1. **Problem 9a:** Given the numbers 33, 33, 66, 77, and 88, find the arithmetic mean, mode, and median. 2. **Arithmetic mean formula:** $$\text{Mean} = \frac{\text{Sum of all numbers}}{\text{Number of numbers}}$$ 3. Calculate the sum: $$33 + 33 + 66 + 77 + 88 = 297$$ 4. Number of numbers is 5. 5. Calculate the mean: $$\text{Mean} = \frac{297}{5} = 59.4$$ 6. **Mode:** The number that appears most frequently. Here, 33 appears twice, others once, so $$\text{Mode} = 33$$ 7. **Median:** The middle number when sorted. Sorted list: 33, 33, 66, 77, 88 Middle number (3rd) is 66, so $$\text{Median} = 66$$ --- 8. **Problem 9b:** Find the number to fill in the blank in [9], [4], [ ], [8], [1] so that the mean is 5. 9. Let the missing number be $x$. 10. Sum of known numbers: $$9 + 4 + 8 + 1 = 22$$ 11. Total numbers: 5 12. Mean formula: $$5 = \frac{22 + x}{5}$$ 13. Multiply both sides by 5: $$5 \times 5 = 22 + x$$ $$25 = 22 + x$$ 14. Solve for $x$: $$x = 25 - 22 = 3$$ --- 15. **Problem 9c:** Fill the blank in [3], [5], [ ], [8], [9] so that mode equals median. 16. Sorted list with $x$: $$3, 5, x, 8, 9$$ 17. Median is the middle number (3rd), so median = $x$. 18. Mode is the most frequent number. 19. To have mode = median = $x$, $x$ must appear at least twice. 20. Since 3,5,8,9 appear once, $x$ must be equal to one of these to create a mode. 21. Try $x=5$: List: 3, 5, 5, 8, 9 Mode = 5 (appears twice), median = 5 (middle number), so condition satisfied. 22. Therefore, $$x = 5$$ --- 23. **Problem 9d:** Fill five natural numbers so that mean = median = mode. 24. Let the numbers be $a, b, c, d, e$ sorted in ascending order. 25. Median is the middle number $c$. 26. Mode is the most frequent number. 27. To have mean = median = mode, choose numbers so that $c$ is the mode and mean. 28. Example: Choose numbers [2, 4, 4, 4, 6] 29. Median is 4 (middle number). 30. Mode is 4 (appears three times). 31. Mean: $$\frac{2 + 4 + 4 + 4 + 6}{5} = \frac{20}{5} = 4$$ 32. So mean = median = mode = 4. 33. Final answer: $$[2], [4], [4], [4], [6]$$