% and the BASIC PERCENT EQUATION

% Percents % • Percent means “parts of 100” or “per 100” • A percent can be written using a percent sign (%), as a fraction, or as a decimal

MSJC ~ Menifee Valley Campus Math Center Workshop Series Janice Levasseur

Converting a % to a Fraction • To convert a percent to a fraction, translate the % symbol: • Remove the % sign and divide by 100 (“per 100”)

Ex: Convert 25% to a fraction 25% 



Ex: Convert 27% to a fraction 27%



27 100

Simplify ?



130 Yes! Simplify? 100 Improper Fraction 30 1 Yes! 100 Simplify? 3 1 Simplify? No . . . 10 done 

1 4

Simplify?

YES!

 Simplify? No . . . done

No . . . done

Ex: Convert 130% to a fraction 130%

25 100

1

Ex: Convert 3% to a fraction 1 1 % 3 3 100

1   100 3

Divide fractions

1 100 1 1     3 1 3 100 

1 Simplify? No . . . 300 done

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Ex: Convert 0.5% to a fraction 0.5%

0.5 Divide decimals? 100 Decimal  Fraction 1   100 2 1 100 1 1     2 1 2 100 1  Simplify? No . . . done 200 

Ex: Convert 27% to a decimal 27% = 27

 100  27.0 = 0.27

Ex: Convert 25% to a decimal 25% = 25

 100  25.0 = 0.25

Ex: Convert

Converting a % to a Decimal • To convert a percent to a decimal, translate the % symbol: • Remove the % sign and divide by 100 (“per 100”)  • Move the decimal point two places to the left

Ex: Convert 130% to a decimal 130% = 130  100  130.0= 1.30

Ex: Convert 0.5% to a decimal 0.5% = 0.5  100  0.5

1 % to a decimal 3

• We are starting with a percent written as a fractional percent • First convert the fractional percent to a fraction (drop the % sign and divide by 100)

1 1 1 1  100    3 3 100 300

Fraction form of the answer

= 0.005

Ex: cont. • Recall: To convert a fraction to a decimal number, divide the numerator by the denominator

1 300

.0033

0.0033

300 1.0000 900 1000 900 100

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Converting a Fraction to a % • To convert a fraction to a percent, reverse the procedure for converting a percent to a fraction: • Multiply by 100 and add the % sign

Ex: Convert ¼ to a percent 1  100%  1  100 % 4 4 1 100  % Simplify? Yes! 4 = 25%

Ex: Convert 1 ½ to a percent

Ex: Convert 5/8 to a percent

1 1  100%  3  100 % 2 2 1 300  % Simplify? Yes! 2

Simplify? 5  100%  5  100 %  500 % Yes! 8 8 1 8 4  62 % Simplify? Yes! 8 1  62 % Simplify? No . . . 2 done

= 150%

Ex: Convert 5/6 to a percent Simplify? 5  100%  5  100 %  500 % Yes! 6 6 1 6 2  83 % Simplify? Yes! 6 1  83 % Simplify? No . . . 3 done

Converting a Decimal to a % • To convert a decimal to a percent, reverse the procedure for converting a percent to a decimal: • Multiply by 100 and add the % sign  • Move the decimal point two places to the right

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Ex: Convert 0.36 to a % 0.36  100%  0.36 = 36%

Ex: Convert 0.01 to a % 0.01  100%  0.01 = 1%

The BASIC PERCENT EQUATION • The Basic Percent Equation is given by Percent x Base = Amount

• The Percent has the percent sign % • The Base always follows the word “of” • The other number is the Amount

Ex: Cont.

Ex: Convert 3.19 to a % 3.19  100%  3.19 = 319%

Ex: Convert 0.005 to a % 0.005

 100%

 0.005 = 0.5%

Ex: 5% of 120 is what? • Identify the three components (remember, the base always follows “of”, the multiplication, and the equals.

Percent = 5% Base = 120 (follows “of”) Amount = Unknown “what”  a • Translate the English statement to the Basic Percent Equation: 5% x 120 = a

Ex: What is 6.3% of 150?

5% x 120 = a

Rewrite the % in working form

• Identify the three components (remember, the base always follows “of”, the multiplication, and the equals.

0.05 x 120 = a

Perform the math operation

Percent = 6.3%

• Now solve the mathematical equation:

6.00 = a Therefore, 5% of 120 is 6.

Base = 150 (follows “of”) Amount = Unknown “what”  a • Translate the English statement to the Basic Percent Equation: a = 6.3% x 150

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Ex: Cont.

Ex: 5% of what is 28?

a = 6.3% x 150

Rewrite the % in working form

• Identify the three components (remember, the base always follows “of”, the multiplication, and the equals.

a = .063 x 150

Perform the math operation

Percent = 5%

• Now solve the mathematical equation:

Base = Unknown “what”  b (follows “of”) Amount = 28

a = 9.45 Therefore, 9.45 is 6.3% of 150.

