A florist had some roses and orchids. She sold $\frac{1}{4}$ of the roses and $\frac{3}{5}$ of the orchids. $\frac{4}{7}$ of the flowers sold were roses. What fraction of the flowers did the florist sell? Express your answer in its simplest form. Ans: ___

1. Determine the ratio of sold roses to sold orchids.

Since $\frac{4}{7}$ of the flowers sold were roses, the remaining $\frac{3}{7}$ were orchids.
Ratio of sold roses to sold orchids = $4 : 3$.

2. Find the total number of units for roses and orchids.

Let the number of sold roses be $4$ units and sold orchids be $3$ units.
Sold roses = $\frac{1}{4}$ of total roses = $4$ units.
Total roses = $4 \times 4 = 16$ units.
Sold orchids = $\frac{3}{5}$ of total orchids = $3$ units.
Total orchids = $3 \times \frac{5}{3} = 5$ units.

3. Calculate the fraction of total flowers sold.

Total flowers sold = $4 + 3 = 7$ units.
Total flowers at first = $16 + 5 = 21$ units.
Fraction of flowers sold = $\frac{7}{21} = \frac{1}{3}$.