At a fruit shop, $\frac{5}{8}$ of the fruits were apples and the rest were oranges. $\frac{2}{5}$ of the apples and 104 oranges were sold. Half of the total fruits at first were left. How many fruits were there at first?

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 = $8 \text{ units} = 8 \times 52 = 416$