### Instruction: Express as a common fraction:         $\cfrac{ \frac{2}{5}+\frac{3}{4} }{ \frac{4}{9}+\frac{1}{6}}$.### Response: This is a complex fraction, which means a fraction where the numerator and denominator are also fractions.
To simplify a complex fraction, I need to multiply the top and bottom by a common denominator of the smaller fractions.
This way, I can get rid of the fractions in the numerator and denominator and end up with a simpler fraction.
To find a common denominator, I can look at the multiples of the denominators 5, 4, 9, and 6.
The smallest number that is divisible by all of them is 180.
So I will multiply the top and bottom of the complex fraction by 180.
This gives me:

\begin{align*}
\cfrac{ \frac{2}{5}+\frac{3}{4} }{ \frac{4}{9}+\frac{1}{6}} \times \cfrac{180}{180} &= \cfrac{ \frac{2}{5} \times 180 + \frac{3}{4} \times 180 }{ \frac{4}{9} \times 180 + \frac{1}{6} \times 180 }\\
&= \cfrac{ 2 \times 36 + 3 \times 45 }{ 4 \times 20 + 1 \times 30 }\\
&= \cfrac{ 72 + 135 }{ 80 + 30 }\\
&= \cfrac{ 207 }{ 110 }
\end{align*}
This is the simplest form of the fraction, since 207 and 110 have no common factors.
So I have expressed the complex fraction as a common fraction.