Overview
This lecture reviews the essential rules for performing operations with fractions: addition, subtraction, multiplication, and division, including the need for common denominators and the use of inverses.
Addition and Subtraction of Fractions
- To add or subtract fractions with the same denominator, add or subtract the numerators and keep the denominator.
- Formula: (a/d) + (b/d) = (a + b)/d ; (a/d) - (b/d) = (a - b)/d
- If denominators differ, convert fractions to equivalent ones with a common denominator before performing the operation.
- Multiply both numerator and denominator by the necessary factors to achieve a common denominator.
- Only modify fractions by multiplying both numerator and denominator by the same number (never by addition or subtraction).
Multiplication of Fractions
- To multiply two fractions, multiply the numerators together and the denominators together.
- Formula: (a/b) × (c/d) = (a × c)/(b × d)
- Do not make denominators the same before multiplying; multiplying in-line is simpler.
- Simplify the fractions by canceling common factors before multiplying when possible.
Division of Fractions and Inverses
- The inverse of a number x is 1/x; the inverse of a/b is b/a.
- Dividing by a number is the same as multiplying by its inverse.
- To divide fractions: (a/b) ÷ (c/d) = (a/b) × (d/c)
- Only invert the second fraction (the divisor) and multiply.
Key Terms & Definitions
- Numerator — the top number in a fraction, indicating how many parts are taken.
- Denominator — the bottom number in a fraction, indicating into how many parts the whole is divided.
- Common Denominator — a shared denominator required for addition/subtraction of fractions.
- Inverse (Reciprocal) — for a fraction a/b, the inverse is b/a.
Action Items / Next Steps
- Practice exercises on adding, subtracting, multiplying, and dividing fractions, especially with different denominators.
- Review and memorize the core formulas for fraction operations.
- Prepare for the test by working on additional fraction operation problems.