Given that three numbers have an average (mean) of 12 and a mode of 8, let's consider the possible scenarios for the numbers.
1. If the mode is 8, at least one of the numbers must be 8.
2. Since the average of the three numbers is 12, the total of the three numbers must be 3 * 12 = 36.
Let's consider different cases to find the number that can be the largest:
Case 1: If all three numbers are 8:
- In this case, 8 + 8 + 8 = 24, which is less than 36. So, not all numbers can be 8.
Case 2: If two numbers are 8:
- If two numbers are 8, the sum of these two numbers and the unknown number must be 36.
- 8 + 8 + x = 36
- 16 + x = 36
- x = 20
- In this case, the largest number can be 20.
Therefore, the largest number that can satisfy the given conditions is 20.
