To find out how many days 4 2/3 bags of oranges will last when Emilio's orange juice stand uses 2/3 of a bag of oranges per day, we need to divide the total amount of oranges by the daily usage.
First, let's convert 4 2/3 bags into an improper fraction:
\[ 4 \frac{2}{3} = \frac{4 \times 3 + 2}{3} = \frac{12 + 2}{3} = \frac{14}{3} \]
Now, we will divide this amount by the daily usage, which is 2/3 bags per day:
\[ \text{Number of days} = \frac{\frac{14}{3}}{\frac{2}{3}} \]
Dividing by a fraction is the same as multiplying by its reciprocal:
\[ \frac{\frac{14}{3}}{\frac{2}{3}} = \frac{14}{3} \times \frac{3}{2} \]
When we multiply the fractions, we can cancel the 3's:
\[ = \frac{14 \times 1}{1 \times 2} = \frac{14}{2} = 7 \]
Thus, the number of days that 4 2/3 bags of oranges will last is
\[ \boxed{7} \]
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