Lesson 4: The Percent Equation

Property

Given positive numbers $A$ and $B$, we say that $A$ is $P$ percent of $B$, that is, $A$ is to $B$ as $P$ is to 100. In terms of fractions:

This can also be written algebraically as:

Examples

• What is $25\%$ of 80? We solve for $A$ in $\frac{A}{80} = \frac{25}{100}$. Multiplying both sides by 80 gives $A = \frac{25}{100} \times 80 = 20$.
• 15 is what percent of 60? We solve for $P$ in $\frac{15}{60} = \frac{P}{100}$. Since $\frac{15}{60} = \frac{1}{4}$, and $\frac{1}{4} = \frac{25}{100}$, $P=25$. So, 15 is $25\%$ of 60.
• 18 is $30\%$ of what number? We solve for $B$ in $\frac{18}{B} = \frac{30}{100}$. This simplifies to $\frac{18}{B} = \frac{3}{10}$. Cross-multiplying gives $3B = 180$, so $B = 60$.

Explanation

Percent just means 'for each hundred.' It’s a special ratio where the second number is always 100. This standardizes fractions and makes it easy to compare different quantities, like scores on tests with different numbers of questions.

Property

Percent problems involve three components: the part ($A$), the whole ($B$), and the percent ($P$). Given any two of these components, the third can be found using the relationship $\frac{A}{B} = \frac{P}{100}$.

1. Find the Part: $A = (\frac{P}{100}) \times B$
2. Find the Percent: $P = (\frac{A}{B}) \times 100$
3. Find the Whole: $B = A \div (\frac{P}{100})$

Examples

• Find the part: A 50 dollar shirt is on sale for $20\%$ off. The discount is $0.20 \times 50 = 10$ dollars.
• Find the percent: You answered 24 out of 30 questions correctly on a test. Your score is $\frac{24}{30} = 0.8$, which is $80\%$.
• Find the whole: 9 students, which is $30\%$ of the class, have brown hair. The total number of students in the class is $9 \div 0.30 = 30$.

Explanation

Every percent problem has three key pieces: the part, the whole, and the percent. If you know any two, you can always find the third. It's like a math detective game where you use the formula to find the missing clue.