First, remember that a fraction represents a division. So, a fraction of fractions is a division of two fractions:

\begin{align*} \dfrac{\left( \dfrac{2}{5} \right)}{\left( \dfrac{8}{5} \right)} = \dfrac{2}{5} \div \dfrac{8}{5} \end{align*}

Now, recall that dividing by $\dfrac{8}{5}$ is the same as multiplying by its reciprocal $\dfrac{5}{8}.$ We can use this fact to simplify the fraction above \begin{eqnarray} \eqalign{ \dfrac{2}{5}\div\dfrac{8}{5}&=\\[5pt] \dfrac{2}{5}\times\dfrac{5}{8}&=\\[5pt] \dfrac{2\times5}{5\times8}&=\\[5pt] \dfrac{10}{40} } \end{eqnarray}

Finally, we simplify:

\[ \dfrac{10}{40} = \dfrac{10\div 10}{40\div 10} = \dfrac{1}{4} \]

Therefore, \[\dfrac{\left( \dfrac{2}{5} \right)}{\left( \dfrac{8}{5} \right)} =\dfrac14.\]

First, remember that a fraction represents a division. So, a fraction of fractions is a division of two fractions:

\begin{align*} \dfrac{\left(-\dfrac{3}{2}\right)}{\left(\dfrac{2}{5}\right)} = \left(-\dfrac{3}{2}\right) \div \dfrac{2}{5} \end{align*}

Now, recall that dividing by $\dfrac{2}{5}$ is the same as multiplying by its reciprocal $\dfrac{5}{2}.$ We can use this fact to simplify the fraction above:

\begin{align*} \left(-\dfrac{3}{2}\right)\div\dfrac{2}{5} &=\\[5pt] \left(-\dfrac{3}{2}\right)\times\dfrac{5}{2} &=\\[5pt] -\dfrac{3 \times 5}{2\times 2}&=\\[5pt] -\dfrac{15}{4} \end{align*}

Therefore,

\[ \dfrac{\left(-\dfrac{3}{2}\right)}{\left(\dfrac{2}{5}\right)} = -\dfrac{15}{4}. \]

First, remember that a fraction represents a division. So, a fraction of fractions is a division of two fractions:

\[ \dfrac{\:\left(-\dfrac{6}{15}\right)\:}{\left(-\dfrac{12}{9}\right)} = \left(-\dfrac{6}{15}\right)\div \left(-\dfrac{12}{9}\right) \]

Now, recall that dividing by $-\dfrac{12}{9}$ is the same as multiplying by its reciprocal $-\dfrac{9}{12}.$ We can use this fact to simplify the fraction above:

\begin{align*} \left(-\dfrac{6}{15}\right)\div \left(-\dfrac{12}{9}\right)&=\\[5pt] \left(-\dfrac{6}{15}\right)\times \left(-\dfrac{9}{12}\right)&=\\[5pt] \dfrac{9 \times 6}{15 \times 12}&=\\[5pt] \dfrac{54}{180} \end{align*}

Finally, we simplify:

\[ \dfrac{54}{180} = \dfrac{54\div 18}{180\div 18} = \dfrac{3}{10} \]

Therefore,

\[ \dfrac{\:\left(-\dfrac{6}{15}\right)\:}{\left(-\dfrac{12}{9}\right)} = \dfrac{3}{10}. \]

First, remember that a fraction represents a division:

\begin{align} \dfrac{3} {\left(\dfrac 7 3\right)} = 3 \div\dfrac {7}{3} \end{align}

Now, recall that $3 = \dfrac{3}{1}$ and dividing by $\dfrac{7}{3}$ is the same as multiplying by its reciprocal $\dfrac{3}{7}.$ We can use these facts to simplify the fraction above:

\begin{align} 3 \div \dfrac 7 3 &= \\[5pt] \dfrac 3 1 \times \dfrac 3 7 &=\\[5pt] \dfrac{3\times 3}{1\times 7} &=\\[5pt] \dfrac{9}{7} \end{align}

Therefore,

\[ \dfrac{3} {\left(\dfrac 7 3 \right)} = \dfrac{9}{7}. \]

First, remember that a fraction represents a division:

\begin{align} \dfrac{7} {\left(-\dfrac 1 2\right)} = 7 \div \left(-\dfrac 1 2\right) \end{align}

Now, recall that $7 = \dfrac{7}{1}$ and dividing by $-\dfrac{1}{2}$ is the same as multiplying by its reciprocal $-\dfrac{2}{1}.$ We can use these facts to simplify the fraction above:

\begin{align} 7 \div \left(-\dfrac 1 2\right) &= \\[5pt] \dfrac 7 1 \times \left(-\dfrac 2 1\right) &=\\[5pt] -\dfrac{7\times 2}{1\times 1} &=\\[5pt] -\dfrac{14}{1} &=\\[5pt] -14 \end{align}

