Overview
This lesson teaches techniques for adding, subtracting, multiplying, and dividing fractions, including finding common denominators and simplifying results.
Adding and Subtracting Two Fractions
- To add or subtract two fractions, find a common denominator by multiplying the denominators together.
- Multiply each numerator by the opposite fraction’s denominator, add or subtract the results, and place over the common denominator.
- Example: ( \frac{3}{5} + \frac{4}{7} = \frac{21 + 20}{35} = \frac{41}{35} ).
- For subtraction, use the same method but subtract the numerators after cross-multiplying.
Adding and Subtracting Three Fractions
- All fractions must have the same denominator to add/subtract; this is called the common denominator.
- The least common denominator (LCD) is the smallest number that each denominator divides evenly into.
- To find the LCD, list multiples of each denominator and choose the smallest common one.
- Alternatively, multiply all denominators to get a common denominator when LCD is difficult to find.
- Adjust each fraction so its denominator matches the common denominator, then add or subtract numerators.
- Example: ( \frac{3}{4} + \frac{5}{3} - \frac{7}{2} ) combined over LCD of 12 becomes ( \frac{9 + 20 - 42}{12} = \frac{-13}{12} ).
Multiplying Fractions
- Multiply across numerators and denominators: ( \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} ).
- For large numbers, factor numerators and denominators to simplify before multiplying.
- Cancel common factors in numerators and denominators for easier computation.
- Example: ( \frac{24}{45} \times \frac{27}{30} ) simplifies to ( \frac{4}{3} ) after canceling factors.
Dividing Fractions
- Use "keep, change, flip": keep the first fraction, change division to multiplication, flip the second fraction.
- Simplify before multiplying if possible by canceling common factors.
- Example: ( \frac{8}{5} \div \frac{12}{7} = \frac{8}{5} \times \frac{7}{12} ) simplifies to ( \frac{14}{15} ).
Key Terms & Definitions
- Fraction — A number expressed as one integer over another (numerator/denominator).
- Denominator — The bottom number of a fraction, indicating the number of equal parts.
- Numerator — The top number of a fraction, showing how many parts are considered.
- Common Denominator — A shared multiple of the denominators of two or more fractions.
- Least Common Denominator (LCD) — The smallest positive common denominator for two or more fractions.
Action Items / Next Steps
- Practice adding, subtracting, multiplying, and dividing given fractions using the techniques described.
- Try the example: ( \frac{8}{5} - \frac{2}{3} + \frac{9}{4} ) using a common denominator.