Question

Every ninth person in line at a movie theater gets free popcorn. Every sixth person gets free apple juice. Shani says that the thirty-sixth customer will be the first customer to get both free popcorn and free apple juice. Is she right? If not,describe her mistake.

Answer 1

We are told that every 9th person gets free popcorn and every 6th person gets free apple juice at the movie theater. We want to show if the 36th person is the first to get both free popcorn and apple juice.

Since every 9th gets free popcorn, then the 9th and 18th and 27th and 36th people get it. We can use numbers and show this as multiples of 9.

9, 18, 27, 36, 45, 54, ...

And we repeat it for every 6th and the free juice.

6, 12, 18, 24, 30, 36,....

Shani thinks 36 comes first, and she's wrong. 18 appears on the list before 36 as something that's common. What are we finding is the lowest common muliple of 6 and 9.

6 = 2 x 3

9 = 3 x 3

The lowest common multiple will have a 2, a 3, and a second 3 (because we don't double count). 2 x 3 x 3 = 18. This matches up to the list of multiples.

Thus, Shani is wrong that the 36th person gets both free items, as she did not account for the 18th person gettting both free items.