>> Welcome to the Cypress College Math Review
on fractions, decimals, and percentages.
Objective one, convert between fractions and decimals.
Fractions and decimals are just two different ways
of writing the same numbers.
When we see 0.7, most people do call it that,
0.7, or zero point seven.
But that's not technically correct.
The correct way to say the number is seven-tenths.
This is the tenths place.
So seven-tenths.
Now close your eyes.
What do you picture when you say seven-tenths?
Yeah, seven over 10.
That's what most people see.
Seven over 10.
So if you simply say a decimal number,
the technically correct way,
then you can easily convert it to fractional form.
I'm not suggesting that you should always read 0.7
as seven-tenths, just when you want to convert it to a decimal.
4.1, so that's four and one-tenth.
So as a fraction, one-tenth.
Four and one-tenth.
This is the hundredths place,
so this would be read the technically correct way
as 36-hundredths.
So as a fraction, that's 36 over 100, 36-hundredths.
Yes, it still needs to be simplified,
but at least we have it in fractional form.
Now to change fractions into decimals, the other way around,
we divide the denominator into the numerator.
Yes, the bottom into the top.
To review, to change decimals into fractions,
say it the technically correct way.
And to change fractions into decimals,
divide the bottom into the top.
Example. Fill in the following chart converting
between fraction and decimal numbers.
Four-fifths.
We divide five into four.
Now five does not go into four, so we put the decimal point
in both places and add a zero.
Five does go into 40 eight times.
There's no remainder, so we're done.
Four-fifths is equal to eight-tenths, 0.8.
So we've changed the fraction into a decimal.
Now for four and two-thirds.
Our answer will be four point something,
so it's the two-thirds that we have to change into a decimal.
Let's divide three into two.
Now three does not go into two,
so we put down the decimal points and a zero.
Three goes into 20 six times, and the remainder is two.
Tack on another zero.
Three goes into 20 six times, leaving a remainder of two.
This process will continue forever.
So we have 0.666, and let's just round that off
to two decimal places.
Let's call it 0.67.
Therefore, four and two-thirds is equal to four
and 67-hundredths, or more simply 4.67.
Now let's change 5.2 into a fraction.
Oh, wait a minute.
We said it wrong.
Let's not say it that way.
Five and two-tenths.
Close your eyes.
Two-tenths, that's two over 10.
So we have five and two-tenths.
Now we need to reduce two-tenths to one-fifth.
So five and two-tenths, or 5.2 is equal to five and one-fifth.
We've changed this decimal number into a mixed number.
But that is a fraction.
Next one, 0.85.
Don't say it that way.
This is a hundredths place, so 85-hundredths.
If you close your eyes,
85-hundredths, that's 85 over 100.
Now let's reduce that.
What goes into both?
Five. Five goes into 85 17 times.
And five goes into 100 20 times.
So this reduces to 17 over 20.
So 85-hundredths or 0.85 is equal to 17-twentieths.
Pause the video and try these problems.
Restart the video when you're finished.
Objective two.
Conversion between decimals or fractions and percents.
The word per cent means parts of 100.
This means that percent can be considered to be a fraction
with a denominator of 100.
Example, write 53% as a fraction and as a decimal.
53% means 53-hundredths.
Close your eyes again.
53-hundredths.
You've already got it in a fractional form.
53 on top with 100 on the bottom.
We have our fraction.
Now let's change it into a decimal.
Recall that when you divide by 10,
you move the decimal place to the left once.
If you divide by 100, you move the decimal place
to the left twice.
53 divided by 100 then, would be 0.53.
Once again, if you say it correctly,
you won't know whether you're referring to the fractional
or the decimal form of the number.
53-hundredths.
So 53% equals 53-hundredths or as a decimal, 53-hundredths.
When the denominator of a fraction is 100,
it's pretty simple to change between the forms.
So in the fraction we have three-hundredths.
