>> Hello, and I'm glad you've joined us for the Cypress College Math Review
on percent applications.
Objective one: Percent equation: 30 is half of 60
and another way to say half is 50%.
So 30 is 50% of 60.
We want to translate this into a mathematical equation.
The word is is our verb.
That's where the equal sign goes.
Of in mathematics is multiplication.
So 30 equals 50% times 60.
Recall that several percentages are easier
to work with as fractions.
Here's some examples: 10% is the same thing as one-tenth;
20% is one-fifth; 25% is the same at one-fourth,
33-1/3% is one-third, but we often just approximate 33%
for one-third; 50% is one-half; 67% is a common approximation
for two-thirds, 75% is three-fourths, and 100% is one.
Example: What's 75% of 400?
What, we'll call that x, is, so equals,
75% of, so multiply, 400.
Now we have an equation.
Seventy-five percent is the same as 0.75.
We could multiply by 0.75, but isn't there an easier way?
Seventy-five percent of something is the same thing
as three-quarters of that thing.
So we can multiply by either 0.75
or three-quarters, whichever is easier.
Use the form of a number, fraction or decimal,
that's easiest for you.
Let's use three-quarters this time.
Our equation becomes x equals three-quarters times 400.
We can cancel four out of four, which gives us one,
four out of 400 gives us 100.
So now we have three times 100, which is 300.
Therefore, 300 is 75% of 400.
Seventy is 40% of what?
Let's translate that into a mathematical equation.
Seventy is, so equals, 40% of, means times, what,
well there we'll put our variable,
x. So 70 equals 40% times x. Let's write 40% as a decimal.
That's probably easier
than using the fractional form for this one.
As a decimal 40% is 0.40.
So 70 equals 0.40 times x. We want to solve for x.
So divide both sides by 0.40, and we get that x equals 175.
Hence 70 is 40% of 175.
Ninety-five is what percent of 190?
Translating that into an equation,
we get 95 equals x times 190.
We can rewrite the right side of our equation as 190x
since order doesn't matter in multiplication.
To solve for x we divide both sides by 190.
95 divided by 190 is 0.5.
Now we change that to a percent, 0.5 is 50%.
Our answer is 95 is 50% of 190.
Let's take another look at this problem.
Ninety-five was what percent of 190?
Ninety-five is part of 190.
We could call 95 the part and 190 the whole in the problem.
So our percent equation could be written
as part equals percent of whole.
The whole is the value that comes after the word of.
Pause the video and try these problems.
Objective Two: Percent Proportion.
One-half equals 50%.
Percent means out of 100.
One-half equals 50 out of 100.
One out of two equals 50 out of 100.
Well that makes sense.
If we were to reduce the fraction 50/100,
we would get one-half.
One out of two.
One is the part of two.
So one out of two represents part out of whole,
50 out of 100 represents 50 percent.
This gives us our percent proportion.
Part out of whole equals percent out of 100.
Example: 20% of 75 is what?
Part is percent of whole.
Here's our percent proportion, the verb is separates the part
from the rest of the pieces.
The part is unknown.
Let's call that x. The percent is 20%.
The whole comes after the word of.
So the whole is 75.
Our percent proportion would be x over 75 equals 20 over 100.
Now to solve, we cross multiply.
X times 100, that's 100x; 75 times 20 is 1500.
Now 100 is being multiplied times x. So we need
to divide both sides by 100 to solve for x. Divide both sides
by 100, 1500 divided by 100 is 15.
Hence 15 is 20% of 75.
Notice that the percent proportion is just another way
to do percent problems.
You can do them using the percent equation
or percent proportion.
Use the method that's easiest for you
or for the problem that you're doing.
Sixty is 15% of what?
Here's our percent proportion.
Part is percent of whole, so our part is 60; the percent is 15,
whole then is the unknown.
We'll call that x. Put these into our proportion.
Sixty over x equals 15 over 100,
part over whole equals percent over 100.
Now let's solve the proportion.
Cross multiply.
Sixty times 100 is 6000, x times 15 is 15x.
Now divide both sides by 15, 6000 divided by 15 is 400.
Hence 60 is 15% of 400.
Example: Eighty is what percent of 500?
Part is percent of whole, so our part is 80;
percent is our unknown, we'll call that x; the whole is 500.
Eighty over 500 equals x over 100.
Now let's solve it.
Cross multiply.
