#1
Introduction to Percent
What is Percent?
The word percent is derived from the Latin words per centum which literally means per
hundred. Percent means how many “hundredths”.
EXAMPLES OF WHERE PERCENT IS USED IN EVERYDAY LIFE
An item is regularly priced at $44.99, but the price has been reduced
by 40%. What is the sale price?
A student receives 125 out of 150 points on an exam. What percent
of the points did the student receive?
A newspaper poll claims that 40% of Americans will vote for a certain
political candidate. What fraction of Americans will vote for this
candidate?
A lawn fertilizer calls for a mixture of water and fertilizer so that the
fertilizer makes up 5% of the solution’s volume. How much fertilizer
should be added to 5 gallons of water?
An employee making $8.00 per hour on a job is given a 4% raise in
pay. What is the increased rate of pay?
Converting Percents to Fraction Form
Many people have difficulty working with percents because percentages cannot usually be
used in calculations unless they are first converted into a fraction or a decimal. After the
percent is converted into fraction or decimal form, it can be used in calculations in the
same manner as any other number.
To write a percent as a fraction, remember that percent means hundredths and use the
following procedure.
PROCEDURE TO CONVERT A PERCENTAGE INTO A FRACTION
1.
Write the percentage over 100 as a fraction.
2.
Reduce the fraction or change the form of the fraction by dividing or
multiplying the numerator and the denominator by the same factor.
Example 1
Write 35% as a fraction. To do this, write the percentage over 100 as a
fraction.
35% = 35 = 7
100
Example 2
20
Write 0.035% as a fraction. To do this, write the percentage over 100 as a
fraction. Then, change the form of the fraction so that the numerator is a
whole number by multiplying by 1000 .
1000
0.035 % =
0.035
100
=
0.035
1000
×
100
1000
=
35
100,000
Converting a Percent to Decimal Form
To convert a percent to decimal form, use the following procedure.
PROCEDURE TO CONVERT A PERCENTAGE INTO A DECIMAL
1.
Convert the percent into fraction form by writing the percentage as
hundredths.
2.
Write the fraction as a decimal by dividing the numerator by the
denominator.
Example 3
Convert 30% into decimal form.
30% =
Example 4
Convert 0.05% into decimal form.
0.05%
Example 5
30
= 30 ÷ 100 = 0.30
100
0.05
100
0.05 ÷ 100
0.0005
Convert 12 1 % into decimal form.
2
12
1
2
%=
1
2
100
12
= 12.5 = 12.5 ÷ 100 = 0.125
100
In this example note that 12
1
was written as 12.5 .
2
#2
Percent and the Rule of Two Places
In the previous section, percents were converted into decimal form by writing the percent
as a fraction with a denominator of 100, and then converting the fraction into a decimal.
When dividing by 100, the decimal point is moved two places to the left. In a similar
manner, a decimal may be converted into a percent by moving the decimal point two
places to the right. Both of these procedures are summarized with the “rule of two
places”.
RULE OF TWO PLACES
To convert a percent into a decimal, move the decimal point two
places to the left.
To convert a decimal into a percent, move the decimal point two
places to the right.
The percent is always the bigger of the two numbers. By
remembering this fact, you will always move the decimal point in the
correct direction.
Example 1
Convert 34% into a decimal.
34% = 0.34
Example 2
Convert 0.50 into a percent.
0.50 = 50.0%
Example 3
The decimal point is moved two places to the right.
Convert 0.50% into a decimal.
0.50% = 0.005
Example 4
The decimal point is moved two places to the left.
The decimal point is moved two places to the left.
Convert 340 into a percent.
340 = 34000%
The decimal point is moved two places to the right.
Note: 340 may be written as the decimal 340.0 .
Note that in each of the previous examples the decimal point was moved two places and the
percent was always the bigger of the two numbers.
HOW TO DETECT AN ERROR IN CONVERTING
In the following example, how would know that an error occurred?
0.025% = 2.5
In this example, the decimal point was moved two places, but the
percent (0.025%) is a smaller number than the decimal (2.5). That is
how we know that the decimal point was moved in the wrong
direction.
