Overview
This lecture covers the concepts of ratio and proportion, including their definitions, properties, and methods for solving for unknowns using cross-multiplication.
Ratios
- A ratio compares two quantities using division, e.g., 4/5, 4:5, or 4 is to 5.
- All representations (fraction, colon, words) express the same relationship between two numbers.
Proportions
- A proportion is an equation stating that two ratios are equal.
- Proportions can be written as a/b = c/d.
- Proportions are true if the fractions reduce to the same value.
Properties of Proportions
- In a proportion a/b = c/d, 'a' and 'd' are called extremes; 'b' and 'c' are called means.
- The product of the extremes equals the product of the means (cross-multiplication).
- Cross-multiplying helps to verify or solve proportions.
Solving for Unknowns in Proportions
- To solve a proportion, cross-multiply and solve for the unknown variable.
- Example: 15/45 = 1/n ⇒ 15n = 45 ⇒ n = 3.
- For distributive property: 7/4 = w/(5000 - w) ⇒ 7(5000 - w) = 4w ⇒ 35000 - 7w = 4w ⇒ 35000 = 11w ⇒ w = 3181.82.
- Example: 4/7 = n/8 ⇒ 4×8 = 7n ⇒ n = 32/7 = 4.57.
- Example: 2/11 = 4/(ab) ⇒ 2ab = 44 ⇒ ab = 22.
- Example: 6/8 = x/5 ⇒ 6×5 = 8x ⇒ x = 30/8 = 3.75.
Key Terms & Definitions
- Ratio — a comparison of two quantities by division.
- Proportion — an equation stating that two ratios are equal.
- Extremes — the first and last terms in a proportion (a and d in a/b = c/d).
- Means — the middle terms in a proportion (b and c in a/b = c/d).
- Cross-multiplication — multiplying extremes and means to find unknowns in proportions.
Action Items / Next Steps
- Practice solving proportions using cross-multiplication.
- Review definitions and properties of ratios and proportions for better understanding.