So three-hundredths is three-hundredths as a decimal.
There we go.
We're good.
0.03. As a percent, well, since percent means out of 100,
three-hundredths, that's 3%.
Decimals. 0.67.
So that's 67-hundredths.
As a fraction, 67-hundredths.
As a percent, 67%.
All right.
We've got 49%.
So percent means out of 100, so 49-hundredths.
As a fraction, 49 over 100.
As a decimal, 49-hundredths or 0.49.
100%. So that's 100 over 100, which reduces to one.
So as either a fraction or a decimal, 100% is one.
We know that the following are equivalent, but how is it
that we convert between fractions
and decimals and percents?
So 53% equals 53-hundredths as either a fraction or a decimal.
53-hundredths.
So it looks like we need to divide by 100.
So to change from a percent into either a fraction
or a decimal, we divide by 100.
And then we leave off the percent symbol.
To change from a fraction or a decimal into a percent,
then we must do the opposite operation and multiply by 100.
And then, of course, we write the percent symbol.
So 53-hundredths times 100.
Now write the percent symbol.
This equals 53%.
53-hundredths times 100,
write the percent symbol, this equals 53%.
Example. Change 4.2 into a fraction and a percent.
Well, first of all, let's say it correctly.
Four and two-tenths.
So four and then a fraction of two over 10.
Now two-tenths reduces to one-fifth,
so we have four and one-fifth.
There we go.
We have it as a mixed number, and that's what we meant
by changing it into a fraction.
Now let's change it into a percent.
So to change it to a percent, we need to multiply by 100
and write the percent symbol.
Multiply by 100 and what does that do?
That moves the decimal two places to the right.
So we have 420%.
Example. Change one-third into a decimal and a percent.
Well, let's first change it into a decimal.
So we need to divide the numerator by the denominator.
So divide one by three.
So three doesn't go into one,
so we put down the decimal places, we put down the zero.
Three goes into 10 three times, that's nine,
leaving one left over.
Ah, we're going to get another repeating decimal.
So we get 0.33333.
Let's just round that to 0.33.
We'll call that our decimal form.
It's an approximation, but that's fine.
Now, for the percent.
We could take the decimal form and multiply by 100,
which means to move the decimal to the right two places, 33%.
Now since we rounded off,
technically it should be 33 and a third percent.
But 33% is an acceptable approximation.
Or we could take the fractional form one-third
and multiply that by 100.
So 100 over three.
Once again, we get approximately 33.
So 33%, and it doesn't matter which way you do it.
Change 0.7% into a decimal and a fraction.
So this is a very small percent, less than one percent.
We're going to change it first to a decimal.
So to change it to a decimal, we need to divide it by 100.
So we're going to take the decimal point,
which is right here, and we're going
to move it left two places.
And we're going to drop the symbol.
So 0.7% is the same thing as 0.007, or what place is that?
The thousandths place, seven-thousandths.
Now let's change it to a fraction.
Divide 0.7 by 100.
Oh, now we've got a decimal inside a fraction.
Gross. So we need to clear that decimal place.
So we're going to multiply it by the number one,
which looks like 10 over 10, because 10 times 0.7 is seven.
There we go.
So we cleared the decimal.
What we do to the top we have to do to the bottom,
so now we multiply the bottom by 10, and we get 1000.
Seven-thousandths.
But you already knew that answer, didn't you?
Because if you look at the decimal answer
and you say it correctly,
seven-thousandths, you're already there.
Pause the video and try these problems.
Objective three.
Percent equation.
50 is 50% of 100.
Let's translate this into an equation.
Is is a verb, so that's where the equals goes.
Of in mathematics means multiply.
So we have 50 is, which is equals, 50% of,
which is multiply, 100.
50 equals 50% times 100.
And recall that 50% is the same thing as 50 over 100.
And 50 over 100 reduces to one-half.
So another way to think of this is 50 is one-half of 100.
