Compound fractions.

 

Simplify \[ \frac{\dfrac{1}{x} }{ 1 + \dfrac{a}{x}} \ \]

Put numerator and denominator over their common denominators.

\[ \begin{align} &= \frac{\dfrac{1}{x} }{\dfrac{x + a}{x}} \end{align} \ \]

Multiply the numerator by the reciprocal of the denominator.

\[ \begin{align} &= \dfrac{1}{x} \times \dfrac{x}{x + a} \end{align} \ \]

Cancel common factors.

\[ = \frac {1} {x + a} \ \]

You will gain most benefit if you work through each problem first and then come back and check your work against the solution.

 

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Now You Try:

Simplify \[ \frac{b+ \dfrac{a}{x-1} }{ a + \dfrac{b}{x-1}} \ \]