PERCENTS
WARNING: There are two prerequisites for learning about percents: knowledge of
decimals and fractions.
There is a series of questions listed below. In order for you to continue,
you should answer yes to all of them. If you are unsure of any of these points,
click on the appropriate question for a review.
DEFINITION OF PERCENT
Imagine an object, either a shape or a number.
Now imagine breaking that object up into 100 smaller pieces.
Guess what? You just performed a percentage problem.
The object that I just referred to, in its entirety, is called the base.
Each piece is one-hundredth of this base.
We could say each piece is one percent of the base.
I'm about to illustrate this concept for you, but first I need to introduce the "%" symbol.
"%" means percent, and as I just wrote 1% = 1/100 of the base.
The base itself constitutes 100%.
In the diagram below you will see a rectangle broken into 100 equal squares.
Each square makes up 1/100 of this rectangle.
The entire rectangle would be considered the base
(i.e. 100% of the rectangle is the rectangle).
Each square is 1/100 of the rectangle, thus it could be said each square is 1% of the
rectangle.
In this next illustration you'll notice that 10 of the squares are colored red.
These squares constitute 10 parts out of the total 100.
Therefore the amount of red squares is 10% of the rectangle.
Frequently we use percents with numbers. In that case the
number that we are taking the "percent of" will be
considered as the base.
Let's say I wanted to know 25% of 1200.
Here 1200 is the base (i.e. 100% of 1200 is 1200)
Each percent would be one-hundredth of 1200.
1200/100 = 120 so 1% of 1200 is 120.
We could write 1% = 120
However, here I'm not interested in 1%, I'm interested in 25%
25% represents 25-hundredths of 1200, so if 1% = 120, then 25%
would equal 25 * 120
25 * 120 = 3000
25% of 1200 is 3000
Let's say I wanted to know 40% of 150.
Here 150 is the base (i.e. 100% of 150 is 150)
Each percent would be 1/100 of 150.
So 1% = 150/100 = 1 1/2 (or if you prefer 1.5)
40% represents 40/100 of 150, so if 1% = 1 1/2,
then 40% would equal 40 * 1 1/2 = 60
40% of 150 is 60
I intend to simplify this process in the third section on this page. However, first you'll
have to get used to converting fractions, percents, and decimals.
FRACTIONS, DECIMALS AND PERCENTS
Decimals, fractions, and percents are three different representations of a number.
You should already know how to change decimals to fractions and back again.
Now I'm throwing percents into this mix.
This section will be broken down into three subsections
Starting with a percent
As I mentions above a percent is one part out of 100.
So we can write any percent as the ratio of "that particular number" over 100.
Thus 35% = 35/100 ; 40% =40/100 ; 65%=65/100 ; etc.
Now obviously the three fractions listed above can be reduced to 7/20, 2/5, and
13/20 respectively.
To change a percent into a decimal, drop the "%" sign and move the decimal point
two places to the left.
35% = 35/100 = 0.35 ; 40% = 40/100 = 0.40 ; and 65% = 65/100 = 0.65
Thus we could write each of the above percentages in the following ways:
- 35% = 7/20 = 0.35
- 40% = 2/5 = 0.40
- 65% = 13/20 = 0.65
Starting with a fraction
The above remarks are fine if you're starting our with a percent, but what if you
are starting with a fraction?
If we started with a fraction we could change it to a decimal by dividing the
numerator by the denominator.
Now we have two different ways to change our fraction to a percent.
We could multiply its equivalent decimal by 100 (move the decimal point two places to the
right)
or
We could change it to a ratio of some number over 100.
(i.e. a/b = ?/100). To solve his problem multiply the fraction by 100 (i.e. ? = 100 * a/b).
That number will be our percent.
Examples
- 1/3
1.000 divided by 3 = 0.333... Thus 1/3 = 0.3333...
1/3 = ?/100.
? = 100 * 1/3 = 100/3 = 33 1/3.
1/3 = 33 1/3%
Or we could have taken the decimal equivalent of 1/3 and moved
the decimal point two places to the right
1/3 = 0.333... Moving the decimal we get 33.333... Thus 1/3 = 33.333... % (which in turn equals 33 1/3%)
- 3/4
3.00 divided by 4 = 0.75 Thus 3/4 = .75
3/4 = ?/100.
? = 100 * 3/4 = 25 * 3/1 = 75
3/4=75%
Remember 3/4 = 0.75. By moving the decimal point in 0.75 two places
to the right we get 75.