The correct conversion is 0.025% = 0.00025 .
Example 5
Convert 343 1 % into decimal form.
2
1
2
In this case, 343 % must first be converted to a decimal percent.
1
343 % = 343.5%
2
Now, move the decimal point two places to the left.
343.5% = 3.435
Example 6
Convert 33 1 % into decimal form.
3
In this problem, the decimal percent is 33.3% .
Moving the decimal point two places results in 0.333 , which we
recognize as 1 .
3
Example 7
Convert 1 1 into a percent.
4
In this problem, the decimal form of 1 1 is 1.25 .
4
Moving the decimal point two places results in 125%.
#3
Writing Ratios as Percents
Converting Fractions or Mixed Numbers into Percents
The procedure to convert a fraction or mixed number into a percent is given here.
PROCEDURE TO CONVERT A FRACTION INTO A PERCENT
1.
Convert the fraction or mixed number into a decimal.
2.
Move the decimal point two places to the right. (Rule of two places)
Example 1
Convert 1 into a percent.
8
1
= 1 ÷ 8 = 0.125
8
0.125 = 12.5%
Example 2
Convert 2 5 into a percent to the nearest tenth of one percent.
6
5
5
2 =2+
6
6
5
2 + = 2 + (5 ÷ 6)
6
= 2 + 0.83 = 2.83
= 283.33 % after the decimal point is moved two places.
This rounds to 283.3 % .
Converting Ratios into Percents
Many percent problems require that you convert a given ratio into a percent. For
example, if you obtain 36 out of 48 points on a quiz, what is your percentage on this
quiz? In this example, the ratio, 36 out of 48 is equivalent to the fraction, 36 . The
48
fraction, 36 is equal to the decimal, 36 ÷ 48 = 0.75 . This decimal is equal to the percent,
75%.
48
To convert a ratio into a percent, use the procedure given on the following page.
PROCEDURE TO CONVERT A RATIO INTO A PERCENT
1.
Write the given ratio as a fraction.
2.
Write the fraction as a decimal by dividing the numerator by the
denominator.
3.
Convert the decimal into a percent by using the rule of two places.
Example 3
If a baseball player has 230 hits out of 824 times at bat, what is the
percentage of hits, rounded to the nearest tenth of a percent?
The ratio is 230 out of 824 may be written as the fraction
230
.
824
230 ÷ 824 = 0.2791 is the decimal equivalent.
0.2791 is equal to 27.91% .
27.91% rounds to 27.9%. Note that the answer had to be calculated out to
a hundredth of a percent in order to round to a tenth of a percent.
Example 4
45 out of 92 dentists surveyed recommend a particular brand of toothpaste.
What percentage of these dentists recommend this toothpaste? Round to
the nearest whole percent.
The ratio, 45 out of 92, may be written as the fraction 45 .
92
45 ÷ 92 = 0.489 is the decimal equivalent.
0.489 is equal to 48.9% .
48.9% rounds to 49%. Note that the answer had to be calculated out to a
tenth of a percent in order to round to the nearest whole percent.
MATH FACT
The words, out of, when used in percent or fraction problems,
indicate division.
#4
Percents, Fractions, and Decimals
In order to work with percents, one must be able to associate the given percent with its
decimal or fraction equivalent. For example, if we wish to find 25% of 400, we could
replace 25% with its fraction equivalent 1 . 25% of 400 = 1 of 400, which is 100.
4
4
Using Percent, Fraction, and Decimal Equivalents
The following are commonly used percents along with fraction and decimal equivalents.
COMMON EQUIVALENTS
Percent
Decimal
Fraction
100%
1.00
1
50%
0.50
1
2
1
33 %
3
0.33
1
3
25%
0.25
1
4
20%
0.20
1
5
10%
0.10
1
10
75%
0.75
3
4
Know these equivalents! You will find percents much more understandable if you are
familiar with these common equivalents. The following examples illustrate the value of
using these common equivalents.
Example 1
On a quiz, you received 30 out of 40 points. What percent of the points
did you receive?
Since 30 out of 40 is equivalent to 3 out of 4 or 3 , the percentage of
4
points you received was 75%.
