Question

Simplify each complex fraction. [ (y+3)/y - 4/(y-1) ] / [ y/(y - 1) + 1/y ]

Accepted Answer

Identify the complex fraction: the numerator is $$\frac{y+3}{y} - \frac{4}{y-1}$$ and the denominator is $$\frac{y}{y-1} + \frac{1}{y}. $$Find a common denominator for the numerator terms, which are $$\frac{y+3}{y}$$ and $$\frac{4}{y-1}. $$The common denominator is $$y(y-1). $$Rewrite each fraction with this common denominator: $$\frac{(y+3)(y-1)}{y(y-1)} - \frac{4y}{y(y-1)}.$$ Combine the numerator fractions into a single fraction by subtracting the numerators over the common denominator: $$\frac{(y+3)(y-1) - 4y}{y(y-1)}.$$ Similarly, find a common denominator for the denominator terms $$\frac{y}{y-1}$$ and $$\frac{1}{y}$$, which is also $$y(y-1). $$Rewrite each fraction: $$\frac{y^2}{y(y-1)} + \frac{y-1}{y(y-1)}.$$ Combine the denominator fractions into a single fraction by adding the numerators over the common denominator: $$\frac{y^2 + (y-1)}{y(y-1)}.$$ Now, rewrite the original complex fraction as a division of two single fractions: $$\frac{\frac{(y+3)(y-1) - 4y}{y(y-1)}}{\frac{y^2 + (y-1)}{y(y-1)}} = \frac{(y+3)(y-1) - 4y}{y(y-1)} \div \frac{y^2 + (y-1)}{y(y-1)}.$$ Recall that dividing by a fraction is the same as multiplying by its reciprocal. So rewrite the expression as: $$\frac{(y+3)(y-1) - 4y}{y(y-1)} 	imes \frac{y(y-1)}{y^2 + (y-1)}.$$ Cancel the common factors $$y(y-1) in $$numerator and denominator, then simplify the remaining expressions by expanding and combining like terms.

Simplify each complex fraction. [ (y+3)/y - 4/(y-1) ] / [ y/(y - 1) + 1/y ]

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Key Concepts

Complex Fractions

Finding a Common Denominator

Dividing Fractions