.
Let's state this question,
and really it's a question in mathematical form.
We're really saying
7/8
is [=]
what [x]
per [x/]
cent [/100]
Now the reason we do this is to emphasize
that the purpose of math later, particularly algebra,
is when you have a verbal question
can you restate it in pure math symbols
as we did in the last chapter in ratios and proportions.
Then in this case, if it's a proportion,
you simply solve the proportion.
And we won't review that.
We will presume that you can already do that.
Then of course in this case,
if we want to divide the unknown further,
we would get the decimal 87.5
So x is 87.5 [ x = 87.5 ]
and this my per [x/] cent [/100] percent right here.
So we are replacing this whole expression [ 7/8 = x/100 ]
by this equivalent expression, [ 87.5% ]
which is a percent expression
equal to our given fractional expression.
Now of course our original way of looking at this
is probably faster.
Think of this [7/8] as a decimal 7 divided by 8.
Dividing it we would get
the decimal equivalent .875,
but now we recall if we want to change a decimal to a percent,
we move the decimal point two places to the percent
and add the percent symbol.
Now this total statement [ 7/8 = .875 = 87.5% ]
really sums up what we've been doing this in chapter so far.
Notice we have a fraction, [7/8]
a pure decimal, [ .875 ]
and a percent. [ 87.5% ]
And the equal signs emphasize [ 7/8 = .875 = 87.5% ]
that we have the same number in every case;
we're simply choosing to write it in a different form.
And again, that's an idea always to keep at the back of your mind
Now in this particular lesson where we want to change a decimal
to a fraction, to a percent, notice the slick and easy way
is simply do the division,
change the fraction to a decimal through that division
with at least two decimal places.
Then move the decimal point two places to the right,
stick the percent sign on, and you're done.
Again, to change a fraction to a percent,
change to a decimal with at least two places
rounded, or else carried out as directed,
then move the decimal point two places to the right,
that is towards the percent, and append the percent symbol [%]
A common difficulty with this type of problem
is simply the matter of following directions.
Always read very carefully how they want this carried out.
An example: Let's say we wish to have this decimal
converted to a percent accurate to the nearest tenth of a percent
and the important thing to realize here it's a tenth of a percent,
not just tenth, but tenth of a percent.
So if we do this division, we would get
this division problem,
and beginning to divide,
there is a temptation to quit here
by saying I have it to the nearest 10th
and that's what they wanted.
And the answer is no.
They didn't want it to the nearest tenth period.
They wanted it to the nearest tenth of a percent,
and a percent is going to be formed
by moving this decimal two places to the right.
So we need at least three decimal places
to be accurate to the nearest tenth of a percent.
So let's go three places and a fourth for round off.
'Accurate' tells me I'm going to round off.
So dividing this further, we would get
So I have a decimal carried out to the nearest four places:
tenth,
hundredth,
thousandths,
ten thousandths.
I wanted this place to realize that I can drop it
and round this one down.
Now to change this to a percent, I move it two places to the right,
and I get 15.6 percent.
See? Accurate to nearest tenth of a percent.
But to have the percent
accurate to the nearest tenth of a percent,
I had to have my decimal accurate to the nearest thousandth.
Three places here to get one place here
because the first two places
was to get me the percent to begin with,
so again, it's a matter of following directions.
We didn't want it rounded to the nearest tenth period,
but to the nearest tenth of a percent.
Let's try a second one again emphasizing
the matter of being very careful with the instructions.
Says, Change to a two-place decimal with fraction,
then to a percent.
So as I do this division,
this says to only go out to two decimal places,
so doing that I get
my two decimal places and a remainder of 20.
But we want that remainder written as a decimal
with fraction at the end of those two places.
So I take my remainder, write it over my divisor,
reduce it,
and write that fractional quotient
against my second place or hundredths place.
So now we have a decimal, a two-place decimal with fraction,
and now we want to change it to a percent.
So now that we do have a decimal point,
we move it two places towards the percent, or to the right,
and stick on the percent symbol [%]
So the fraction 37/80 and the percent 46 and 1/4 percent [46 1/4%]
are in fact, the same number written differently.
