Simplifying using the LCD
Simplify complex rational expressions in Grade 10 by finding the least common denominator, multiplying through to clear fractions, and reducing the resulting expression.
Key Concepts
Multiply the numerator and the denominator by the Least Common Denominator (LCD) of all the fractions in the complex fraction. Then simplify.
$$\frac{5 + \frac{2}{x}}{4 \frac{1}{y}} = \frac{(5 + \frac{2}{x}) \cdot xy}{(4 \frac{1}{y}) \cdot xy} = \frac{5xy + 2y}{4xy x}$$.
$$\frac{\frac{4}{a} + \frac{1}{a 1}}{\frac{2a}{a 1}} = \frac{(\frac{4}{a} + \frac{1}{a 1}) \cdot a(a 1)}{(\frac{2a}{a 1}) \cdot a(a 1)} = \frac{4(a 1) + a}{2a \cdot a} = \frac{5a 4}{2a^2}$$.
Common Questions
How do you find the LCD of rational expressions?
Factor each denominator completely. The LCD is the product of each unique factor raised to the highest power it appears in any single denominator.
How do you simplify (3/x + 2/y) / (1/xy)?
The LCD of all fractions is xy. Multiply the entire complex fraction by xy/xy: numerator becomes 3y+2x, denominator becomes 1. Result: 3y+2x.
Why does multiplying by the LCD work to simplify complex fractions?
Multiplying numerator and denominator by the same expression (the LCD) is equivalent to multiplying by 1, which doesn't change the fraction's value but clears all inner denominators.