Eighty times 100 is 8000, 500 times x. Now divide both sides
by 500, 8000 divided by 500 is 16.
Therefore 80 is 16% of 500.
Sixty is twice 30.
Another way to say that is 60 is two times 30.
If we convert 200% into either decimal or fraction,
you get the number two.
So two is equal to 200%.
We can then rewrite our sentence as 60 is 200% of 30.
The part is 60; the percent is 200%, and the whole is 30.
In this case, the part is bigger than the whole.
The words part and whole can definitely be confusing.
The whole is the original,
what you started with, or what it is out of.
There are many cases where you end up with more
than what you started with.
Example: What percent of 200 is 450.
Four hundred fifty is our part.
It's by itself on the other side of the verb.
Percent is our unknown, that's x. The whole comes
after the word of, so the whole is 200.
Since the part is bigger than the whole, definitely,
450 is bigger than 200,
our percent will be bigger than 100%.
So we have 450 over 200 equals x over 100.
Now we solve.
Cross multiply, 45,000 equals 200x, divide both sides
by 200 and we get 225.
Therefore 450 is 225% of 200.
Pause the video and try these problems.
Objective Three: Percent applications.
Example: McKenzie has 9% of her earnings automatically deposited
into her savings account.
This way she puts $270 into her savings account each month.
What's her monthly earnings?
Let's use the percent equation for this problem.
We should reword the problem to make it sound
like part is percent of whole.
McKenzie's savings are 9% of her earnings.
So savings are 9% of earnings.
The savings was the $270.
So 270 equals 9% of earnings.
Our unknown is her earnings.
We'll call that x. 270 equals 9% times x.
As a decimal, 9% is .09.
Be sure and move the decimal place twice.
270 equals 0.09x.
To solve, we divide both sides by .09,
and when we divide, we get $3000.
So McKenzie makes $3000 per month.
Ethan had budgeted $320 for textbooks.
But he ended up spending $375.
The amount that he spent was what percent of his budget?
Round your answer to the nearest tenth of a percent.
Again let's use the percent equation.
Let's reword the sentences.
The amount that he spent was what percent
of the budgeted amount?
$375 is what he spent.
The percent is our unknown.
We'll call that x, and he had budgeted $320.
So 375 equals 320 times x. Solve the equation.
Divide both sides by 320 and we get approximately 1.171875.
We need to round our answer to the nearest tenth
of a percent, but don't round yet.
Change this to a percent first.
So 1.171875 as a percent would be 117.1875%.
I move the decimal place to the right twice.
Now we want to round to the nearest tenth of a percent.
A tenth of a percent.
Well the tenth's place is where this one is,
so we look next door to the eight.
Is that a five or greater?
Yes. Therefore the one becomes a two.
So we rounded up to 117.2%.
Ethan spent 117.2% of what he had budgeted.
Owen earned a score of 85% on his test.
There were 80 problems on the test.
How many did he get correct?
For this problem, we'll use the percent proportion.
It doesn't matter.
I just wanted to show both methods.
Let's rewrite it.
The number he got correct was 85% of the total number
of problems on the test.
So the number correct was 85% of 80.
Part is the unknown.
That's our x. Percent is 85 and the whole is 80.
So x over 80 equals 85 over 100.
Now we solve it.
Cross multiply, 100x equals 6800.
Divide both sides by 100, 6800 divided by 100 is 68,
so Owen did 68 problems correctly on the test.
A food label indicates one serving
of cereal contains 250 calories
of which 90 calories are provided by fat.
What percent fat does the food source contain?
This is a little confusing since the word of appears twice
and neither time does it mean multiply
with the whole coming after it.
Let's sort this out.
90 is the fat calories, 250 is the total calories,
so 250 is the whole and 90 is the part.
Percent's the unknown.
We'll call that x. In our percent proportion,
90 over 250 equals x over 100.
Now we solve.
Cross multiply, 9000 equals 250x.
Divide both sides by 250.
So x is 36.
The cereal is 36% fat.
A food label indicates that each serving will have 80 grams
of carbohydrates.
That translates into 320 food calories or kilocalories.
If there are a total of 608 kilocalories in each serving,
what percent is carbohydrates?
Wait a minute, 80 grams, 320 kilocalories, 608 kilocalories.
We've got an extra number.
But 80 grams translates into 320 kilocalories.
So we don't use the 80 number.
We're comparing 320 kilocalories
of carbohydrates to 608 kilocalories.
That's the total.
