Caylan is making baked goods for a charity bake sale. He places a tray of scones, a tray of
muffins, and a tray of cookies in the oven. The scones bake for 15 minutes, the muffins bake
for 12 minutes, and the cookies bake for 10 minutes. When one tray is done, he removes it
and replaces it with another tray of the same item.
The Correct Answer and Explanation is:
If Caylan continuously bakes trays of each item in the oven, we need to find the time intervals for each type of baked good and consider how these intervals align. Specifically, each tray of scones bakes for 15 minutes, each tray of muffins for 12 minutes, and each tray of cookies for 10 minutes.
Solution:
The goal is to determine how frequently each item would be removed from the oven so that Caylan can put in a new tray immediately afterward. Since each item has its own unique bake time, each item will finish at different intervals, forming a repeating cycle. To find when all items will be ready at the same time, we can look for the Least Common Multiple (LCM) of the bake times (15, 12, and 10 minutes).
- Prime Factors:
- 15 = ( 3 \times 5 )
- 12 = ( 2^2 \times 3 )
- 10 = ( 2 \times 5 )
- LCM Calculation:
To calculate the LCM, we take the highest power of each prime factor present:
- For 2, the highest power is ( 2^2 ).
- For 3, the highest power is ( 3 ).
- For 5, the highest power is ( 5 ). So, ( \text{LCM} = 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60 ).
- Interpretation:
Every 60 minutes, all three items will be removed from the oven at the same time, creating a synchronized point where Caylan can restart all three trays. However, between these synchronized points, each item will follow its individual cycle:
- Every 15 minutes, a new tray of scones is baked.
- Every 12 minutes, a new tray of muffins is baked.
- Every 10 minutes, a new tray of cookies is baked.
Explanation:
The LCM helps to identify intervals when multiple cycles align, which is particularly useful in kitchen settings where timing is essential. By understanding the intervals, Caylan can efficiently manage his baking process, ensuring a steady supply of freshly baked items for the charity bake sale. With this method, he will have a tray of each item ready at least every 60 minutes, ensuring variety and continuous output.