To find out how many hamburgers are sold when 63 hot dogs are sold, we can set up a ratio based on the information given:
The ratio of hot dogs to hamburgers is \( 7:10 \). This means that for every 7 hot dogs sold, 10 hamburgers are sold.
First, we can determine how many groups of 7 hot dogs are in 63 hot dogs:
\[ \text{Number of groups} = \frac{63 \text{ hot dogs}}{7 \text{ hot dogs/group}} = 9 \text{ groups} \]
Now that we know there are 9 groups of hot dogs, we can find out how many hamburgers are sold by multiplying the number of groups by the number of hamburgers per group:
\[ \text{Hamburgers sold} = 9 \text{ groups} \times 10 \text{ hamburgers/group} = 90 \text{ hamburgers} \]
Therefore, if 63 hot dogs are sold, the number of hamburgers sold will be \( \boxed{90} \).