The average marks obtained by 22 candidates in an examination are 45. The average marks of the first ten are 55 and that of the last eleven are 40. The number of marks obtained by the 11th candidate is

### Concept & Strategy When a single item in a sequence separates two distinct, non-overlapping groups, we can find its value by subtracting the sum of the two groups from the total sum of all items. $$\text{Value of Missing Item} = \text{Total Sum} - (\text{Sum of Group 1} + \text{Sum of Group 2})$$ ### Step-by-Step Solution * **Given:** * Total candidates ($n$) = $22$, Average marks = $45$. * First group: $10$ candidates, Average marks = $55$. * Second group: $11$ candidates, Average marks = $40$. * **Calculation:** 1. Find the total marks of all $22$ candidates: $22 \times 45 = 990$. 2. Find the total marks of the first $10$ candidates: $10 \times 55 = 550$. 3. Find the total marks of the last $11$ candidates: $11 \times 40 = 440$. 4. Notice that the first $10$ and last $11$ candidates account for $21$ distinct candidates. The only one left out is the 11th candidate. 5. Marks of the 11th candidate = Total Marks - (Marks of first $10$ + Marks of last $11$). 6. Marks = $990 - (550 + 440) = 990 - 990 = 0$. ### Exam Strategy & Shortcut **Net Deviation Method:** Compare the subgroups to the main average ($45$). First $10$ candidates: deviation is $+10$ ($55 - 45$). Total deviation = $10 \times (+10) = +100$. Last $11$ candidates: deviation is $-5$ ($40 - 45$). Total deviation = $11 \times (-5) = -55$. Net deviation of these $21$ candidates = $+100 - 55 = +45$. For the total group of $22$ to have an average of $45$ (meaning a total deviation of $0$), the 11th candidate must perfectly balance this $+45$ deviation. Therefore, the 11th candidate must be $45$ below the average. $45 - 45 = 0$. ### Common Pitfall Students might assume a calculation error when they get an answer of $0$ and waste time recalculating. Zero is a perfectly valid mathematical outcome for a test score or a missing value in average problems. ### Final Answer **Therefore, the correct answer is 0.**