To find the ratio of the number of gold coins to the number of silver coins, we first need to determine how many silver coins Michelle has.
Michelle has a total of 37 coins, and 13 of them are gold. Therefore, the number of silver coins can be calculated as follows:
\[ \text{Number of silver coins} = \text{Total coins} - \text{Number of gold coins} \] \[ \text{Number of silver coins} = 37 - 13 = 24 \]
Now, we can express the ratio of the number of gold coins to the number of silver coins:
\[ \text{Ratio of gold coins to silver coins} = \frac{\text{Number of gold coins}}{\text{Number of silver coins}} = \frac{13}{24} \]
Thus, the ratio of the number of gold coins to the number of silver coins is
\[ \boxed{\frac{13}{24}} \]