### Instruction: Express as a common fraction:         $\cfrac{ \frac{2}{5}+\frac{3}{4} }{ \frac{4}{9}+\frac{1}{6}}$.### Response: To simplify a fraction, I need to make sure the numerator and the denominator have the same denominator, so I can add or subtract them easily.
For the numerator, the common denominator of 5 and 4 is 20, so I multiply the first fraction by 4/4 and the second fraction by 5/5 to get: $\cfrac{ \frac{2}{5}+\frac{3}{4} }{ \frac{4}{9}+\frac{1}{6}}=\cfrac{ \frac{8}{20}+\frac{15}{20} }{ \frac{4}{9}+\frac{1}{6}}=\cfrac{ \frac{23}{20} }{ \frac{4}{9}+\frac{1}{6}}$.
For the denominator, the common denominator of 9 and 6 is 18, so I multiply the first fraction by 2/2 and the second fraction by 3/3 to get: $\cfrac{ \frac{23}{20} }{ \frac{4}{9}+\frac{1}{6}}=\cfrac{ \frac{23}{20} }{ \frac{8}{18}+\frac{3}{18}}=\cfrac{ \frac{23}{20} }{ \frac{11}{18}}$.
Now I have a fraction of fractions, which I can simplify by flipping and multiplying the denominators: $\cfrac{ \frac{23}{20} }{ \frac{11}{18}}=\cfrac{23}{20} \cdot \cfrac{18}{11}$.
To multiply fractions, I multiply the numerators and the denominators: $\cfrac{23}{20} \cdot \cfrac{18}{11}=\cfrac{23 \cdot 18}{20 \cdot 11}$.
I can cancel out any common factors between the numerator and the denominator to simplify further: $\cfrac{23 \cdot 18}{20 \cdot 11}=\cfrac{23 \cdot 9}{10 \cdot 11}=\cfrac{207}{110}$.