So to take 50% of something,
that's the same thing as taking half of it.
What is 25% of 400?
So let's translate this into an equation.
What, well, we'll call that X. Is, well, that's equals.
25% of, that's multiply, 400.
So we have X equals 25% times 400.
We could change 25% into a decimal
and multiply 0.25 times 400, and if we had a calculator out,
that's probably what I would do.
But isn't there an easier way if we're doing it by hand?
Well, 25% of something is the same as, well, as a fraction,
25% is 25 over 100, which reduces to one-fourth.
So 25% of something is the same thing
as one-fourth of that thing.
Think of money.
A quarter, 25 cents.
25 cents is a quarter of a dollar, isn't it?
So we could just take one-quarter of that thing,
in other words, multiply it by one-fourth.
In general, use the form of the number, either fraction
or decimal, that's easiest.
So X equals one-fourth times 400.
Let's do that.
We can cancel the four.
Four goes into four once.
Four goes into 400 100 times.
So X equals 100.
100 is 25% of 400.
Here are some common percentages
that would be good to know as fractions.
10% is one-tenth, 20% is one-fifth, 25% is one-fourth,
33 and a third percent, or commonly rounded off to 33%,
is one-third, 50% is a half, 66 and two-thirds,
or commonly rounded off to just 67% is two-thirds,
75% is three-quarters, and 100% is one.
When working with these percentages,
it probably would be easier to work with the fractional form
of a number instead of the decimal form.
Example. What's 33% of $6000?
Well, what, we'll call that X, is, that's equals, 33%,
of is times, and then 6000.
So X equals 33% times 6000.
What form of the number is easier here?
One-third or 0.33?
Probably one-third.
So X equals one-third times 6000.
Three goes into three once, three goes into 6000 2000 times.
So 33% of $6000 is $2000.
What's 21% of 400?
Let's translate this into an equation.
What, that's X, is, equals, 21%, of, times, 400.
X equals 21% times 400.
For this problem, the decimal form, which is 21 hundredths,
or 0.21, is the easiest to use.
So we're going to multiply 400 times 0.21 and we get 84.
So 21% of 400 is 84.
Example. What is 75% of $400?
What, that's X, is, equals, 75%, of, times, 400.
So X equals 75% times 400.
Well, 75% of something is the same thing
as three-quarters of that thing.
So I think the fractional form would be easier to use here.
X equals three-quarters times 400.
We cancel the four out of the numerator and the denominator,
so four goes into four once, four goes into 400 100 times.
So now we have three times 100, which is 300.
75% of $400 is $300.
Example. What is 142% of 5000?
So we have X equals 142% times 5000.
For this problem, the decimal form, which is 1.42,
is the easiest to use.
Recall that you divide by 100, which moves the decimal
to the left two places.
That's how I got 1.42.
So we take 5000 times 1.42.
That gives us $7100.
So 142% of 5000 is 7100.
Does that make sense?
It's always a good idea to check to make sure
that your answer is reasonable.
142%. Well, that's 100% plus 42%, isn't it?
Well 100% of 5000 is, well, 5000.
100% of anything is that same number.
42% of it, well, that's almost half of it.
Yeah, that's almost 50% of it.
And what's half of it?
Half of 5000 is 2500.
So 142% of 5000 should be a little under 5000 plus 2500.
Well, that's 7500.
Our answer is 7100, a little under 7500, so it makes sense.
It's reasonable.
Pause the video and try these problems.
Objective for estimating percentages.
It's often helpful to be able to estimate percentages.
This is common when you're trying to figure
out a tip or gratuity.
Example. Estimate a 15% gratuity for a bill of $31.70.
Well, first of all, it's a good idea to just round off your bill
to make it easier to work with.
Let's round $31.70 off to $32.
15%, well, that's 10% plus 5%, so let's break it up that way.
What's 10% of $32?
