Understanding and Using Percentages

Learning Objectives

Convert between decimals, fractions, and percents.

Calculate the percent of a number.

Solve problems involving discounts and sales tax.

Calculate percent change (increase or decrease).

A Special Kind of Ratio

A percent is one of the most common ways we encounter math in our daily lives, from shopping to school grades. The word 'percent' comes from Latin and means 'per hundred'. A percent is simply a ratio that compares a number to 100. The symbol for percent is %.

Conversions

Because a percent is just a ratio, we can easily convert it to a fraction or a decimal.

Percent to Decimal: To convert a percent to a decimal, divide by 100 (or simply move the decimal point two places to the left).

`75% = 75.0 / 100 = 0.75`

`8% = 8.0 / 100 = 0.08`

Decimal to Percent: To convert a decimal to a percent, multiply by 100 (or simply move the decimal point two places to the right).

`0.25 = 0.25 × 100 = 25%`

`0.9 = 0.90 × 100 = 90%`

Percent to Fraction: To convert a percent to a fraction, write the percent as a ratio over 100 and then simplify.

`60% = 60/100 = 6/10 = 3/5`

Finding the Percent of a Number

This is the most common type of percent problem. The key is to remember that in math, the word 'of' almost always means multiply.

Rule: To find the percent of a number, first convert the percent to a decimal, then multiply.

Example: What is 20% of 80?

1.Convert 20% to a decimal: `20% = 0.20`

2.Multiply: `0.20 × 80 = 16`

3.So, 20% of 80 is 16.

Real-World Applications

Discounts: A discount is a percentage decrease in price.

Example: A $50 sweater is on sale for 30% off. How much is the discount?

Discount = 30% of $50 = 0.30 × 50 = $15.

The sale price is $50 - $15 = $35.

Sales Tax: Sales tax is a percentage increase in price.

Example: You buy a shirt for $20 and the sales tax is 5%. How much is the tax?

Tax = 5% of $20 = 0.05 × 20 = $1.

The total cost is $20 + $1 = $21.

Percent Change

We can also use percents to describe how much a quantity has changed.

Percent Change = (Amount of Change / Original Amount) × 100%

Example: The price of a video game went from $40 to $50. What is the percent increase?

1.Amount of Change = New Price - Original Price = $50 - $40 = $10.

2.Percent Change = ($10 / $40) × 100%

3.`0.25 × 100% = 25%`

4.The price increased by 25%.

Key Terms

**Percent: A ratio that compares a number to 100.
**Decimal: A number expressed in a system that uses a decimal point to separate the whole number part from the fractional part.
**Fraction: A numerical quantity that is not a whole number, expressed as a numerator divided by a denominator.
**Discount: A deduction from the usual cost of something.
**Sales Tax: A tax on sales or on the receipts from sales.