11 natural numbers are listed sequentially. The mean of the 11 numbers is 50. If the average of first 6 numbers is 49 and that of the last 6 numbers is 52, then the sixth number is
Show Hint
Whenever a set of numbers overlaps, the sum of the overlapping groups minus the total sum will directly yield the value of the overlapping element.
You can also use deviations from the mean to solve this mentally.
The first \( 6 \) numbers have a deviation of \( -1 \) each (total \( -6 \)).
The last \( 6 \) numbers have a deviation of \( +2 \) each (total \( +12 \)).
The net deviation is \( +6 \), which means the overlapping term is \( 50 + 6 = 56 \).
Step 1: Find the total sum of all 11 numbers. Mean = 50, so total sum = 11 x 50 = 550. Step 2: Find the sums of the two overlapping groups. Sum of first 6 numbers (mean = 49) = 6 x 49 = 294. Sum of last 6 numbers (mean = 52) = 6 x 52 = 312. Step 3: Find the 6th number, which appears in both groups. 6th number = (294 + 312) - 550 = 606 - 550 = 56. \[ \boxed{56} \]