Question

Caylan is making baked goods . The scones bake for 15 minutes, the muffins bake for 12 minutes and the cookies for 10 minutes. When one tray is done, he removes it and replaces it with another tray of the same item. How many minutes after Caylan puts the trays in the oven will he first remove the scones, muffins and cookies at the same time?

Answer 1

To determine when Caylan will first remove the scones, muffins, and cookies at the same time, we need to find the least common multiple (LCM) of the baking times for each type of baked good.

The baking times are:
- Scones: 15 minutes
- Muffins: 12 minutes
- Cookies: 10 minutes

Let's break down each time into its prime factors:

- **15**: The prime factorization is \(3 \times 5\).
- **12**: The prime factorization is \(2^2 \times 3\).
- **10**: The prime factorization is \(2 \times 5\).

Next, we will determine the LCM by taking the highest power of each prime that appears in the factorizations:

- For the prime factor **2**, the highest power is \(2^2\) (from 12).
- For the prime factor **3**, the highest power is \(3^1\) (from both 15 and 12).
- For the prime factor **5**, the highest power is \(5^1\) (from both 15 and 10).

Now, we multiply these together to find the LCM:

\[
LCM = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5
\]

Calculating this step-by-step:

1. \(4 \times 3 = 12\)
2. \(12 \times 5 = 60\)

Thus, the least common multiple is \(60\) minutes.

This means that Caylan will first remove the trays of scones, muffins, and cookies at the same time after **60 minutes**.