Mean of Student Scores

Let the 10 scores in non-decreasing order be $x_1 \le x_2 \le \dots \le x_{10}$. Sum of lowest 9 scores: $x_1 + x_2 + \dots + x_9 = 9 \times 42 = 378$. Sum of highest 9 scores: $x_2 + x_3 + \dots + x_{10} = 9 \times 47 = 423$. Subtracting these two equations gives: $x_{10} - x_1 = 423 - 378 = 45$ Total sum $S = x_1 + x_2 + \dots + x_9 + x_{10} = 378 + x_{10}$. To maximize $S$, we maximize $x_{10}$. Since $x_9 \le x_{10}$ and $x_9 = 378 - (x_1 + \dots + x_8)$: The maximum and minimum values of $x_{10}$ depend on the bounds for $x_1$ and $x_{10}$. Notice that the total sum can also be written as $S = 423 + x_1$. Difference between max mean and min mean = $\frac{\Delta S}{10} = \frac{x_1^{\text{max}} - x_1^{\text{min}}}{10}$. Since $x_1 \le x_2 \le \dots \le x_9$, $x_1 \le 42$ and $x_9 \ge 42$. Also $x_2 \le \dots \le x_{10}$, so $x_2 \le 47$ and $x_{10} \ge 47$. Since $x_{10} = x_1 + 45$, and $x_1 \le 42$, $x_{10} \le 87$. Also, $x_9 \le x_{10} \implies x_9 \le x_1 + 45$. To find the difference directly: Mean difference $= 47 - 42 - 1 = 4$.