Let's rewrite it.
Kilocalories of carbohydrates equals what percent
of total kilocalories?
As an equation, that would be 320 equals x times 608.
320 equals 608x.
Divide both sides by 608 to solve, and we get 0.525.
We want to change that to a percent,
so we move the decimal place to the right twice.
So as a percent, that's 52.5%.
Therefore the food is 52.5% carbohydrates.
Pause the video and try these problems.
Objective Four: Calculating tax and discounts.
Let's say you want to buy an item that cost $200.
The tax rate is 5%.
How much tax will you pay?
The tax is our part, and the tax rate is the percent,
and the original cost of the item is our whole,
part is percent of whole.
So tax equals tax rate times the original price.
That's our formula to find tax.
For this problem, tax equals 5% of $200.
Now we change 5% to a decimal.
That would be .05.
Tax equals .05 times 200, and we multiply, and we get 10.
So the tax is $10.
Remember that the tax is not 5%.
That's the tax rate.
Unfortunately, in our language, we commonly call it 5% tax.
You will see this all over the place,
but remember tax is money.
It is not percent.
Tax is a percent of a quantity, not the percent itself.
If someone tells you that the tax is 5%, which is .05,
you will not be paying five cents tax.
Tax is 5% of the selling of the selling price.
How much would you end up paying for the item?
If you take the original price plus the tax,
we get the final price.
For this item, the original price was $200.
The tax was $10.
$200 plus $10 is $210.
So you would pay $210 for the item.
Robin bought a printer priced at $267.95.
The sales tax rate was 7.5%.
How much did she pay for the printer?
The original price plus the tax equals the final price.
Recall that we calculate
by multiplying the tax rate times the original price.
The original price was $267.95.
The tax rate was 7.5%.
As a decimal, that would be 0.075.
So here's our equation.
Order of operations says to multiply first.
That gives us 20.10.
This is the tax.
The original price was $267.95 and the tax is $20.10.
We add to get the final price.
The final price is $288.05.
Louis purchased 10 pounds of steak.
His final bill was $73.05.
Sales tax was 8%.
What was the original price of the steak?
Here's our formula.
We calculate tax by multiplying the tax rate times the
original price.
What are we given?
Well his final bill was $73.05.
So that's the final cost.
Sales tax was 8%.
Remember this is often how we say it
in our language, but this is wrong.
The sales tax rate is 8%.
Our unknown is the original price.
Let's call that x. We have our equation all set up now.
What do we do over here on the left though?
We have x plus .08x.
Let's see.
You can add like terms, right?
If I have 2x plus 3x, that would give me 5x.
So you add the numbers in front, they're the coefficients.
So what's in front of this x?
It's a one.
So one plus .08.
So we need to add 1 plus .08.
Line up the decimal places and we get 1.08.
Our equation is 1.08x equals 73.05.
Now divide both sides by 1.08 and we get 67.64
after we round to the nearest cent.
Therefore the original price of the steak was $67.64.
If you added 100% and 8%, you would get 108%.
So here's our percent equation.
Part is percent of whole.
The final price equals 108% of the original price, 108%.
So the 100% is the original price and 8%
of the original price is the tax.
So 108% would include the original price plus the tax.
We could have started here instead.
The final cost was $73.05.
That equals 108% of the original price.
108% is 1.08 as a decimal.
Divide both sides by 1.08 and we get the same answer.
It's just another way to think of the problem.
Janice bought a dress for $37.80.
The dress was on sale for 20% off.
What was the original price before the discount?
For sales, you take the original price and subtract the discount
to get the final cost.
Makes sense?
You pay less.
Just like for tax, to find the amount of the discount,
you multiply the discount rate times the original price.
So here's our formula.
What do we know?
She spent $37.80.
So that's the final price.
The discount rate was 20%.
We don't know the original price.
Let's call that x. So original price, x, discount rate 20%
or .2, original price x again, equals final price, 37.8.
Once again, we need to combine like terms.
We have 1x minus 0.2x.
We need to subtract .2 from one.
Line up the decimal places, and we get .8.
Now we have .8x equals 37.8.
Divide both sides by .8 and we get 47.25
after we round to the nearest cent.
Hence the original price of the dress was $47.25.
Is there another way that we could have done this problem?
Originally, you pay 100% of a produce of course.
If it's discounted by 20%,
that's 100% minus 20%, that gives you 80%.
So the final price equals 80% of the original cost.
