The average of $11$ numbers is $50$. If the average of the first six is $49$ and that of the last six is $52$, find the sixth number.

**Set up.** Total of all $11$ numbers $= 11 \times 50 = 550$. Sum of first six $= 6 \times 49 = 294$. Sum of last six $= 6 \times 52 = 312$. **Key observation.** The sixth number is counted in both groups (it's the last of the first six *and* the first of the last six). So $294 + 312 = 606 = \text{sum of all 11} + \text{sixth number}$. **Solve.** Sixth number $= 606 - 550 = 56$. **Why option A (52) is tempting.** Picking the average of the second group directly ignores the overlap; the sixth number is *systematically pulled toward whichever group has the higher mean* relative to the overall mean.