3.2 Solve Percent Applications

Percentages are a powerful tool for understanding proportions and changes in real-world situations. They help us make sense of discounts, population shifts, and financial calculations. From retail to data analysis, percentages are everywhere.

In this section, we'll dive into percent applications. We'll learn how to translate percent equations, calculate changes, and use simple interest. We'll also explore discounts, markups, and data analysis using percentages.

Percent Applications

Translation of percent equations

• Grasp the connection between percent, part, and whole
  • Percent represents a proportion out of 100 (25%)
  • Part signifies a segment of the entire amount (50 out of 200)
  • Whole denotes the total quantity (200 students)
• Apply the percent equation to determine an unknown value
  • $\text{Percent} = \frac{\text{Part}}{\text{Whole}} \times 100$
  • Manipulate the equation to solve for the missing variable (solve for Part or Whole)
• Recognize key phrases in word problems to identify the unknown value
  • "What percent" signals solving for percent (What percent of the class is female?)
  • "How much" or "What amount" indicates solving for the part (How much money did she save?)
  • "Out of" or "total" suggests solving for the whole (Out of the total population, 150 people voted)

Calculation of percent changes

• Percent increase
  • Signifies a rise in value compared to the initial amount (population growth)
  • Formula: $\text{Percent Increase} = \frac{\text{Increase}}{\text{Original}} \times 100$
  • Example: Price increases from $50 to$60, percent increase = $\frac{10}{50} \times 100 = 20\%$
• Percent decrease
  • Represents a drop in value relative to the original quantity (price reduction)
  • Formula: $\text{Percent Decrease} = \frac{\text{Decrease}}{\text{Original}} \times 100$
  • Example: Salary decreases from $80,000 to$75,000, percent decrease = $\frac{5,000}{80,000} \times 100 = 6.25\%$
• Utilize these formulas in real-life scenarios
  • Changes in population size (growth or decline)
  • Fluctuations in prices, salaries, or inventory quantities (discounts or markups)

Simple interest using percentages

• Simple interest is computed using the principal, rate, and time
  • Principal (P) represents the starting sum invested or borrowed ($1,000)
  • Rate (R) stands for the yearly interest rate as a decimal (5% = 0.05)
  • Time (T) is the duration in years the funds are invested or borrowed (3 years)
• Apply the simple interest formula: $I = P \times R \times T$
  • I symbolizes the interest earned or paid ($150)
• Determine the total amount (A) by combining the principal and interest
  • $A = P + I$
  • Example: $1,000 +$150 = $1,150 total amount after 3 years

Retail and Data Analysis Applications

Discounts and markups in retail

• Discount
  • A price reduction from the original cost (20% off)
  • Calculate the discount amount: $\text{Discount} = \text{Original Price} \times \text{Discount Percentage}$
    • Example: $100 \times 0.20 =$20 discount
  • Determine the sale price: $\text{Sale Price} = \text{Original Price} - \text{Discount}$
    • Example: $100 -$20 = $80 sale price
• Markup
  • A price increase from the cost to set the selling price (50% markup)
  • Calculate the markup amount: $\text{Markup} = \text{Cost} \times \text{Markup Percentage}$
    • Example: $50 \times 0.50 =$25 markup
  • Determine the selling price: $\text{Selling Price} = \text{Cost} + \text{Markup}$
    • Example: $50 +$25 = $75 selling price

Data analysis with percentages

• Compute the percentage of each group within a dataset
  1. Find the sum of all categories (500 total students)
  2. Divide each category value by the total and multiply by 100
     • Example: 200 female students ÷ 500 total students × 100 = 40% female
• Interpret percentages for comparisons and conclusions
  • Identify the highest or lowest categories (highest: 45% science majors)
  • Compare the proportional size of different groups (30% more females than males)
• Use percent concepts in real-world data analysis
  • Demographic information (age distribution, income levels)
  • Election outcomes and survey results (60% approval rating)
  • Market share and sales figures (25% market share for a product)

Percentage Relationships and Conversions

• Understand the relationship between percentages, ratios, and proportions
  • A percentage is a ratio expressed as a fraction of 100
  • Proportions can be used to solve percentage problems
• Convert between percentages, decimals, and fractions
  • Percentage to decimal: divide by 100 (25% = 0.25)
  • Decimal to percentage: multiply by 100 (0.75 = 75%)
  • Fraction to percentage: divide numerator by denominator and multiply by 100 (3/4 = 75%)
• Identify the base value in percentage calculations
  • The base value is the whole amount to which a percentage is applied
  • Example: In "15% of 80," 80 is the base value