1. **Problem statement:** Find the values of $x$ and $y$ given: - The median of the numbers $1, 3, x, 3y, 10, 15$ arranged in ascending order is 8. - The mean of the numbers $3, 2x, y, 7, 11$ is $7 \frac{3}{5}$. 2. **Median formula and explanation:** For an even number of terms, the median is the average of the two middle numbers after sorting. 3. **Step 1: Median condition** There are 6 numbers: $1, 3, x, 3y, 10, 15$. Sorted ascending, the median is the average of the 3rd and 4th terms. Given median = 8, so: $$\frac{\text{3rd term} + \text{4th term}}{2} = 8 \implies \text{3rd term} + \text{4th term} = 16$$ 4. **Step 2: Mean condition** Mean of $3, 2x, y, 7, 11$ is $7 \frac{3}{5} = \frac{38}{5}$. Mean formula: $$\frac{3 + 2x + y + 7 + 11}{5} = \frac{38}{5}$$ Simplify numerator: $$3 + 7 + 11 = 21$$ So: $$\frac{21 + 2x + y}{5} = \frac{38}{5} \implies 21 + 2x + y = 38$$ Therefore: $$2x + y = 17$$ 5. **Step 3: Analyze median order** Since $1 < 3 < 10 < 15$, $x$ and $3y$ must fit between these. The 3rd and 4th terms sum to 16. Possible pairs for (3rd, 4th) are $(x, 3y)$ or $(3y, x)$ depending on their values. 6. **Step 4: Try $x \leq 3y$** If $x$ is 3rd and $3y$ is 4th: $$x + 3y = 16$$ From mean equation: $$2x + y = 17$$ 7. **Step 5: Solve system** From $x + 3y = 16$, express $x$: $$x = 16 - 3y$$ Substitute into $2x + y = 17$: $$2(16 - 3y) + y = 17$$ $$32 - 6y + y = 17$$ $$32 - 5y = 17$$ $$-5y = 17 - 32 = -15$$ $$y = 3$$ 8. **Step 6: Find $x$** $$x = 16 - 3(3) = 16 - 9 = 7$$ 9. **Step 7: Verify order** Numbers: $1, 3, 7, 3(3)=9, 10, 15$ Sorted ascending: $1, 3, 7, 9, 10, 15$ Median = $(7 + 9)/2 = 8$ correct. Mean check: $$3 + 2(7) + 3 + 7 + 11 = 3 + 14 + 3 + 7 + 11 = 38$$ Mean = $38/5 = 7.6 = 7 \frac{3}{5}$ correct. **Final answer:** $$x = 7, \quad y = 3$$