• Translate the English statement to the Basic Percent Equation: 5% x b = 28

Ex: Cont. • Now solve the mathematical equation: 5% x b = 28

Rewrite the % in working form

0.05 x b = 28

Solve the equation by dividing both sides by 0.05

b = 560

Ex: What % of 32 is 20? • Identify the three components (remember, the base always follows “of”, the multiplication, and the equals.

Percent = Unknown “what”  p Base = 32 (follows “of”) Amount = 20 • Translate the English statement to the Basic

Therefore, 5% of 560 is 28.

Percent Equation: p x 32 = 20

Ex: Cont. • Now solve the mathematical equation: p x 32 = 20

Solve the equation by dividing both sides by 32

p = 0.625

This is the decimal form of the percent. Rewrite using the % sign

Ex: The human body contains 206 bones. The fingers and the toes contain a total of 56 small bones, or phalanges. What percent of the bones of the body are phalanges?

p = 0.625 = 62.5% Therefore, 62.5% of 32 is 20.

Find the sentence that will be your equation (percent, base, amount, of, “is”)

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• Identify the three components (remember, the base always follows “of”, the multiplication, and the equals: What percent of the bones of the body are phalanges?

Percent = Unknown “what”  p Base = Bones of the body (follows “of”) Amount = phalanges Multiplication “of” Equals “is” • Write a hybrid sentence – half in math, half in English: p x bones of the body = phalanges

Ex: The new 8-mile nature trail is 125% of the length of the original trail. How long was the original trail? Find the sentence that will be your equation (percent, base, amount, of, “is”)

• Translate the hybrid sentence into the Basic Percent Equation p x bones of the body = phalanges p x 206 = 56

• Now solve the mathematical equation: p x 206 = 56

Solve the equation by dividing both sides by 206

p = 0.2718 . . . This is the decimal form of the percent. Rewrite using the % sign p = 0.2718 . . . = 27.2% Therefore, about 27.2% of bones are phalanges.

• Identify the three components (remember, the base always follows “of”, the multiplication, and the equals: The new 8-mile nature trail is 125% of the length of the original trail.

Percent = 125% Base = Original trail length = b (follows “of”) Amount = New 8-mile trail Multiplication “of” Equals “is” • Write a hybrid sentence – half in math, half in English: 125% x original trail = new trail

• Translate the hybrid sentence into the Basic Percent Equation 125% x original trail = new trail 125% x b = 8

• Now solve the mathematical equation: 1.25 x b = 8

Solve the equation by dividing both sides by 1.25

b = 6.4 Therefore, the original trail was 6.4 miles.

Ex: A medical supply company charges 5% of the order total as a shipping and handling charge. If the shipping and handling charge is $38.75, what was the cost of the order? Find the sentence that will be your equation (percent, base, amount, of, “is”)

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• Identify the three components (remember, the base always follows “of”, the multiplication, and the equals: 5% of the order total as a shipping and handling charge.

Percent = 5% Base = Order total = b (follows “of”) Amount = S&H charge Multiplication “of” Equals “is”

• Translate the hybrid sentence into the Basic Percent Equation 5% x order total = S & H 5% x b = 38.75

• Now solve the mathematical equation: 0.05 x b = 38.75

Solve the equation by dividing both sides by 0.05

b = 775

• Write a hybrid sentence – half in math, half in

Therefore, the cost of the order was $775.00.

English: 5% x order total = S & H

More Practice: Ex: 26.1% of the 364 patients admitted in an emergency room are admitted due to an auto accident injury. How many patients are admitted due to an auto accident injury? Find the sentence that will be your equation (percent, base, amount, of, “is”)

• Identify the three components (remember, the base always follows “of”, the multiplication, and the equals: 26.1% of the 364 patients admitted in an emergency room are admitted due to an auto accident injury.

Percent = 26.1% Base = Admitted patients (follows “of”) Amount = Admitted due to auto accident = a Multiplication “of” Equals “is” • Write a hybrid sentence – half in math, half in English: 26.1% x admitted patients = admitted due to auto accident

• Translate the hybrid sentence into the Basic Percent Equation

More Practice Ex: 45 is what % of 30?

26.1% x admitted patients = admitted due to auto accident

• Identify the three components (remember, the base always follows “of”, the multiplication, and the equals.

26.1% x 364 = a

• Now solve the mathematical equation: .261 x 364 = a

Solve the equation by multiplying

95.004 = a

Percent = Unknown “what”  p Base = 30 (follows “of”) Amount = 45 • Translate the English statement to the Basic

Therefore, 95 patients were admitted due to an auto accident injury.

Percent Equation: 45 = p x 30

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Ex: Cont. • Now solve the mathematical equation: 45 = p x 30 p = 1.5

Solve the equation by dividing both sides by 30 This is the decimal form of the percent. Rewrite using the % sign

p = 1.5 = 150% Therefore, 45 is 150% of 30.

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