Learn on PengiAoPS: Introduction to Algebra (AMC 8 & 10)Chapter 6: Ratios and Percents

Lesson 4: Percent

In this Grade 4 AMC Math lesson from AoPS: Introduction to Algebra, students learn the definition of percent as a ratio per hundred and apply the equation a/b = x/100 to solve problems involving finding a percent of a number, determining what percent one number is of another, and working backwards from a percentage. Students also practice converting between fractions, decimals, and percents to simplify calculations, and explore how to increase or decrease a number by a given percentage using the expressions x(1 + k/100) and x(1 - k/100).

Section 1

Definition of a percent

Property

A percent is a ratio whose denominator is 100.

Examples

57 out of 100 students is 57%, which is the ratio $\frac{57}{100}$ .

25% means a ratio of $\frac{25}{100}$ .

Section 2

Calculating a percent of a number

Property

To calculate a percent of a number:

Change the percent to a decimal fraction.
Multiply the number by the decimal fraction.

To change a percent to a decimal fraction, divide the percent by 100, or move the decimal point two places to the left.

Examples

To find 8% of 300, convert 8% to a decimal by moving the decimal point two places left: $8\% = 0.08$ . Then multiply: $0.08 \times 300 = 24$ .
A phone costs 850 dollars and sales tax is 7.5%. Convert 7.5% to a decimal: $0.075$ . The tax is $0.075 \times 850 = 63.75$ dollars.
Calculate 150% of 60. First, change 150% to a decimal: $1.50$ . Then, multiply: $1.50 \times 60 = 90$ .

Explanation

To use a percent in a calculation, you must first convert it into a form your calculator understands, like a decimal. Moving the decimal point two places left is a quick shortcut for dividing by 100.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Definition of a percent

Property

A percent is a ratio whose denominator is 100.

Examples

57 out of 100 students is 57%, which is the ratio $\frac{57}{100}$ .

25% means a ratio of $\frac{25}{100}$ .

Section 2

Calculating a percent of a number

Property

To calculate a percent of a number:

Change the percent to a decimal fraction.
Multiply the number by the decimal fraction.

To change a percent to a decimal fraction, divide the percent by 100, or move the decimal point two places to the left.

Examples

To find 8% of 300, convert 8% to a decimal by moving the decimal point two places left: $8\% = 0.08$ . Then multiply: $0.08 \times 300 = 24$ .
A phone costs 850 dollars and sales tax is 7.5%. Convert 7.5% to a decimal: $0.075$ . The tax is $0.075 \times 850 = 63.75$ dollars.
Calculate 150% of 60. First, change 150% to a decimal: $1.50$ . Then, multiply: $1.50 \times 60 = 90$ .

Explanation

To use a percent in a calculation, you must first convert it into a form your calculator understands, like a decimal. Moving the decimal point two places left is a quick shortcut for dividing by 100.

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Definition of a percent

Property

A percent is a ratio whose denominator is 100.

Examples

57 out of 100 students is 57%, which is the ratio $\frac{57}{100}$ .

25% means a ratio of $\frac{25}{100}$ .

Section 2

Calculating a percent of a number

Property

To calculate a percent of a number:

Change the percent to a decimal fraction.
Multiply the number by the decimal fraction.

To change a percent to a decimal fraction, divide the percent by 100, or move the decimal point two places to the left.

Examples

To find 8% of 300, convert 8% to a decimal by moving the decimal point two places left: $8\% = 0.08$ . Then multiply: $0.08 \times 300 = 24$ .
A phone costs 850 dollars and sales tax is 7.5%. Convert 7.5% to a decimal: $0.075$ . The tax is $0.075 \times 850 = 63.75$ dollars.
Calculate 150% of 60. First, change 150% to a decimal: $1.50$ . Then, multiply: $1.50 \times 60 = 90$ .

Explanation

To use a percent in a calculation, you must first convert it into a form your calculator understands, like a decimal. Moving the decimal point two places left is a quick shortcut for dividing by 100.