The average of 11 results is 60. If the average of first six results is 58 and that of last six is 63, find the $6^{\text{th}}$ result.

### Concept & Formula This is a classic **Overlapping Averages** problem. When a central item is counted twice (once in the first half and once in the second half), you can find its exact value by subtracting the total sum of all distinct items from the combined sum of the overlapping subsets. $$ \text{Value of Overlapping Item} = (\text{Sum of Part 1} + \text{Sum of Part 2}) - \text{Total Sum} $$ ### Step-by-Step Solution * **Given:** * Average of all $11$ results = $60$ * Average of the first $6$ results = $58$ * Average of the last $6$ results = $63$ * **Calculation / Deduction:** 1. Calculate the sum of all $11$ results: $\text{Total Sum} = 11 \times 60 = 660$ 2. Calculate the sum of the first $6$ results: $\text{Sum of First } 6 = 6 \times 58 = 348$ 3. Calculate the sum of the last $6$ results: $\text{Sum of Last } 6 = 6 \times 63 = 378$ 4. Determine the $6^{\text{th}}$ result: Notice that $6$ results + $6$ results = $12$ results. The $6^{\text{th}}$ result is counted in both groups. $6^{\text{th}} \text{ Result} = (\text{Sum of First } 6 + \text{Sum of Last } 6) - \text{Total Sum}$ $6^{\text{th}} \text{ Result} = (348 + 378) - 660$ $6^{\text{th}} \text{ Result} = 726 - 660 = 66$ ### Exam Strategy & Shortcut Instead of calculating massive sums, use the **Net Deviation Method**. The overall average is $60$. The first $6$ results deviate by $-2$ each ($58 - 60$). Total deviation $= 6 \times (-2) = -12$. The last $6$ results deviate by $+3$ each ($63 - 60$). Total deviation $= 6 \times (+3) = +18$. Net deviation $= -12 + 18 = +6$. The $6^{\text{th}}$ result must be the overall average plus this net deviation: $60 + 6 = 66$. This mental math completely avoids large multiplications like $6 \times 63$. ### Common Pitfall Students often get confused by the overlap and subtract the overlapping subsets from each other rather than summing them and subtracting the master total. Always visualize the set: $\{1..6\}$ and $\{6..11\}$. Adding them gives $\{1..11\} + \{6\}$. Subtracting the master sum $\{1..11\}$ leaves just $\{6\}$. ### Final Answer Therefore, the correct answer is **66**.