Question

The average of 11 numbers is 50. The average of the first 6 numbers is 49 and the average of the last 6 numbers is 52. Find the 6th number.

• A. 54
• B. 55
• C. 56
• D. 58

Hint:

Think in terms of deviations from the overall average of 50 rather than working with the raw totals.

Answer 1

The Correct Option is C

Step 1: Take deviations from the overall average of 50. First-6 average $=49\Rightarrow$ each of these 6 numbers has an average deviation of $-1$, so the sum of first-6 deviations $=-6$.
Step 2: Last-6 average $=52\Rightarrow$ each of these 6 numbers has an average deviation of $+2$, so the sum of last-6 deviations $=+12$.
Step 3: Adding the first-6 and last-6 lists together covers all 11 numbers once each EXCEPT the 6th number, which is counted twice (once in "first 6", once in "last 6"). Combined deviation sum $=-6+12=+6$.
Step 4: Since the true average of all 11 numbers is exactly 50, the sum of deviations of all 11 numbers (counted once) is 0. So the extra $+6$ found above must be exactly the deviation of the double-counted 6th number.
Step 5: Deviation of the 6th number $=+6\Rightarrow$ 6th number $=50+6=56$.
\[\boxed{56}\]