### Instruction: Express as a common fraction:         $\cfrac{ \frac{2}{5}+\frac{3}{4} }{ \frac{4}{9}+\frac{1}{6}}$.### Response: To express a complex fraction as a common fraction, I need to simplify the numerator and the denominator separately, and then divide them as fractions.
To simplify the numerator, I need to find a common denominator for $\frac{2}{5}$ and $\frac{3}{4}$.
The least common multiple of 5 and 4 is 20, so I multiply the first fraction by $\frac{4}{4}$ and the second by $\frac{5}{5}$ to get equivalent fractions with a common denominator.
This gives me $\frac{8}{20}+\frac{15}{20}=\frac{23}{20}$ for the numerator.
Similarly, to simplify the denominator, I need to find a common denominator for $\frac{4}{9}$ and $\frac{1}{6}$.
The least common multiple of 9 and 6 is 18, so I multiply the first fraction by $\frac{2}{2}$ and the second by $\frac{3}{3}$ to get equivalent fractions with a common denominator.
This gives me $\frac{8}{18}+\frac{3}{18}=\frac{11}{18}$ for the denominator.
Now I have a simpler complex fraction: $\cfrac{\frac{23}{20}}{\frac{11}{18}}$.
To divide two fractions, I need to flip the second fraction and multiply.
This gives me $\frac{23}{20}\times\frac{18}{11}=\frac{23\times 18}{20\times 11}$.
To simplify this fraction further, I look for common factors in the numerator and the denominator.
I notice that both 18 and 20 are divisible by 2, so I divide them by 2 to get $\frac{23\times 9}{10\times 11}$.
I don't see any other common factors, so this is the simplest form of the fraction.