Mean Median 966Af9

1. **State the problem:** We have a list $S = \{1, 2, x, x+1, x+1, 15\}$ with $2 \leq x \leq 14$. The mean of $S$ is one more than the median of $S$. We need to find the value of $x$. 2. **Recall formulas:** - Mean $= \frac{\text{sum of all elements}}{\text{number of elements}}$ - Median for an even number of elements is the average of the two middle numbers when the list is sorted. 3. **Sort the list:** Since $x \geq 2$, the order is $1, 2, x, x+1, x+1, 15$. 4. **Find the median:** The list has 6 elements, so median is average of 3rd and 4th elements: $$\text{median} = \frac{x + (x+1)}{2} = \frac{2x + 1}{2}$$ 5. **Find the mean:** Sum of elements: $$1 + 2 + x + (x+1) + (x+1) + 15 = 1 + 2 + x + x + 1 + x + 1 + 15 = 20 + 3x$$ Number of elements = 6, so $$\text{mean} = \frac{20 + 3x}{6}$$ 6. **Set up the equation:** Mean is one more than median: $$\frac{20 + 3x}{6} = 1 + \frac{2x + 1}{2}$$ 7. **Solve the equation:** Multiply both sides by 6: $$20 + 3x = 6 \times 1 + 6 \times \frac{2x + 1}{2} = 6 + 3(2x + 1) = 6 + 6x + 3 = 9 + 6x$$ Rewrite: $$20 + 3x = 9 + 6x$$ Subtract $3x$ from both sides: $$20 = 9 + 3x$$ Subtract 9 from both sides: $$20 - 9 = 3x$$ $$11 = 3x$$ Divide both sides by 3: $$x = \frac{11}{3} = 3 \frac{2}{3}$$ 8. **Check if $x$ is in the allowed range:** $2 \leq 3 \frac{2}{3} \leq 14$ is true. 9. **Final answer:** $x = 3 \frac{2}{3}$ which corresponds to option B.