Calculate the 6th Number Using Mean Properties

This problem involves finding a specific number within a dataset when given the overall mean and the means of overlapping subsets. We need to determine the value of the 6th number.

Understanding Mean and Sum Relationships

• The mean of a set of numbers is calculated by dividing the sum of the numbers by the count of the numbers.
• Conversely, the sum of a set of numbers can be found by multiplying the mean by the count of the numbers.
• Let the 11 numbers be represented as $n_1, n_2, \dots, n_{11}$.

Calculating the Total Sum

• The mean of all 11 numbers is given as 44.
• The total sum of these 11 numbers ($S_{1-11}$) is: $S_{1-11} = \text{Mean}_{1-11} \times 11 = 44 \times 11 = 484$

Calculating the Sums of the Subsets

• The mean of the first 6 numbers ($n_1$ to $n_6$) is 39.
• The sum of the first 6 numbers ($S_{1-6}$) is: $S_{1-6} = \text{Mean}_{1-6} \times 6 = 39 \times 6 = 234$
• The mean of the last 6 numbers ($n_6$ to $n_{11}$) is 48.
• The sum of the last 6 numbers ($S_{6-11}$) is: $S_{6-11} = \text{Mean}_{6-11} \times 6 = 48 \times 6 = 288$

Finding the 6th Number

When we add the sum of the first 6 numbers ($S_{1-6}$) and the sum of the last 6 numbers ($S_{6-11}$), the 6th number ($n_6$) is included in both sums. Therefore, adding these two sums results in the total sum of all 11 numbers plus an extra count of the 6th number.

The relationship is expressed as:

This simplifies to:

Which can be written using our calculated sums:

• To find the 6th number ($n_6$), we rearrange the formula: $n_6 = S_{1-6} + S_{6-11} - S_{1-11}$
• Now, substitute the calculated sum values: $n_6 = 234 + 288 - 484$ $n_6 = 522 - 484$ $n_6 = 38$

Final Result

The value of the $6^{\text{th}}$ number is 38.