6.1 Understand Percent

Understanding Percent

A percent is a way of expressing a number as a part out of 100. The word itself comes from "per centum," meaning "per hundred." So 25% literally means 25 out of 100, or $\frac{25}{100}$.

Percents make it easy to compare different quantities on the same scale. Whether you're figuring out a discount, reading a poll, or calculating a tip, you're working with percents.

Percent in Real-World Contexts

You'll run into percents constantly outside of math class. Here are some of the most common places:

• Discounts and sales tell you how much is taken off the original price (20% off means you save 20 out of every 100 dollars)
• Tax rates express the portion of a purchase or income that goes to the government (an 8% sales tax adds 8 cents for every dollar you spend)
• Interest rates describe the cost of borrowing money or the return on savings (a 5% annual interest rate on $200 earns you $10 in a year)
• Survey results show what fraction of people chose a certain answer (65% of participants agreed)
• Probability uses percents to describe how likely something is (a 30% chance of rain)

In every case, the percent is doing the same job: telling you a part-to-whole relationship scaled to 100.

Converting Between Percents, Decimals, and Fractions

These three forms all represent the same value. Being able to switch between them is one of the most useful skills in this unit.

Percent → Decimal: Divide by 100 (move the decimal point two places left).

Decimal → Percent: Multiply by 100 (move the decimal point two places right), then add the % symbol.

Fraction → Percent: Divide the numerator by the denominator, then multiply by 100.

Percent → Fraction: Write the percent over 100, then simplify.

Applications of Percent Concepts

Finding a percent of a quantity: Convert the percent to a decimal, then multiply.

Finding what percent one quantity is of another: Divide the part by the whole, then multiply by 100.

Calculating percent change: Use this formula:

$\frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100$

If the result is positive, it's a percent increase. If negative, it's a percent decrease.

Applying successive percents: Convert each percent to a decimal multiplier and multiply them in order. Don't just add the percents together.

Solving for the original amount: If you know the result after a percent change, divide by the decimal multiplier.

Ratios, Proportions, and Percentages

A ratio compares two quantities and can be written as a fraction. Percents are really just ratios with a denominator of 100.

A proportion is an equation that says two ratios are equal. You can use proportions to solve percent problems by setting up something like $\frac{\text{part}}{\text{whole}} = \frac{x}{100}$ and cross-multiplying.

A few more terms to know:

• Base refers to the whole amount that a percent is applied to. In "30% of 50," the base is 50.
• Percentage points measure the straight difference between two percents. If approval goes from 40% to 55%, that's a 15 percentage-point increase (not a 15% increase).
• Markup is a percent increase added to the cost of an item to set its selling price. If a store buys a shirt for $20 and marks it up 50%, the selling price is $30.