Lesson 58: Solving Percent Problems with Equations

New Concept

A percent of a whole is a part. This relationship is expressed with the fundamental percent equation:

What’s next

This card is just the foundation. Soon, we'll use this equation to solve for any missing part, whether it's the percent, the whole, or the part itself.

Property

To solve percent problems with an equation, use the formula: `

Examples

• What is 40% of 200? $0.40 \times 200 = 80$
• Thirty is 25% of what number? $0.25 \times W = 30 \implies W = \frac{30}{0.25} = 120$
• Ten is what percent of 50? `$P \times 50 = 10 \implies P = \frac{10}{50} = 0.20$, which is 20%.

Explanation

Think of this as a universal recipe for percent puzzles! If you know any two of the ingredients (the percent, the total, or the piece), you can use this simple equation to magically find the one you're missing. It turns tricky word problems into straightforward algebra you can totally crush.

Property

For any percent problem, you can convert the percent to a fraction or a decimal. The best choice depends on which form makes the calculation easier and more accurate.

Examples

• For $33\frac{1}{3}\%$ off a 27 dollars shirt, use a fraction: $\frac{1}{3} \times 27 = 9$ dollars saved.
• For an 8.25% tax on a 20 dollars item, use a decimal: $0.0825 \times 20 = 1.65$ dollars in tax.

Explanation

It’s like choosing a superpower! For friendly percents like $33\frac{1}{3}\%$, using a fraction like $\frac{1}{3}$ is great for quick mental math. For tricky ones like 7.75%, using a decimal and a calculator is your best bet. Choose your power wisely to save time!

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.