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 :

### Concept & Strategy When two subgroups overlap by exactly one item (e.g., first $6$ and last $6$ out of $11$ total items), the overlapping item is counted twice when you add the sums of the subgroups. To find the overlapping item, subtract the true total sum from the combined sum of the subgroups. $$\text{Overlapping Item} = (\text{Sum of Group 1} + \text{Sum of Group 2}) - \text{Total Sum}$$ ### Step-by-Step Solution * **Given:** * Total numbers ($n$) = $11$, Average = $10.9$. * First $6$ numbers, Average = $10.5$. * Last $6$ numbers, Average = $11.4$. * **Calculation:** 1. Find the total sum of all $11$ numbers: $11 \times 10.9 = 119.9$. 2. Find the sum of the first $6$ numbers: $6 \times 10.5 = 63.0$. 3. Find the sum of the last $6$ numbers: $6 \times 11.4 = 68.4$. 4. Add the sums of the two groups: $63.0 + 68.4 = 131.4$. 5. This combined sum ($131.4$) contains all $11$ numbers, but the 6th number is counted twice. 6. Middle number = Combined Sum - True Total Sum = $131.4 - 119.9 = 11.5$. ### Exam Strategy & Shortcut **Deviation Method:** Compare subgroup averages to the overall average ($10.9$). First $6$: Deviation is $10.5 - 10.9 = -0.4$. Total deviation = $6 \times (-0.4) = -2.4$. Last $6$: Deviation is $11.4 - 10.9 = +0.5$. Total deviation = $6 \times (+0.5) = +3.0$. Net deviation = $-2.4 + 3.0 = +0.6$. In an overlapping sequence, you simply add the net deviation to the overall average to find the overlapping number. Middle number = $10.9 + 0.6 = 11.5$. ### Common Pitfall Confusing the overlapping formula with the excluded item formula. If subgroups overlap (first $6$ + last $6 = 12$ > $11$), you subtract the total sum from the subgroup sum. If there is a gap (first $5$ + last $5 = 10$ < $11$), you subtract the subgroup sum from the total sum. ### Final Answer **Therefore, the correct answer is 11.5.**