Simplifying Compound/Complex Fractions
Complex/Compound Fractions are essentially fractions of fractions, or fractions of rational expressions.
Complex/Compound fractions can be simplified by first simplifying the fractions in the numerator and denominator of the big fraction, then dividing the two fractions using Keep Change Flip.
Find and simplify the result of
$$\frac{1+\frac{1}{2}}{1-\frac{1}{2}}$$
Start by simplifying the numerator and denominator of the “big” fraction separately, finding a common denominator for each. Viz:
$$\begin{align}\frac{1+\frac{1}{2}}{1-\frac{1}{2}}&=\frac{\frac{2}{2}+\frac{1}{2}}{\frac{2}{2}-\frac{1}{2}}\end{align}$$
Simplify the sum and difference of fractions in the numerator and denominator of the “big” fraction, respectively. Viz:
$$\begin{align}\frac{\frac{2}{2}+\frac{1}{2}}{\frac{2}{2}-\frac{1}{2}}&=\frac{\frac{3}{2}}{\frac{1}{2}}\end{align}$$
Now, to divide the results, perform a “Keep-change-flip” as usual.
$$\begin{align}\frac{\frac{3}{2}}{\frac{1}{2}}&=\frac{3}{2}\cdot \frac{2}{1}\\&=\frac{3}{1}\\&=3\end{align}$$
\(\frac{x+1}{x-1}\)
\(\frac{(-1)(x+3)x}{3(x-1)(x+1)}\) or \(\frac{-x^2-3x}{3(x-1)(x+1)}\)
\(\frac{-2x}{x+2}\)
\(\frac{y^3-y-1}{y-1}\)
\(\frac{-2(2p+5)}{5p(p+5)}\)