The average age of seven boys sitting in a row facing North is $26$ years. If the average age of the first three boys is $19$ years and the average age of the last three boys is $32$ years, what is the age of the boy who is sitting in the middle of the row?

### Concept & Strategy When given the average of an entire group and the averages of non-overlapping subgroups that exclude exactly one item, the value of that excluded item can be found by subtracting the sum of the subgroups from the total sum. $$\text{Middle Item} = \text{Total Sum} - (\text{Sum of Group 1} + \text{Sum of Group 2})$$ ### Step-by-Step Solution * **Given:** * Total boys = $7$, Average age = $26$ years. * First $3$ boys, Average age = $19$ years. * Last $3$ boys, Average age = $32$ years. * **Calculation:** 1. Calculate the total age of all $7$ boys: $7 \times 26 = 182$ years. 2. Calculate the total age of the first $3$ boys: $3 \times 19 = 57$ years. 3. Calculate the total age of the last $3$ boys: $3 \times 32 = 96$ years. 4. Notice that the first 3 boys and the last 3 boys cover $6$ distinct boys. The only one left out is the 4th boy (the middle one). 5. Sum of the $6$ boys = $57 + 96 = 153$ years. 6. Age of the middle boy = Total Age - Sum of the $6$ boys. 7. Age = $182 - 153 = 29$ years. ### Exam Strategy & Shortcut **Deviation Method:** Compare the subgroup averages to the main average ($26$). First 3 boys: deviation is $19 - 26 = -7$ per boy. Total deviation = $3 \times (-7) = -21$. Last 3 boys: deviation is $32 - 26 = +6$ per boy. Total deviation = $3 \times (+6) = +18$. Net deviation of the $6$ boys = $-21 + 18 = -3$. In order to balance the overall average back to zero deviation, the middle boy must be $+3$ relative to the average. Middle boy = $26 + 3 = 29$. This involves minimal calculation! ### Common Pitfall Getting bogged down by the "facing North" detail. This is extraneous information meant to confuse you into thinking this is a seating arrangement/logical reasoning puzzle rather than a pure mathematics question. ### Final Answer **Therefore, the correct answer is 29 years.**