1.9 Percentages

Percentages are a way of expressing a number as a fraction of 100. The symbol for percent is "\%". Percentages are widely used in comparing ratios, increases and decreases, discounts, profits, and more.

Key Concepts:

• To convert a fraction or decimal to a percentage:

  Multiply by 100. \[ \frac{3}{4} = 0.75 = 75\% \]

• To convert a percentage to a decimal:

  Divide by 100. \[ 40\% = \frac{40}{100} = 0.4 \]

• Percentage Increase/Decrease:

  \[ \text{Percentage Change} = \frac{\text{Change}}{\text{Original Value}} \times 100\% \]

• Finding a percentage of a number:

  \[ \text{e.g., } 25\% \text{ of } 200 = \frac{25}{100} \times 200 = 50 \]

Examples:

Example 1: Converting a Fraction to a Percentage

Convert \( \frac{5}{8} \) to a percentage.

Solution:

\[ \frac{5}{8} \times 100 = 62.5\% \]

Example 2: Finding Percentage of a Quantity

Find 30\% of 250.

Solution:

\[ \frac{30}{100} \times 250 = 75 \]

Example 3: Calculating Percentage Increase

A quantity increases from 40 to 50. Find the percentage increase.

Solution:

\[ \text{Increase} = 50 - 40 = 10, \quad \text{Percentage Increase} = \frac{10}{40} \times 100 = 25\% \]

Example 4: Calculating Percentage Decrease

A value drops from 80 to 60. Find the percentage decrease.

Solution:

\[ \text{Decrease} = 80 - 60 = 20, \quad \text{Percentage Decrease} = \frac{20}{80} \times 100 = 25\% \]