Lesson 3-2: Understand the Percent Equation

Property

To solve percentage problems, you can use the equation:

The percent must be written as a decimal or a fraction. You can represent the unknown value with a variable.

Examples

• What is $20\%$ of $50$?

Let $x$ be the unknown part.
$x = 0.20 \times 50$
$x = 10$

• $12$ is what percent of $60$?

Let $p$ be the unknown percent.
$12 = p \times 60$
$p = \frac{12}{60} = 0.20$, which is $20\%$.

• $15$ is $30\%$ of what number?

Let $w$ be the unknown whole.
$15 = 0.30 \times w$
$w = \frac{15}{0.30} = 50$

Explanation

This skill applies your ability to write algebraic expressions to real-world percentage scenarios. By translating the words "is," "of," and "what" into mathematical symbols ($=$, $\times$, variable), you can create a solvable equation. Remember to convert the percentage to its decimal form for calculation by dividing by 100. This method allows you to find any missing part of a percentage problem: the part, the whole, or the percent itself.

Property

When you know the part and the whole, set up the equation $P \times \text{Whole} = \text{Part}$ and solve for $P$. Then, convert your decimal or fraction answer to a percent.

Examples

• Sarah correctly answered 17 of 20 questions. What is her score? $P \times 20 = 17 \implies P = \frac{17}{20} = 0.85$, or 85%.
• A baker sold 30 out of 40 muffins. What percent of muffins were sold? `$P \times 40 = 30 \implies P = \frac{30}{40} = 0.75$, or 75%.

Explanation

Ever wonder what grade you got on a quiz? This is exactly how you find out! Just divide the number of questions you got right by the total number of questions. The decimal you get is your grade, which you can show off as a shiny new percentage.

Property

To find the part of a whole in a real-world context, use the percent equation:

Remember to convert the percent to a decimal (by dividing by $100$) before multiplying.

Examples

• Tip: A meal costs $40$ dollars. If you want to leave a $15\%$ tip, what is the amount of the tip? Convert $15\%$ to $0.15$, then multiply: $\text{part} = 0.15 \times 40 = 6$ dollars.
• Discount: A jacket is originally priced at $80$ dollars and is on sale for $25\%$ off. What is the discount amount? Convert $25\%$ to $0.25$, then multiply: $\text{part} = 0.25 \times 80 = 20$ dollars.
• Commission: A salesperson earns a $5\%$ commission on a car sold for $20,000$ dollars. How much is the commission? Convert $5\%$ to $0.05$, then multiply: $\text{part} = 0.05 \times 20000 = 1000$ dollars.

Explanation

To find the "part" in a real-world scenario like calculating a tip, tax, or discount, you can use the percent equation directly. First, always convert the given percent into a decimal by dividing it by $100$. Then, multiply this decimal by the "whole" amount (such as the original price or total bill) to find the specific part you are looking for. This straightforward multiplication helps you quickly determine extra costs, earnings, or savings in everyday situations.