Overview
This lecture covers section 3.4 on percents, focusing on converting between decimals, fractions, and percents, and solving percent-related word problems using both equations and proportions.
Understanding Percents
- "Percent" means "per 100" (Cent = 100), represented as ÷100 or as the symbol %.
- The percent sign (%) can be replaced with ÷100 to form fractions or decimals.
Converting Decimals, Percents, and Fractions
Decimal to Percent
- Move the decimal point two places to the right, then add the percent sign.
- Example: 0.25 → 25%; 1.8 → 180%; 7 → 700%.
Percent to Decimal
- Move the decimal point two places to the left, then remove the percent sign.
- Example: 37% → 0.37; 2.9% → 0.029; 6% → 0.06.
Percent to Fraction
- Replace the percent sign with ÷100, then reduce the fraction.
- Example: 56% → 56/100 → 14/25; 125% → 125/100 → 5/4; 0.4% → 0.4/100 → 1/250.
Fraction to Percent
- Divide the numerator by the denominator to get a decimal, then convert to percent by moving the decimal two places right.
- Example: 3/5 → 0.6 → 60%; 2¼ → 2.25 → 225%.
Solving Percent Problems
Using Equations
- Translate “what is x% of y” into the equation: (percent as decimal) × y = value.
Using Proportions
- Set up a proportion: (percent/100) = (part/whole).
- Cross-multiply and solve for the unknown.
Examples
- What is 30% of 180? 30/100 = x/180 → x = 54.
- 40% of what number is 25? 40/100 = 25/x → x = 62.5.
- 12.4 is what percent of 80? x/100 = 12.4/80 → x = 15.5.
Word Problems
- Identify the part (is/numerator) and the whole (of/denominator).
- Example 1: 18% of a class is 380 students. 18/100 = 380/x → x = 2,111 students (rounded).
- Example 2: Broker gets $1,370 commission at 3%. 3/100 = 1,370/x → x = $45,667 selling price.
- Example 3: Athlete loses 2 out of 55 liters of water in an hour. x/100 = 2/55 → x ≈ 3.6% loss.
Key Terms & Definitions
- Percent (%) — Means “per hundred” or out of 100.
- Decimal — Number representation with base-10 and a decimal point.
- Fraction — A ratio of two numbers, numerator over denominator.
- Proportion — An equation stating two ratios are equal.
- Part/Whole — The part is a portion of the total (whole).
Action Items / Next Steps
- Practice conversion exercises from decimals, percents, and fractions.
- Complete assigned word problems using both equations and proportions.
- Review homework or quiz on percent word problems as announced.