Thus 3/4 = 75%
- 7/8
7.000 divided by 8 = 0.875. Thus 7/8 = 0.875
7/8 = ?/100
?=100 * 7/8 = 25 * 7/2 = 175/2 = 87 1/2
7/8 = 87 1/2%
Remember 7/8 = 0.875. Multiplying 0.875 by 100 we get
87.5
7/8 = 87.5% (which in turn equals 87 1/2%)
Starting with a decimal
Now it's time to consider the last of our three cases.
To change the decimal into a fraction place its corresponding whole number over the
appropriate power of ten (If the decimal has one decimal place, put it over 10 ; if it
has two decimal places, put it over 100, if it has three decimal places, put it over 1000, etc.).
Now reduce this fraction to lowest terms.
To change the decimal to a percent, multiply it by 100 (move the decimal two places to the right)
Examples
- 0.50
0.50 * 100 = 0.500 *100 = 50.0 = 50 Thus 0.5 = 50%
0.50 = .5 = 5/10 = 1/2 (since .5 has one decimal place we place 5 over 10)
0.50 = 1/2 = 50%
- 0.025
0.025 * 100 = 02.5 = 2.5 Thus 0.25 = 2.5%
0.025= 25/1000 = 1/40 (since .025 has three decimal places we place 25 over 1000)
- 0.05
0.05 * 100 = 0.050 * 100 = 5.0 = 5 Thus 0.05 = 5%
0.05 = 5/100 = 1/20 (since .05 has two decimal places we place 5 over 100)
Percents are probably the most useful concept within this entire tutorial.
No matter where you go, you will see percents used.
I will attempt to illustrate that in the examples and exercises within this unit.
But first you need to practice converting fractions, decimals, and percents.
Below you'll see a table.
On each row only one figure is supplied, either a percentage, a decimal, or a fraction.
Fill in the missing data, and then check your answers.
Note: Make sure your fractions are in lowest terms.
| PERCENT |
DECIMAL |
FRACTION |
CHECK THE ANSWERS ON THIS ROW |
RESET THE BOXES IN THIS ROW |
|
|
|
|
|
|
|
|
|
PERCENTS OF A NUMBER
There is basically only one equation (or formula) throughout percents.
BASE * PERCENT = AMOUNT
The BASE is again the number (or object) that we are working on. It constitutes 100%
Each PERCENT is 1/100 of the base.
The percent can be expressed as either a fraction or a decimal.
To see this more clearly, let's look a few examples
Example 1
What is 25% of 1200?
1200 is the base.
25 is the percent. 25% = 25/100 = 1/4 or 25% = 25/100 = 0.25
1200 * 1/4 = 3000
or
1200 * 0.25 = 3000
25% of 1200 is 3000
Example 2
What is 40% of 150?
150 is the base.
40 is the precent. 40% = 40/100 = 0.40 or 40% = 40/100 = 2/5
150 * 0.4 = 60.
or
150 * 2/5 = 60.
40% of 150 is 60
What are some real-life examples of percents? Well consider the following:
When dining out it is customary to leave a 15% tip. So how much would you tip
if the bill came to $35.00?
A store charges a 5% sales tax. So much will the tax be on a $2.00 item?
Let's attack them one at a time.
So your dinner came to $35.00 and you want to tip 15%.
$35.00 is the base
15% is the percent. 15% = 15/100 = 0.15 or 15% = 15/100 = 3/20
35 * 0.15 = 5.25
or
35 * 3/20 = 7 * 3/4 = 21/4 = 5 1/4 (which in turn would equal 5.25)
So your tip should come to $5.25
A product costs $2.00 and there is a 5% sales tax.
$2 is the base.
5% is the percent. 5% = 5/100 = 0.05 or 5% = 5/100 = 1/20
2 * 0.05 = 0.10
or
2 * 1/20 = 1 * 1/10 = 1/10 (which in turn equals 0.10)
The tax comes to $0.10
In the next section we'll continue this discussion by going to a clearance sale and
you will get a raise (congratulation).
But first practice what you have just learned.
In each exercise give your answer as a decimal.
In order for the computer to check your answer, please do not include any commas.
So if your answer was 25,340 enter it as 25430.
Exercise 1
A museum is selling your artwork. Whenever they sell one of your paintings, you
receive 75% of the purchase price.
How much will you get from a $125 sale?
Go to the top of this page
Go to the percent home page
Go to the next problem.
Go to the next section
Exercise 2
An election is coming between John Smith and Bill Jones.
According to a recent survey 55% of the people surveyed support John Smith.