Example 2
You are in a store shopping for a computer and there is a “10% off” sale.
The price of the computer you are interested in is $2000. How can you
quickly calculate the price of the computer?
You can quickly calculate the reduced price by subtracting 10% of $2000
from $2000. Since 10% is equal to 1 , you subtract 1 of $2000, which is
10
10
equal to the following:
$2000
( 1 of $2000) = $2000 - $200 = $1800
10
Thus, the reduced price is $1800.
Example 3
Fill in the following table.
Percent
Decimal
Fraction
45%
0.006
125%
4
5
The rule of two places and the methods for converting fractions to
decimals can be used to do these conversions.
Percent
Decimal
Fraction
45%
0.45
45
100
9
20
0.6 %
0.006
6
1000
3
500
125%
1.25
25
100
125
100
80%
0.8
1
4
5
4.5
Applications of Percents
Taking a Percent of a Given Number
One of the most common real-life applications of percent consists of taking a percentage
of a given quantity. The procedure to take a percentage is given here.
PROCEDURE TO TAKE A PERCENT OF A GIVEN NUMBER
1.
Convert the percent into decimal form.
2.
Multiply the decimal by the given number.
Example 1
What is 43.2% of 58?
The decimal form of 43.2% is 0.432 .
43.2% of 58
= 0.432 of 58
= 0.432 × 58
= 25.056
Example 2
What is 0.5% of 16?
The decimal form of 0.5% is 0.005 .
0.5% of 16
= 0.005 of 16
= 0.005 × 16 = 0.08
MATH REMINDER
The word “of”, when used with fractions, percents, and decimals
means to multiply.
Percent Applications With Ratios
A common real-life application of percents is converting a ratio into a percent. For example,
if a student scores 68 on a 90 point exam, that ratio could be converted into a percent by
dividing 68 by 90 and then converting the decimal answer into a percent. The procedure for
converting a ratio into a into a percent was given in Section 4.3 and is given again here.
PROCEDURE FOR CONVERTING A RATIO INTO A PERCENT
1.
Write the ratio as a fraction.
2.
Write the fraction as a decimal by dividing the numerator by the
denominator.
3.
Convert the decimal into a percent by moving the decimal point two
places.
Example 3
A student receives 46 out of 50 points on an exam. What percent is this?
46 out of 50 = 46
50
46
50
= 46 ÷ 50 = 0.92
0.92 = 92%
Example 4
A poll of 5000 people revealed that 3400 were in favor of a certain political
candidate. What percent of the people were in favor of the candidate?
3400 out of 5000 = 3400
5000
3400
5000
= 3400 ÷ 5000 = 0.68
0.68 = 68%
Example 5
If there are 750 total points possible in a math class, and a student obtains 644
points, what percent of the points did the student obtain? Round to the
nearest whole number percent.
644 out of 750 =
640
750
640
750
= 644 ÷ 750 = 0.85866
0.85866 = 85.866 % which rounds to 86%.
MATH REMINDER
The words “out of”, usually mean “divided by” when used in percent
or fraction problems.
Percent Increase and Decrease
When an annual budget is increased from 10 million dollars to 14 million dollars, we can
calculate the percent increase by finding what percentage the difference, 4 million dollars,
is out of the original amount, 10 million dollars. Since 4 million out of 10 million is 0.4, the
percent increase is 40%. Percent increase and percent decrease are always calculated using
the procedure given here.
PROCEDURE TO CALCULATE PERCENT INCREASE OR DECREASE
1.
Obtain the difference between the “original” amount and the “new”
amount.
2.
Divide the difference by the original amount.
3.
Convert the decimal answer into a percent.
Example 6
If the population of a certain city increased from 23,000 to 28,000 people,
what is the percent increase? Round the answer to the nearest whole percent.
Here, the difference is 28,000 - 23,000 = 5,000.
5,000 out of 23,000
= 5,000 ÷ 23,000
= 0.217
0.217 = 21.7 %
22% increase
Example 7
A member of a health club weighed 210 pounds before joining the club, and
lost 15 pounds after the first three months of membership. What was the
percent decrease, to the nearest tenth of one percent, in the member’s weight?
