在一家水果店里，所有水果中 $\frac{5}{8}$ 是苹果，其余是橙子。售出了 $\frac{2}{5}$ 的苹果和 104 个橙子。最后剩下起初总水果数的一半。 起初共有多少个水果？

1. Express the initial number of apples and oranges in units

Let the total number of fruits at first be $8$ units.
Apples at first = $5$ units.
Oranges at first = $8 - 5 = 3$ units.

2. Calculate the number of apples sold and left in units

Apples sold = $\frac{2}{5} \times 5 \text{ units} = 2 \text{ units}$.
Apples left = $5 - 2 = 3 \text{ units}$.

3. Determine the total fruits left and oranges left in units

Total fruits left = $\frac{1}{2} \times 8 \text{ units} = 4 \text{ units}$.
Oranges left = Total fruits left $-$ Apples left = $4 - 3 = 1 \text{ unit}$.

4. Find the number of oranges sold in units and equate to the given value

Oranges sold = Oranges at first $-$ Oranges left = $3 - 1 = 2 \text{ units}$.
We are given that $104$ oranges were sold, so:
$2 \text{ units} = 104$
$1 \text{ unit} = 104 \div 2 = 52$

5. Calculate the total number of fruits at first

Total fruits at first = $8 \text{ units} = 8 \times 52 = 416$