How do you simplify  ((x/3)-6)/(10+(4/x))?

Jul 28, 2015

$\frac{{x}^{2} - 18 x}{30 x + 12}$

Explanation:

Multiply the top (numerator) and bottom (denominator) of the overall compound fraction by the least common multiples of the bottoms of the fractions in the numerator and denominator, which is $3 x$. Use the distributive property in the top and bottom and cancel appropriately to simplify.

$\frac{\left(\frac{x}{3}\right) - 6}{10 + \left(\frac{4}{x}\right)} = \frac{\left(\frac{x}{3}\right) - 6}{10 + \left(\frac{4}{x}\right)} \cdot \frac{3 x}{3 x} = \frac{{x}^{2} - 18 x}{30 x + 12}$

Alternatively, you can add the fractions in the original numerator and denominator by getting common denominators and then divide the fractions by inverting and multiplying:

$\frac{\left(\frac{x}{3}\right) - 6}{10 + \left(\frac{4}{x}\right)} = \frac{\frac{x}{3} - \frac{18}{3}}{\frac{10 x}{x} + \left(\frac{4}{x}\right)} = \frac{\frac{x - 18}{3}}{\frac{10 x + 4}{x}}$

$= \frac{x - 18}{3} \cdot \frac{x}{10 x + 4} = \frac{{x}^{2} - 18 x}{30 x + 12}$