Question

The average of 11 numbers is 10.9. If the average of the first six numbers is 10.5 and that of the last six numbers is 11.4, then the middle number is

• A. 11.5
• B. 11.4
• C. 11.3
• D. 11.0

Answer 1

The Correct Option is A

The correct option is (A): 11.5
Explanation: Let the 11 numbers be \( N_1, N_2, N_3, \ldots, N_{11} \).
The average of these 11 numbers is 10.9, so their total sum is:
\[\text{Total sum} = 11 \times 10.9 = 119.9\]
The average of the first six numbers \( N_1, N_2, N_3, N_4, N_5, N_6 \) is 10.5, so their total sum is:
\[\text{Sum of first six} = 6 \times 10.5 = 63\]
The average of the last six numbers \( N_6, N_7, N_8, N_9, N_{10}, N_{11} \) is 11.4, so their total sum is:
\[\text{Sum of last six} = 6 \times 11.4 = 68.4\]
Now, \( N_6 \) is counted in both the sum of the first six and the last six. Therefore, the total sum can be expressed as:
\[\text{Total sum} = \text{Sum of first six} + \text{Sum of last six} - N_6\]
Substituting the known sums:
\[119.9 = 63 + 68.4 - N_6\]
This simplifies to:
\[119.9 = 131.4 - N_6\]
Solving for \( N_6 \):
\[N_6 = 131.4 - 119.9 = 11.5\]
Thus, the middle number \( N_6 \) is Option A: 11.5.