Another example: Change this number to a percent
accurate to the nearest tenth of a percent.
The word 'accurate' tells me I'm going to round off.
It's to be accurate tenth but tenth of a percent,
so we know as we divide this fraction and begin to get a decimal,
I need two places to get to the percent.
Then I need a tenth of that percent.
Then I need one more place to round up or down.
So tenth of a percent means, in fact, decimal wise
I need to divide four places if that's possible.
Now what about the 1? Well this is 1 and 5/16.
As a decimal we would say
"One, and" and the decimal point becomes the 'and'
so we simply do this division and stick it on against that 'and' [.]
Doing that we get
this,
so we have 1 and [ 1. ]
5/16 and our 5/16 has become .3125.
Now we wish to change this to a percent,
so we move the decimal point two places towards the percent,
and we get 131 point, [ 131. ]
but we want it to the nearest tenth of a percent.
The digit just to its right is 5 or more [5]
so we drop it and round this up,
which gives me this [ 131.3% ] as my percent.
Now remember 1 is the same thing as 100 percent.
So this is more than one,
so our percent equivalent should be more than 100 percent,
and this is. [ 131.3% > 100% ]
So remembering this [ 1 = 100% ] sometimes gives you
sort of a quick and easy way to check
to see if your answer is at least in the ballpark.
As a matter of fact,
here are six more fractions or whole numbers
it's well to memorize.
1/4 is 25 percent. [ 1/4 = 25% ]
1/2, 50 percent. [ 1/2 = 50% ]
3/4, 75 percent. [ 3/4 = 75% ]
The whole number 1, 100 percent, and also the phrase "all of it,"
if we were to say, "All of our group went to the fair,"
we're saying 100 percent of our group went to the fair.
1 and 1/2 percent is 150 percent, [ 1 1/2 = 150% ]
and 2, which is twice 1, would actually be
twice 100 percent or 200 percent, [ 2 = 200% ]
but also very helpful just remember the word 'twice.'
If something costs 'twice' as much as it used to,
it cost 200 percent as much as it did originally.
So remember, twice or 200 percent is the number 2.
All of something or 100 percent is the number 1.
By using these certain key equivalents,
we can check to see if certain answers
seem to be at least in the ball park.
For instance, our last problem.
This is somewhat more than one, [ 1 5/16 ]
so we know our answer should be somewhat more than 100 percent
and indeed it was somewhat more than 100 percent.
So at least if we had made a mistake and had 13 percent,
we would know by this comparison that we're out of the ball park.
That's not reasonable.
So again, these are handy things to remember.
If you're to be a business, accounting, or bookkeeping major,
then you'll have a list
of about a dozen-and-a-half fraction to percents
which ultimately must be memorized.
Let's take what we've just done
and then backtrack and review the last two lessons
and begin to tie the three together
because a common error made by students at this point
is to become somewhat confused
about what is happening at any one moment.
To begin with, if you're going
from a decimal representation to a percent,
we have stated that the key was the words "to a percent."
And that you simply move the decimal point
two places to or towards where the percent is going to be,
and then you just stick the percent sign on.
Now remember percent means hundredths,
so we saying roughly 13 hundredths,
and roughly we do have 13 hundredths here, don't we?
So always check back
"into the ball park" as we've been calling it,
to see if this seems reasonable
with what we know about what percent means,
that is percent means per hundredth.
Okay, so if you have only one decimal place or even less,
the key still is, you move two places towards the percent.
So you actually move it two places towards the percent,
and if in so doing if you create extra places,
you fill those places with placeholder zeros
and stick that percent [%] on.
Again checking our 'ball park' thinking, two hundred means 2.
230 means more than 2, and this is indeed more than 2.
Again, if you're changing TO a percent,
you move the decimal point two places towards the percent
and stick the percent sign on.
Now if you get an unusual number,
this is quite a small percent, then so be it.
