We never actually divide fractions.  When given a division problem, you multiply by the reciprocal of the 2nd fraction. 

So, \(\displaystyle \dpi{100} 9\div \frac{1}{7}\) becomes \(\displaystyle \dpi{100} 9\times \frac{7}{1}\).

This is the same as \(\displaystyle \dpi{100} 9\times7=63\)

Simply use the long division process. The quotient will be: 

\(\displaystyle 0.17 \div 32 = 0.0053125\)

\(\displaystyle \frac{3}{4}\div\frac{1}{3}=\frac{3}{4}\times\frac{3}{1}=\frac{9}{4}\)

Rewrite this as a division, dividing the numerator by the denominator:

\(\displaystyle \frac{\frac{2}{7}}{\frac{9}{14}} = \frac{2}{7} \div \frac{9}{14} = \frac{2}{7} \times \frac{14}{9}= \frac{2}{1} \times \frac{2}{9}=\frac{2 \times 2}{1 \times 9}= \frac{4}{9}\)

Rewrite as a division, then solve:

\(\displaystyle \frac{\frac{2}{5}}{\frac{7}{10}} = \frac{2}{5} \div \frac{7}{10} = \frac{2}{5} \times \frac{10}{7} = \frac{2}{1} \times \frac{2}{7} = \frac{4}{7}\)

Rewrite as a division, then solve:

\(\displaystyle \frac{ 2 \frac{1}{5}}{7 \frac{1}{3}}\)

\(\displaystyle = 2 \frac{1}{5} \div 7 \frac{1}{3}\)

\(\displaystyle = \frac{1+ 2 \times 5}{5} \div \frac{1+ 7 \times 3}{3}\)

\(\displaystyle = \frac{11}{5} \div \frac{22}{3}\)

\(\displaystyle = \frac{11}{5} \times \frac{3}{22}\)

\(\displaystyle = \frac{1}{5} \times \frac{3}{2} = \frac{1 \times 3}{5 \times 2} = \frac{3}{10}\)

To divide any number by a fraction, we can multiply that number by the reciprocal of the fraction. That means:

\(\displaystyle \frac{a}{b}\div \frac{c}{d} = \frac{a}{b}\times \frac{d}{c}=\frac{ad}{bc}\)

So we have:

\(\displaystyle \sqrt{3}\div \frac{3}{4}= \frac{\sqrt{3}}{1}\times \frac{4}{3}\)

\(\displaystyle =\frac{4\sqrt{3}}{3}\)

Multiply each term by 6:

\(\displaystyle \frac{2+\frac{5}{6}}{2-\frac{5}{6}} = \frac{12+5}{12-5}\)

\(\displaystyle =\frac{17}{7}\)

A mixed number represents the sum of an integer and a fraction. In order to evaluate this problem first we need to change the mixed number ( \(\displaystyle 8\frac{1}{3}\)) to an improper fraction. Improper fractions are fractions whose numerator is greater than the denominator. So we can write:

\(\displaystyle 8\frac{1}{3}=\frac{8\times 3+1}{3}=\frac{25}{3}\)

Now we should evaluate   \(\displaystyle \frac{25}{3}\div \frac{1}{6}\) and we have:

\(\displaystyle \frac{25}{3}\div \frac{1}{6}=\frac{25}{3}\times \frac{6}{1}=50\)

We can first find a common denominator for the expression in the numerator, which is \(\displaystyle x\). This gives us:

\(\displaystyle \frac{\frac{4}{x}}{\left(\frac{1}{x}-\frac{4x^2}{x}\right)}=\frac{\frac{4}{x}}{\left(\frac{1-4x^2}{x}\right)}\)

\(\displaystyle =\frac{4}{x}\times \left(\frac{x}{1-4x^2}\right)\)

\(\displaystyle =\frac{4}{(1-4x^2)}\)