Final answer:
To determine the number of games it would take to sell 864 hotdogs, given that 12 hotdogs are sold in three games, we set up a proportion and find that it would take 216 games to sell 864 hotdogs at the same rate.
Step-by-step explanation:
The question is asking for a proportion to estimate the number of games needed to sell a certain amount of hotdogs. Given that 12 hotdogs are sold in three games, we can set up the proportion as follows:

To solve for \(x\), we cross-multiply:
12 * x = 3 * 864

x = 216
Therefore, it will take 216 games to sell 864 hotdogs at the same rate.