The sample size was 12000.
How many people support John Smith?
Go to the top of this page
Go to the percent home page
Go to the next problem.
Go to the next section
Exercise 3
At a restaurant your bill comes to $22.40. You are planning to leave a 15% tip.
How much should the tip be?
Go to the top of this page
Go to the percent home page
Go to the next problem.
Go to the next section
Exercise 4
Your state has a flat-rate income tax of 2.5%
You earned $32,350 last year.
How much will your state income tax be?
Go to the top of this page
Go to the percent home page
Go to the next problem.
Go to the next section
Exercise 5
The interest rate on your savings account is 1.75%
How much interest will your get on $25,000?
Go to the top of this page
Go to the percent home page
Go to the next section
INCREASES AND DISCOUNTS
Remember our tip example above when you left a $5.25 tip on a $35.00 bill?
Well your total expenditure for the night came to $35 + $5.25 = $40.25.
Remember our tax example above when a $2 item had a $0.10 tax? If we added the
tax to the cost of the item, the final cost would be $2.10.
These were examples of using percentages for an increase.
In general whenever you have to do an increase, such as the ones above,
multiply your base number by the percentage, and then add this product to the base figure.
Another glorious example of increases involves getting a raise.
Suppose you earn $35,460 annually. And you're about to get a 7% raise. How much will
you be earning?
What we have to do is take 7% of 35,460 and then add this to the
original base figure of 35,460.
7% of 35,460 = 0.07 * 35,460 = 2482.20
35460 + 2482.20 = 37942.20
You will be earning $37,942.20
The opposite of increasing with percentages is to discount (or decrease) using percents.
To discount a number, you will again multiply this number
by the percentage, but this time you'll
subtract it from the original number.
Probably the most common example of discounting comes regarding sales.
Suppose an item that usually sells for $25 is on sale at 40% off. What will its
sale price be?
Here the base number is 25.
40% of $25.00 = 25 * 40/100 = 25 * 2/5 = 5 * 2/1 = 10
So the price is being discounted by $10.00
25 - 10 = 15.
Thus the sale price is $15.00
Let's look at one more example of decreases (i.e. discounts)
The popularity of a presidential candidate is going down. According to a poll,
after the last debate the number of people who said they would vote for him dropped by 12%.
Before the debate 110,250 (out of 200,00) supported him.
How many people support him now?
Basically we have to discount 110,250 by 12%.
110,250 * 0.12 = 13230
110,250 - 13,230 = 97,020
97,020 people currently support the candidate.
On the next page you'll actually figure out percents, but first it's time for practice.
In each of the exercises listed below, you'll be asked to increase or
decrease a number using percentages.
To help you along, there will be two steps to each problem.
First take the percentage of you base number.(Step 1)
And then add it to you base number, or subtract it from your base number.(Step 2).
Give your answers as decimals
In order for the computer to check your answer, please do not include any commas.
So if your answer was 25,340 enter it as 25430.
Exercise 1
Your going to a 60% off sale to get an item that usually sells for $125.50.
How much will you pay for it?
Go to the top of this page
Go to the percent home page
Go to the next problem
Go to the next page
Exercise 2
Good news! Your bills are actually going down!
Your insurance bill is being reduced by 3% (hey, it's not much, but every little bit counts)
You currently pay $85 a month.
How much will you be paying?
Go to the top of this page
Go to the percent home page
Go to the next problem.
Go to the next page
Exercise 3
Not everything is so rosey. Your insurance may be going down, but
your gas bill is going up. (Isn't that always the way-whenevr one bill goes down and
another goes up)
The gas company just announced that they are raising rates by 4.5%.
Your average bill comes to $58.00.
What will you be paying?
Go to the top of this page
Go to the percent home page
Go to the next problem.
Go to the next page
Exercise 4
In the last contract, your union negotiated a 6% raise.
You currently make $722.50 per week.
What will your paycheck look like after the raise?
Go to the top of this page
Go to the percent home page
Go to the next problem.
Go to the next page
Exercise 5
According to the ratings, on the week of May 13, 2002 a certain television
series received a 4.6 share.
(each share represents a million households)
The very next week its popularity fell by 15%.
(Entertainment is such a fickle business-isn't it)
What was its share the week of May 20th?
Go to the top of this page
Go to the percent home page
Go to the next problem.
Go to the next page
Exercise 6
An item costs $1.60. There is a 5% sales tax. How much will it cost including the tax?
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Go to the percent home page
Go to the next page
