Percentages

Introduction

Deciding which is the larger of two fractions is often difficult when fractions do not have the same denominator. For example, if John pays 13/23 of his income in taxes and Mary pays 17/29 of her income in taxes, it may not be immediately clear who pays a higher fraction in taxes. A protocal where all fractions have the same denominator would eliminate this difficulty. Percentages do this by assigning all fractions a denominator of 100.

Converting Percentages to Fractions

To convert percentages to fractions, we need only know that the meaning of the character % is "over 100" or $\frac{}{100}$ .

Examples

13% = 13 over 100 = $\frac{13}{100}$ .
20% = 20 over 100 = $\frac{20}{100} = \frac{1}{5}$ .
117% = 117 over 100 = $\frac{117}{100}$ .

Click below to practice the relationship between fractions and percentages.

Solving Simple Equations with Percentages

Simple expressions with percentages have the form a is n% of b which is equivalent to $a = \frac{n}{100} \cdot b$ .

Steps

Simple Expressions with percentages can be solved with the following steps:

Step 1: Write the basic percentage expression.

$a = \frac{n}{100} \cdot b$

Step 2: Identify Known Variables.

Step 3: Substitute.

Step 4: Solve for Unknown

Click here if you have problems with Step 4.

Examples

Example 1

If 35 is 25% of x, find x.

Step 1: Write the Basic Percentage Expression.

$a = \frac{n}{100} \cdot b$

Step 2: Identify Known Variables.

a = 35, n = 25, and b = x

Step 3: Substitute.

$35 = \frac{25}{100} \cdot x$

Step 4: Solve for Unknown

x = 140

Example 2

48 is x% of 60, find x

Step 1: Write the Basic Percentage Expression.

$a = \frac{n}{100} \cdot b$

Step 2: Identify Known Variables.

a = 48, b = 60, and n = x

Step 3: Substitute.

$48 = \frac{x}{100} \cdot 60$

Step 4:

Solve for Unknown

x = 80

Example 3

x is 30% of 1500, find x.

Step 1: Write the Basic Percentage Expression.

$a = \frac{n}{100} \cdot b$

Step 2: Identify Knownn Variables.

n = 30, b = 1500, and a = x.

Step 3: Substitute.

$x = \frac{30}{100} \cdot 1500$

Step 4:

Solve for Unknown

x = 450.

Solving Simple Word Problems with Percentages

Basic word problems with percentages also have the form $a = \frac{n}{100} \cdot b$ . The only difference is that words are used for a and b.

Steps

Word problems with percentages can be solved with the following steps:

Step 1: Write the Basic Percentage Expression.

$a = \frac{n}{100} \cdot b$

Step 2: Substitute words for a and b.

Step 3: Identify Known Variables.

Step 4: Substitute.

Step 5:

Solve for Unknown

Click here if you have problems with Step 5.

Examples

Example 1

30% of students in a class are men, if the total of students in the class are 20. How many men are in the class?

Step 1: Write the Basic Percentage Expression.

$a = \frac{n}{100} \cdot b$

Step 2: Substitute words for a and b.

$men = \frac{30}{100} \cdot total class$

Step 3: Identify Known Variables.

n = 30, total class = 20 and men is unknown.

Step 4: Substitute.

$men = \frac{30}{100} \cdot 20$

Step 5:

Solve for Unknown

men = 6.

Conclusion: there are six men in the class.

Example 2

Walter left a $3.50 tip for his dinner, which cost $25. What percent of the total bill did Walter leave for a tip?

Step 1: Write the Basic Percentage Expression.

$a = \frac{n}{100} \cdot b$

Step 2: Substitute words for a and b.

$tip = \frac{n}{100} \cdot total bill$

Step 3: Identify Known Variables.

tip = 3.5, total bill = 25, and n is unknown.

Step 4: Substitute.

$3.5 = \frac{n}{100} \cdot 25$

Step 5:

Solve for Unknown

n = 14.

Conclusion: Walter left 14% of the price of dinner as a tip.

Example 3

5% of the employees at M&D Company called in sick with the flu. If 10 employees called in sick, How many employees are there at the company?

Step 1: Write the Basic Percentage Expression.

$a = \frac{n}{100} \cdot b$

Step 2: Substitute words for a and b.

$Employees sick = \frac{n}{100} \cdot total employees$

Step 3: Identify Known Variables.

employees sick = 10, n = 5, and total employees is unknown.

Step 4: Substitute.

$10 = \frac{5}{100} \cdot total employees$

Step 5:

Solve for Unknown

total employees = 200.

Conclusion: M&D Company has 200 employees.