We start with the ratio of hot dogs to hamburgers sold, which is given as 7 hot dogs for every 10 hamburgers. We can express this ratio mathematically as:
\[ \frac{7 \text{ hot dogs}}{10 \text{ hamburgers}} = \frac{d}{h} \]
where \( d \) is the number of hot dogs sold and \( h \) is the number of hamburgers sold.
Now, if we know that 63 hot dogs are sold (\( d = 63 \)), we can set up a proportion based on the given ratio:
\[ \frac{7}{10} = \frac{63}{h} \]
To solve for \( h \), we can cross-multiply:
\[ 7h = 10 \times 63 \]
Calculating \( 10 \times 63 \):
\[ 10 \times 63 = 630 \]
Now our equation is:
\[ 7h = 630 \]
Next, we can solve for \( h \) by dividing both sides by 7:
\[ h = \frac{630}{7} \]
Calculating \( \frac{630}{7} \):
\[ h = 90 \]
Thus, if 63 hot dogs are sold, the number of hamburgers sold will be:
\[ \boxed{90} \]
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