Final answer:
Patsy would need to choose between cheerleading and play practice on September 29th, as this is the next date when both practices fall on the same day, calculated by finding the least common multiple of the practice schedules.
Step-by-step explanation:
To determine when Patsy has to choose between cheerleading and play practice, we need to find the least common multiple (LCM) of the two practice schedules, since she has cheerleading every fourth day and play practice every sixth day.
First, list the multiples of both 4 and 6 until we find a common one:
- Multiples of 4: 4, 8, 12, 16, 20, 24, 28, ...
- Multiples of 6: 6, 12, 18, 24, 30, ...
The lowest common multiple of 4 and 6 is 12, but since our sequence started on September 5th and 12 is not a multiple of both 4 and 6, we need to keep going. Continuing our list, 24 is the next common multiple but still not satisfying both conditions. At 24 days, the cycle would repeat itself, and Patsy would again have both practices on the same day.
If these practices both start on September 5th, to find the next date she has both practices, we add 24 days to September 5th. Because September has 30 days, adding 24 days to September 5th results in September 29th. Hence, Patsy would need to choose between cheerleading and play practice on September 29th.