Well, 10%, that's one-tenth, so we're dividing $32 by 10.
That moves the decimal one place to the left, so we get 3.2.
So 10% of 32 is 3.2.
5% of 32, well, that's half of 10%, right?
5% is half of 10%?
So what's half of 3.2?
That's 1.6.
So 15% would be 10% plus 5%, so we need to take the 3.2,
which was 10% of it, plus the 1.6, which is 5% of it.
3.2 plus 1.6, that's 4.8.
I left off the extra zero just to make it easier to work with.
So the tip should be around $4.80.
Example. Estimate a 20% gratuity for a bill of $78.95.
Well, let's round $78.95 off to $80.
That makes it easier to work with.
20%. Well, that's twice 10%, yes?
So let's find out what 10% of it is, and then we'll double it.
10% of $80, well, that's one-tenth of $80.
Divide 80 by 10 and we get $8.
So 10% of $80 is $8.
20%, then, would be twice that.
What's twice $8?
$16. So the tip should be around $16.
Pause the video and try these problems.
Objective five.
Calculating tax.
Find the sales tax on an E6B computer
if the selling price is $26 and the tax rate is 8%.
So translating that into an equation,
we have tax is 8% of 26.
So tax equals 8% times 26.
Let's use the decimal form of 8%, which is 0.08.
26 times 0.08 is 2.08.
So the tax is $2.08.
A ticket agent must calculate a 10% airport tax
on a ticket of $250.
How much in tax will the customer have to pay?
So tax is 10% of 250.
So tax equals 10% times 250.
When we change it into an equation,
10% is 10 over 100, which is one-tenth.
Let's use this form,
the fractional form, for 10% this time.
Tax equals one-tenth times 250.
We change 10% into one-tenth.
Canceling, we get 25.
The airport tax would be $25.
Pause the video and try these problems.
Objective six.
Percent applications.
All employees of Zezic Airlines are being asked
to take a 5% pay cut.
Your salary is $25,000 per year.
How much of a pay cut will you receive?
Translating that into an equation, pay cut is 5%
of the total, $25,000.
So pay cut equals 5% times 25,000.
If you choose to use a fraction, 5 over 100 times 25,000.
Cancel by dividing top and bottom by 100.
And we have 5 times 250, which gives us 1,250.
If you choose to use a decimal,
5% is five-hundredths, or 0.05 as a decimal.
Multiply 25,000 times 0.05 and you get 1,250.
Your pay cut will be $1,250.
What will your new salary be?
Well, we need to take the original salary
and subtract the pay cut, which was $1,250.
So 25,000, which was the original salary,
minus 1,250 gives us 23,750.
Your new salary will be $23,750.
But there actually is a more straight-forward way
to calculate your new salary.
Since you're getting a pay cut of 5% of your salary,
your new salary will be 100% minus 5%,
which is 95% of your old salary.
New salary equals 95% of old salary.
95% as a decimal is 95-hundredths, or 0.95.
And if we multiply 25,000 times 0.95,
we get the same answer, $23,750.
Example. All employees
at Fly High Airlines are now getting a 2% raise.
Your salary is $25,000 per year.
How much of a raise will you get?
So pay raise is 2% of 25,000.
So pay raise equals 2% times 25,000.
Let's change 2% into a decimal, 0.02 or two-hundredths.
Now we multiply 25,000 times 0.02 and we get $500.
Your raise will be $500.
What will your new salary be?
We need to take your original salary and add in the pay raise.
So we take the $25,000 and add in 500,
and your new salary will be $25,500.
Once again, there is a more straight-forward way
to find your new salary.
Since you are getting a raise of 2% of your salary,
your new salary will be 100% plus 2%,
which is 102% of your old salary.
So your new salary will be 102% of your old salary.
Let's change 102% into a decimal, 1.02.
25,000 times 1.02 equals 25,500.
Once again, your new salary will be $25,500.
Pause the video and try these problems.