The difference is given as 15 pounds.
15 out of 210
= 15 ÷ 210
= 0.0714
0.0714 = 7.14%
7.1% decrease
To calculate the difference, when given a percent increase or decrease, always multiply the
percent increase or decrease by the original amount. This procedure is given here.
HOW TO FIND THE DIFFERENCE WHEN A PERCENT INCREASE OR DECREASE IS
GIVEN
1.
Convert the percent increase or decrease into a decimal.
2.
Multiply the decimal form of the percent increase or decrease by the
original amount.
Note: (Percent Increase or Decrease) × Original = Difference
Example 8
Last years budget of $1,000,000 is increased by 4% for this year. How much
is this years budget increased?
The decimal form of 4% is 0.04 .
0.04 × 1,000,000 = 40,000
This years budget is increased by $40,000.
Often when we purchase an item, there is sales tax added to the price of the item. For
example, if we purchase a $400 stereo and a 6% sales tax is added to the price, we find the
increased price by taking 6% of $400 and adding it to $400. Thus, in this example, the price
with tax is $400 plus 0.06 times $400 which is $400 plus $24 or $424. The procedure for
finding a new amount after a percent increase or decrease is given on the following page.
HOW TO FIND AN AMOUNT THAT IS INCREASED OR DECREASED BY A PERCENTAGE
1.
Convert the percent increase or decrease into a decimal.
2.
Multiply the decimal form of the percent by the original amount to
calculate the difference.
3.
Add or subtract the difference from the original amount, depending
on whether this is a percent increase or decrease.
Note: New Amount = Original ± Difference
Example 9
A scientific calculator is priced at $12.95 . There is a 6% sales tax that is
added to this price at the checkout. What is the total price at the checkout?
6% is equal to the decimal 0.06 .
The difference = 0.06 × 12.95 = 0.777 which rounds to $0.78 .
The price at the checkout is $12.95 + $0.78 = $13.73
Example 10 A stereo shop advertises that all prices are marked 25% off. If a stereo
normally sells for $599, what is the sale price?
The original price of $599 is decreased by 25%.
25% is equal to the decimal 0.25 .
The difference is equal to 0.25 × 599 = $149.75 .
The sale price is $599
$149.75 = $449.25 .
SUGGESTIONS FOR PERCENT APPLICATION PROBLEMS
Always convert percents into decimals before using them in
calculations.
Example:
45% of 50 = 0.45 × 50 = 22.5
When calculating a percent, obtain a decimal answer. Then, convert
the decimal into a percent.
Example:
30 out of 80 = 30 ÷ 80 = 0.375 = 37.5%
Estimation
You will find it beneficial to be able to estimate an answer without actually calculating it for
the following two reasons:
1.
In many real-life situations, you do not have paper and pencil or a calculator available
to actually calculate an answer, yet you wish to obtain an approximate answer to a
given problem.
2.
If you estimate the answer before calculating an exact answer of a percent problem,
you can compare your answer to your estimate. If the two figures are not close in
value, then you know that there was probably an error in your calculations.
Example 11 210 out of 415 people polled were found to favor a particular political
candidate. What percent is this close to?
Since 210 out of 415 is very close to 1 of 415, about 50% of the people
2
favored the candidate.
Example 12 The original price of the automobile was $12,000. The price was reduced by
$1,100. About how many percent was the price reduced?
One-tenth of $12,000 is $1,200. Since $1,100 is very close to $1,200, the
price was reduced by approximately 1 or 10%.
10
Example 13 Estimate what 19.9% of 99 is.
Since 19.9% is very close to 20%, 19.9% may be approximated with 1 .
5
Since 99 is very close to 100, a good estimate of this answer is 1 of 100
5
which is 20.
Example 14 Estimate what 49.8% of 0.2 is by replacing the given percent with a fraction
that is approximately equal to the percent and then multiplying the fraction
by 0.2 .
Since 49.8% is very close to 50%, a good estimate of this answer would be
1
of 0.2 which is 0.1 .
2