That definitely sounds easier.
Final price 37.8 equals 80% times original price,
x. 37.8 equals .8 times x and divide
and we get the same answer.
Let's say that you've been asked by your employer to go
around to all the price tags in the store
and reduce the prices by 20%.
You could go around and figure out 20% of each price,
that gives you the discount.
Then you'd need to subtract that discount from the original price
to get the discounted price.
Or taking away 20% of the price is the selling it
for 80% of the price.
Wouldn't it be quicker to just multiply each price by .8?
Noah bought toner for his printer.
He paid $29.16.
The price tag read $27.00.
What was the sales tax rate?
Since original price plus tax equals final price,
we can subtract the original price from both sides
of that equation and solve for tax.
So tax equals the final price minus the original price.
In our problem, tax equals $29.16 minus $27.00
and that's $2.16.
So the tax was $2.16.
Now we need to find the sales tax rate.
Let's use the percent proportion this time.
The part is tax, which is $2.16, the percent is our unknown,
and the whole is the original price, $27.00.
Always the original.
So we have 2.16 over 27 equals x over 100.
Cross multiply, divide and we get eight.
The sales tax rate was 8%.
Yank's hardware store just put their paint on sale for 15% off.
If the original price for a gallon of latex enamel was $35,
what's the sale price?
Recall that the original price minus the discount gives you the
final cost.
To calculate the discount,
we multiply the discount rate times the original price.
What do we know?
We know the original price was $35
and we know the discount rate is 15%.
We do not know the final price.
Order of operations tells us
to multiply first and then subtract.
Therefore the sale price is $29.75.
Pause the video and try these problems.
Objective Five: Percent increase or decrease.
We often want to find out how much something has gone
up or down.
In 1970, the average price for a loaf of bread was 25 cents.
In 1980, the price was 50 cents.
In 1990, the price was 70 cents.
So the price rose by 25 cents between 1970 and 1980.
And it rose by 20 cents by 1980 and 1990.
About the same, right?
Not really.
The price rose by 100% between 1970 and 1980, but it only went
up by 40% between 1980 and 1990.
It's usually more useful to look at the percent increase
or decrease and not the actual increase or decrease.
To find the percent increase, we take the amount of the increase
and divide it by the original.
Same idea for percent decrease.
Example: The price of oatmeal has gone
up from $4.50 per box to $6.75 per box.
Find the percent increase.
Here's our formula.
We need to find the amount of the increase first.
So we take $6.75 and subtract $4.50.
That gives us $2.25.
That's the amount of the increase or how much the price
of oatmeal has gone up.
We divide that by the original, always the original,
so that would be $4.50.
And we get 0.5.
0.5 as a percent is 50%.
Therefore the price of oatmeal went up by 50%.
Tessie had been paying $65 a month
for electricity at her apartment.
Now she is more careful about turning off lights
and unplugging appliances that are not being used.
Her electric bill has gone down to $50 a month.
What's the percent decrease in her bill.
First we find the amount of the decrease, so we subtract $50
from $65, and that's $15.
To find the percent decrease, we divide the $15
by the original, which was $65.
That gives us approximately .23 or 23%.
Tessie electric bill went down by 23%.
Kiley and Juanita paid a total of $65 for their meal.
Their bill is $54.25.
What was the tip rate that they left for the server?
Well the tip rate is the same as the percent increase.
What they paid above the original.
The amount of increase we subtract $65 minus $54.25,
which is $10.75.
Now we want to find the tip rate.
So we're going to take the $10.75 and divide it
by the original, which was $54.25.
When we divide and we round, we get approximately .20 or 20%.
Kiley and Juanita left a 20% gratuity.
Ray made $1,365 on an investment that he made.
His broker told him that he had cleared 15% on his money.
How much had Ray invested?
Here's our formula.
What are we given?
Well $1,365 is not the original investment or the amount
of money he had at the end.
That's the amount of money that he made as profit.
That's the amount that his money grew.
So that is the amount of the increase.
We're given the amount of the increase
and the percent increase.
We wish to find the original investment.
The percent increase is 15% or .15,
the amount of increase is $1,365,
the original investment is unknown.
So we'll call that x. We have a fraction
on the right, but not on the left.
We can write .15 as .15 over one.
You can put anything over one.
That doesn't change it.
Now we cross multiply, then we divide.
So x equals 9100.
So Ray had invested $9,100.
Pause the video and try these problems.