Arithmetic Mean Calculation

The problem states that the arithmetic mean of five different natural numbers is $12$. We need to find the largest possible value for one of these numbers.

Sum of Numbers

The formula for the arithmetic mean is:

$ \text{Mean} = \frac{\text{Sum of numbers}}{\text{Count of numbers}} $

Given Mean = $12$ and Count = $5$. Therefore, the sum of the five numbers is:

$ \text{Sum} = \text{Mean} \times \text{Count} = 12 \times 5 = 60 $

Maximizing the Largest Number

Let the five different natural numbers be $n_1, n_2, n_3, n_4, n_5$, such that $n_1 < n_2 < n_3 < n_4 < n_5$.

To maximize the largest number ($n_5$), we must choose the smallest possible values for the other four numbers ($n_1, n_2, n_3, n_4$).

Since the numbers must be different natural numbers (which start from 1), the smallest possible values are:

• $n_1 = 1$
• $n_2 = 2$
• $n_3 = 3$
• $n_4 = 4$

Finding the Largest Value

Now, substitute these minimum values into the sum equation:

$ n_1 + n_2 + n_3 + n_4 + n_5 = 60 $

$ 1 + 2 + 3 + 4 + n_5 = 60 $

$ 10 + n_5 = 60 $

Solving for $n_5$:

$ n_5 = 60 - 10 $

$ n_5 = 50 $

The largest possible value among the five different natural numbers is $50$. This ensures all numbers ($1, 2, 3, 4, 50$) are different natural numbers and their mean is $12$.