The rule, you always movement decimal point
two places towards the percent is inviolable,
so never change that rule just because your results look strange.
That's just the nature of your results.
Okay, so from going from decimal to a percent
the key was to move towards the percent, two places of course.
And if you had a percent to begin with,
and you wanted to go back to the pure decimal,
now note I keep saying 'pure' decimal
because in fact these percents have a decimal already
along with the percent.
Don't they? Including this one.
Because remember whole numbers
always have a decimal point to the right end.
It's just we don't bother writing it.
On this problem we said the key to help remember this
was, that you really want to get away from this percent.
So concentrate on the thought
that I want to get away from the use of this percent.
Then in that case we'd drop the percent
move it two places away.
Of course that's just the reverse of what we did previously.
Again, drop it, move it two places away,
and if in so doing it you create extra places,
you insert place holder zeros,
and then not to be concerned because it looks rather strange.
It's this terribly small number.
That's just the way it comes up, and this is the correct answer.
And again, if you want to get away from the use of the percent,
you drop the percent and move it to two places away.
I've created an extra place, so I insert the place holder zero,
and of course there is no percent.
And finally our current lesson, changing fractions to percents.
Since we've already covered changing decimals to a percent,
and prior to that fractions to decimals,
we simply go through decimals first
and then from decimals to percents.
And to change to a decimal,
we just remember that this fraction bar
is another way of saying 'divided by.'
So we essentially have a division problem.
7 divided by 13. [ 7 ÷ 13 ]
But many fractions, when you change to decimals,
will in fact be non-terminating decimals;
that is they're going to go forever.
So at some point they must tell you where to quit,
whether you are going to round off at a designated spot,
or whether you're going to have a fraction
with that decimal at a designated spot.
But remember that
if we're going to go from this decimal to a percent,
ultimately we're going to move the decimal point
two places towards the percent.
There's two places that we need.
So always when you're dividing, divide at least two places,
decimal places,
so that you can move the decimal point those two places
to go a percent.
But more specifically, watch for the directions
because usually the directions,
depending on what you're going to use this number for,
will specify where they want you to stop.
Let's say that you were requested
to round to the nearest hundredth of a percent.
Here we cautioned you that you not stop in your thoughts
at the word hundredth
because that would tell me to stop here, and round here.
But they want not the nearest hundredth decimal wise
but the nearest hundredth of a percent.
So I will have to move the decimal point two places
to get to a percent,
then two more places to get to the nearest hundredth of a percent
because the percent decimal point will be here,
so we go one more place to decide do I round up or down.
So in order to have this accurate
to the nearest hundredth of a percent,
I'm going to have to perform this division out five places,
not the two that a beginner might think of
because this is the nearest hundredth period,
the nearest hundredth of a decimal,
but we want the hundredth of a percent
that is going to end up being.
So doing this division we get
this decimal equivalent of this fraction,
changing it to a percent,
I move the decimal point two places towards the percent.
So it's going to end up here at 53.8
but I want it rounded to the nearest hundredth of a percent,
so the digit just before that is 6, which is 5 or more,
so I will add up and make that 85, and since it is a percent,
I stick the percent symbol on to indicate that.
So 53 point 58 percent. [ 53.58% ]
Now let's see if this seems reasonable,
that is what we've been calling 'in the ball park.'
50 percent is a little bit, is half.
So 53 percent is just a trifle over a half.
And 7/13 is a trifle over a half,
so this, at least, seems reasonable
assuming that our division was done properly.
Here's a very real problem I extracted from a business document.
This actually was needed in a business transaction.
We want to change the fraction 3/17 to a percent,
but a percent that has a fraction at the tenth of a percent.
This is involving everything:
the decimal, percent, and fraction.
So we start off first concentrating on the fact
that I wish to change it to a percent,
which means I'm going to perform this division.
So let's actually begin to do that,
then we'll think our way further through this
to see how far into this division I go.
So let's actually do this rather slowly together
so you can actually see the process and thinking that takes place.
So this reads 3 divided by 17
I'm going to insert quite a few zeros.
