Complex fractions
Simplify complex fractions in Grade 9 algebra by multiplying numerator and denominator by the LCD or converting stacked fractions to division problems involving rational expressions.
Key Concepts
Property A complex fraction is a fraction that contains one or more fractions in the numerator or the denominator. It can be written as division: $\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{b} \div \frac{c}{d}$.
Explanation Think of a complex fraction as a division problem in a clever disguise! To solve it, just rewrite the big fraction bar as a division symbol. Then, you can use the classic “keep, change, flip” trick by multiplying by the reciprocal of the denominator. It is that simple!
Examples $\frac{\frac{a}{x}}{\frac{b}{a+x}} = \frac{a}{x} \div \frac{b}{a+x} = \frac{a}{x} \cdot \frac{a+x}{b} = \frac{a(a+x)}{xb}$ $\frac{\frac{3}{4}}{\frac{5}{7}} = \frac{3}{4} \div \frac{5}{7} = \frac{3}{4} \cdot \frac{7}{5} = \frac{21}{20}$.
Common Questions
What is a complex fraction?
A complex fraction is a fraction where the numerator, denominator, or both contain fractions themselves. For example, (1/x) / (2/x²) is a complex fraction with fractional expressions in both positions.
What are the two main methods to simplify a complex fraction?
Method 1: Find the LCD of all inner fractions, multiply every term by it to clear denominators, then simplify. Method 2: Rewrite the complex fraction as a division problem (numerator ÷ denominator) and multiply by the reciprocal.
How do you simplify (1/a + 1/b) / (1/a - 1/b)?
Find the LCD (ab), multiply every term by ab: (b + a) / (b - a). The result is (a + b)/(b - a). Multiplying by the LCD is the most efficient method when multiple fractions appear in the complex fraction.