Therefore,

\[ \dfrac{7} {\left(-\dfrac 1 2 \right)} = -14. \]

First, remember that a fraction represents a division:

\[ \dfrac{\:\:(-9)\:\:}{\left(\dfrac{1}{2}\right)} = (-9)\div \dfrac{1}{2} \]

Now, recall that $-9 = -\dfrac{9}{1}$ and dividing by $\dfrac{1}{2}$ is the same as multiplying by its reciprocal $\dfrac{2}{1}.$ We can use these facts to simplify the fraction above:

\begin{align} (-9)\div \dfrac{1}{2}&=\\[5pt] \left(-\dfrac {9}{ 1}\right) \times \dfrac{2}{1}&=\\[5pt] -\dfrac{9 \times 2}{1 \times1}&=\\[5pt] -\dfrac{18}{1}&=\\[5pt] -18 & \end{align}

Therefore,

\[ \dfrac{\:\:(-9)\:\:}{\left(\dfrac{1}{2}\right)} = -18. \]

We can use either of the following two methods.

Method 1 - The Long Way

First, remember that a fraction represents a division:

\[ \dfrac{\left(\dfrac 8 5 \right)}{2} = \dfrac 8 5 \div 2 \]

Now, recall that dividing by $2$ is the same as multiplying by its reciprocal $\dfrac{1}{2}.$ We can use this fact to simplify the fraction above:

\begin{align} \dfrac 8 5 \div 2 &=\\[5pt] \dfrac 8 5 \times\dfrac 1 2 &=\\[5pt] \dfrac{8\times 1}{5\times 2} &=\\[5pt] \dfrac{8}{10} \end{align}

Finally, we simplify:

\[ \dfrac{8}{10} = \dfrac{8 \div 2}{10 \div 2} = \dfrac{4}{5} \]

Therefore,

\[ \dfrac{\left(\dfrac 8 5 \right)}{2}= \dfrac{8}{10}. \]

Method 2 - The Quick Way

The result will have the same numerator as the top fraction, and the denominator is just the product of the two denominators:

\begin{align*} \dfrac{\left(\dfrac 8 5 \right)}{2}&=\\[5pt] \dfrac{8}{5 \times 2} &=\\[5pt] \dfrac{8}{10} &=\\[5pt] \dfrac{8\div 2}{10\div 2} &=\\[5pt] \dfrac{4}{5}& \end{align*}

We can use either of the following two methods.

Method 1 - The Long Way

First, remember that a fraction represents a division:

\[ \dfrac{\left(- \dfrac 7 2 \right)}{5} = \left(-\dfrac 7 2\right) \div 5 \]

Now, recall that dividing by $5$ is the same as multiplying by its reciprocal $\dfrac{1}{5}.$ We can use this fact to simplify the fraction above:

\begin{align} \left(-\dfrac 7 2\right) \div 5 &=\\[5pt] \left(-\dfrac 7 2\right) \times \dfrac 1 5 &=\\[5pt] -\dfrac{7\times 1}{2\times 5} &=\\[5pt] -\dfrac{7}{10} \end{align}

Therefore,

\[ \dfrac{\left(- \dfrac 7 2 \right)}{5} =- \dfrac{7}{10}. \]

Method 2 - The Quick Way

The result will have the same numerator as the top fraction, and the denominator is just the product of the two denominators:

\begin{align*} \dfrac{\left(- \dfrac 7 2 \right)}{5}&=\\[5pt] \dfrac{-7}{2\times 5} &=\\[5pt] -\dfrac{7}{10} \end{align*}

We can use either of the following two methods.

Method 1 - The Long Way

First, remember that a fraction represents a division:

\[ \dfrac{\left(-\dfrac 2 5 \right)}{(-5)} = \left(-\dfrac 2 5\right) \div (-5) \]

Now, recall that dividing by $-5$ is the same as multiplying by its reciprocal $-\dfrac{1}{5}.$ We can use this fact to simplify the fraction above:

\begin{align} \left(-\dfrac 2 5\right) \div (-5) &=\\[5pt] \left(-\dfrac 2 5\right) \times \left(-\dfrac 1 5\right) &=\\[5pt] \dfrac{2\times 1}{5\times 5} &=\\[5pt] \dfrac{2}{25} \end{align}

Therefore,

\[ \dfrac{\left(-\dfrac 2 5 \right)}{(-5)} = \dfrac{2}{25}. \]

Method 2 - The Quick Way

The result will have the same numerator as the top fraction, and the denominator is just the product of the two denominators:

\begin{align*} \dfrac{\left(-\dfrac 2 5 \right)}{(-5)}&=\\[5pt] \dfrac{-2}{5\times (-5)} &=\\[5pt] \dfrac{-2}{-25} &=\\[5pt] \dfrac{2}{25} \end{align*}