Now to find out how many zeros,
that is how far must I go through the division,
let's just put in four or five to begin with
hoping we won't have to do all of them,
but then reminding ourselves
that we're going to ultimately change to a percent,
which means that the decimal point
is going to move two places towards the percent.
But then this says I want a fraction,
which means at some point I'm going to take my remainder,
write it over my divisor, but the question is where?
And it says at the tenth of a percent,
so if my decimal point is going to move
from here to here to get to the percent,
then one more place will give me the tenth of a percent,
and at that point, I want to quit and take the fraction
and have a fraction at the tenth of a percent,
which will actually be at the thousandth of the original decimal.
Do you see that? It can be very confusing
if you're not thinking
very, very slowly, carefully, and methodically.
So now, beginning our division:
17 into 30 is 1. [ 30 ÷ 17 = 1 ]
17 into 130 is 7. [ 130 ÷ 7 = 7 ]
That's 49, carry the 4.
That's 11.
17 into 110 is 6. [ 110 ÷ 17 = 6 ]
That's 42, carry the 4.
Now my remainder is 8, so I write the remainder over the divisor.
Now I've changed this fraction to a decimal
with fraction at the nearest thousandth.
Now changing that decimal to a percent,
I move the decimal point two places
towards where I want the percent to be,
and I get 17.6 and 8/17,
and because I moved it towards the percent,
I stick the percent sign on, so I have a percent,
but it's a percent with a fraction at the nearest tenth.
Now again let's check our ball park reasonableness of this.
17 percent is a little bit more than 10 percent,
and 10 percent is 1/10, and this [ 3/17 ] certainly
is a little bit more than 1/10,
so that at least seems to be within reasonable magnitude.
Did you follow this?
If not you might want to rewind this tape
and watch that portion once again.
Let's say that all of that was somewhat confusing to you,
but you were good at solving proportions.
Then let me show you how knowing how to solve proportions
can help you make all the decisions we made more easily perhaps.
First restate this question that I'm using words for here
in mathematics, that is, as a proportion.
Now what do I mean by that? Well, watch.
What are we really saying here?
We're saying 3/17 is what percent? Well, let's say that.
3/17
is
what?
And 'what' in algebra is always a variable.
Something we don't know
so we use a letter to stand for that temporarily.
But how do we say percent? Well, there's my per [/]
And there's my cent. [/100]
3/17 is what percent? [ 3/17 = x/100 ]
And I know that when I get done,
this will be a percent because that's exactly what I'm asking.
Okay. So now we simply solve a proportion,
which we've been reviewing the last couple of lessons.
So we know that we would multiply these two,
which obviously is 300, and divide by 17.
And my answer will be in percent form.
So notice as I divide this the decimal point now is here.
I'll move it straight up and that's going to be in percent form,
so let's just go ahead and stick the percent sign on it.
But now the problem says
it wants me to go to the nearest tenth of that percent,
so I'm going to go one more place
and then give my remainder as a fraction
at that tenth of a percent.
So I just follow through
on what the proportion really is directing me to do.
Now 17 into 30 again,
and see I'm just repeating my previous work.
And I'm all done.
When you begin to think in mathematical sentence,
and in particular in this case proportions,
you don't have to worry
about all that remembering our rules we used here.
You can actually state the problem mathematically,
and the mathematics itself
will direct your attention and your moves.
Like this one.
Write 5/6 as a percent accurate to the nearest tenth of a percent
This is telling you what they want.
This is merely telling you how accurate they want it.
So I want 5/6 is what percent?
And if you are allowed to use a calculator,
and if you know how to solve for percents,
you're going to multiply this, which is 500
divide by 6,
and that is the percent answer,
but they want it accurate to the nearest tenth of a percent,
which is this.
This is percent form already.
So we say 83 point and there is .33
so the digit just to the right of the tenth is less than 4,
so I round down.
So this is 3, and it's already a percent,
so I simply stick on the % to indicate this.
Easy? I hope so.
This is your host Bob Finnell, we'll see you at the next lesson.