The average of 6 numbers is 7. The average of three numbers of them is 5. What will be the average of remaining numbers?

### Concept & Logic This problem is solved by understanding that the sum of a whole group is equal to the sum of its distinct parts. We can find the sum of all 6 numbers, subtract the sum of the known 3 numbers, and then find the average of the remaining 3 numbers. $$Sum = Average \times N$$ ### Step-by-Step Solution * **Calculation (Total Group):** Find the sum of all 6 numbers. Total Sum = $6 \times 7 = 42$ * **Calculation (Subset 1):** Find the sum of the 3 specified numbers. Sum of 3 numbers = $3 \times 5 = 15$ * **Deduction (Subset 2):** Find the sum of the remaining numbers. Sum of remaining numbers = (Total Sum) - (Sum of 3 numbers) Sum of remaining numbers = $42 - 15 = 27$ * **Calculation (Final Average):** Since there were 6 numbers initially and we removed 3, there are 3 numbers remaining. Average of remaining 3 numbers = $\frac{27}{3} = 9$ ### Exam Strategy & Shortcut **Balance / Deviation Method:** The overall average is 7. Three numbers have an average of 5, which is a deviation of $-2$ from the main average. Total deficit created by these three numbers = $3 \times (-2) = -6$. For the overall average to remain 7, the remaining three numbers must compensate for this deficit of $-6$. Therefore, they must create a surplus of $+6$ spread across the 3 remaining numbers. Required surplus per number = $\frac{+6}{3} = +2$. So, the average of the remaining numbers must be $7 + 2 = 9$. This requires virtually no calculation! ### Common Pitfall A common error is confusing sums with averages. Students might find the remaining sum (27) and mistakenly look for it in the options, or they might try to subtract the averages directly (7 - 5 = 2), which violates the rules of arithmetic means. Always convert averages to sums before adding or subtracting groups. ### Final Answer **Therefore, the correct answer is 9.**