Notes on Percentages

Percentages

Learning Objectives

  • Convert between percent, decimal, and fractional values.

  • Calculate the total, percent, or part.

  • Solve application problems involving percentages.

  • Calculate a flat tax.

Introduction to Percentages

  • The term "percent" originates from the Latin phrase "per centum," meaning "by the hundred."

  • It is composed of two words: "per" and "cent," which denotes a ratio comparing a part to a whole basis of 100.

  • Definition of Percent: A percent is a ratio or fraction whose denominator is 100.

    • If we express a part as some percent of a whole, the relationship is defined as:
      percent=partwhole×100\text{percent} = \frac{\text{part}}{\text{whole}} \times 100

Conversion Between Forms

Percent to Decimal and Fraction

  • To convert a percentage to decimal or fractional form:

    • Percent to Decimal: Drop the percent symbol and divide by 100.

    • Example: 40% converts to decimal as:
      40÷100=0.4040 \div 100 = 0.40

    • Fractional Form: Convert to fraction by writing n over 100:

    • Example: 40% can be expressed as:
      40100=25\frac{40}{100} = \frac{2}{5}

Example Calculation

  • Example 1: 243 out of 400 people like dogs. What percent is this?

    • Solution: The part is 243 and the whole is 400. Thus:
      percent=243400×100=60.75%\text{percent} = \frac{243}{400} \times 100 = 60.75\%

Mathematical Relationships

Finding Percent of a Whole

  • Using the formula: part=percent×whole\text{part} = \text{percent} \times \text{whole}

  • Example 2: Find 70% of 3,500.

    • Calculation:
      part=0.70×3500=2450\text{part} = 0.70 \times 3500 = 2450

Finding the Whole from Percent and Part

  • Rearranging the formula provides:
    whole=partpercent\text{whole} = \frac{\text{part}}{\text{percent}}

  • Example 3: What is 35% of 90?

    • Here, let n be the part:
      n=0.35×90=31.5n = 0.35 \times 90 = 31.5

Applications of Percentages

Sales Tax Calculation

  • Sales Tax is a percentage of the purchase price:
    Sales Tax=Tax Rate×Purchase Price\text{Sales Tax} = \text{Tax Rate} \times \text{Purchase Price}

  • Example 4: Calculate tax on a $140 purchase with a tax of 9.4%:
    Sales Tax=0.094×140=13.16\text{Sales Tax} = 0.094 \times 140 = 13.16

  • Total Price: Total Price=Purchase Price+Sales Tax\text{Total Price} = \text{Purchase Price} + \text{Sales Tax}

    • Therefore:
      Total Price=140+13.16=153.16\text{Total Price} = 140 + 13.16 = 153.16

Tips Calculation

  • Tips work similarly, typically calculated on pre-tax amounts: Tip=Tip Rate×Purchase Price\text{Tip} = \text{Tip Rate} \times \text{Purchase Price}

    • Example 5: A bill of $68.50 and a tip of 18%:
      Tip=0.18×68.50=12.33\text{Tip} = 0.18 \times 68.50 = 12.33

Percent Increase and Decrease

Definitions and Formulas

  • Percent Increase Formula:
    Percent Increase=New Amount−Original AmountOriginal Amount×100%\text{Percent Increase} = \frac{\text{New Amount} - \text{Original Amount}}{\text{Original Amount}} \times 100\%

  • Percent Decrease Formula:
    Percent Decrease=Original Amount−New AmountOriginal Amount×100%\text{Percent Decrease} = \frac{\text{Original Amount} - \text{New Amount}}{\text{Original Amount}} \times 100\%

Example Calculations

  • Example 6: New fees increased from $26 to $36:

    • Calculation:
      Percent Increase=36−2626×100=38.5%\text{Percent Increase} = \frac{36 - 26}{26} \times 100 = 38.5\%

  • Example 7: Car's value dropped from $7400 to $6800:

    • Calculation:
      Percent Decrease=7400−68007400×100≈8.1%\text{Percent Decrease} = \frac{7400 - 6800}{7400} \times 100 \approx 8.1\%

Flat and Progressive Taxes

Flat Rate Tax

  • A flat rate tax is a system where regardless of income, everyone pays the same percentage. Examples include states like Colorado, Illinois, Indiana, etc.

    • Example 8: A taxable income of $39,500 at a flat tax of 4.95% results in:
      Tax=39,500×0.0495=1,995.25\text{Tax} = 39,500 \times 0.0495 = 1,995.25

Progressive Taxation

  • In progressive taxation, the tax rate increases as the income increases. This is often structured in brackets.

    • Example 9: A taxable income of $60,000 calculated across different brackets reveals the total tax amount.

Exercises and Applications

Practical Exercises

  1. Determine the sales tax for a $250 purchase at 8.7%.

  2. Calculate what number is 18% of 75.

  3. What is the total cost for a family meal of $75 before tax with an 8.7% sales tax?

Complex Percentages Problems

  1. Find the percentage of change for stock prices that fluctuate with a +/- 30% change.

  2. A store has a clearance of 60% and an additional 30% off. What is the final price percentage of the original price?

Summary

  • Mastery of percentages involves understanding conversions, calculations, and real-world applications, such as taxes and tips. By recognizing these patterns, one can effectively manage financial tasks and interpret quantitative data efficiently.