1. **Problem a:** Find digits 1 to 6 used once in the form $a \times bc / d$ to get the greatest possible answer without making an improper fraction in the mixed number. 2. To maximize $a \times \frac{bc}{d}$, maximize $a$, $bc$, and minimize $d$ while ensuring $\frac{bc}{d}$ is a proper fraction (less than 1). 3. Choose $d=6$ (largest denominator to keep fraction proper), then $bc$ must be less than 6. 4. Possible two-digit numbers less than 6 are only 12, 13, 14, 15, 16 but 16 uses 6 already as denominator, so try $bc=15$ (digits 1 and 5). 5. Remaining digit for $a$ is 4 (digits used: 1,5,6), so expression is $4 \times \frac{15}{6}$. 6. Simplify $\frac{15}{6} = \frac{5}{2} = 2\frac{1}{2}$, which is improper fraction, so not allowed. 7. Try $bc=14$, $\frac{14}{6} = \frac{7}{3}$ improper fraction again. 8. Try $bc=12$, $\frac{12}{6} = 2$ improper fraction. 9. Try $d=5$, then $bc$ less than 5, possible $bc=14$ (digits 1 and 4), $a=6$ (digits 6,1,4,5 used). 10. $\frac{14}{5} = 2\frac{4}{5}$ improper fraction again. 11. Try $d=4$, $bc$ less than 4, possible $bc=13$, $a=6$. 12. $\frac{13}{4} = 3\frac{1}{4}$ improper fraction. 13. Try $d=3$, $bc$ less than 3, possible $bc=12$, $a=6$. 14. $\frac{12}{3} = 4$ improper fraction. 15. Try $d=2$, $bc$ less than 2, no two-digit number less than 2. 16. Try $d=1$, fraction $bc/1$ is whole number, improper fraction. 17. So try $a=5$, $d=6$, $bc=14$, $\frac{14}{6} = 2\frac{1}{3}$ improper. 18. Try $a=3$, $d=6$, $bc=15$, $\frac{15}{6} = 2\frac{1}{2}$ improper. 19. Try $a=2$, $d=6$, $bc=15$, $\frac{15}{6} = 2\frac{1}{2}$ improper. 20. Try $a=1$, $d=6$, $bc=25$, invalid digits. 21. Try $a=6$, $d=5$, $bc=14$, $\frac{14}{5} = 2\frac{4}{5}$ improper. 22. Try $a=4$, $d=5$, $bc=13$, $\frac{13}{5} = 2\frac{3}{5}$ improper. 23. Try $a=3$, $d=5$, $bc=14$, $\frac{14}{5} = 2\frac{4}{5}$ improper. 24. Try $a=2$, $d=5$, $bc=14$, $\frac{14}{5} = 2\frac{4}{5}$ improper. 25. Try $a=1$, $d=5$, $bc=24$, $\frac{24}{5} = 4\frac{4}{5}$ improper. 26. Try $a=6$, $d=3$, $bc=14$, $\frac{14}{3} = 4\frac{2}{3}$ improper. 27. Try $a=5$, $d=3$, $bc=14$, $\frac{14}{3} = 4\frac{2}{3}$ improper. 28. Try $a=4$, $d=3$, $bc=15$, $\frac{15}{3} = 5$ improper. 29. Try $a=3$, $d=2$, $bc=14$, $\frac{14}{2} = 7$ improper. 30. Try $a=2$, $d=1$, $bc=36$, invalid digits. 31. Try $a=1$, $d=2$, $bc=36$, invalid digits. 32. Try $a=6$, $d=1$, $bc=25$, invalid digits. 33. Try $a=5$, $d=1$, $bc=26$, invalid digits. 34. Try $a=4$, $d=1$, $bc=25$, invalid digits. 35. Try $a=3$, $d=1$, $bc=26$, invalid digits. 36. Try $a=2$, $d=1$, $bc=35$, invalid digits. 37. Try $a=1$, $d=1$, $bc=36$, invalid digits. 38. Since no proper fraction with two-digit numerator and single-digit denominator less than 1 is possible with digits 1-6 once, try $a \times \frac{b}{c}$ with $b$ single digit. 39. Try $a=6$, $b=5$, $c=6$ invalid repeated digit. 40. Try $a=6$, $b=5$, $c=4$, $\frac{5}{4} = 1\frac{1}{4}$ improper. 41. Try $a=5$, $b=4$, $c=6$, $\frac{4}{6} = \frac{2}{3}$ proper. 42. Expression: $5 \times \frac{4}{6} = 5 \times \frac{2}{3} = \frac{10}{3} = 3\frac{1}{3}$ improper. 43. Try $a=4$, $b=3$, $c=6$, $\frac{3}{6} = \frac{1}{2}$ proper. 44. Expression: $4 \times \frac{3}{6} = 4 \times \frac{1}{2} = 2$ proper. 45. Try $a=6$, $b=1$, $c=3$, $\frac{1}{3}$ proper. 46. Expression: $6 \times \frac{1}{3} = 2$ proper. 47. Try $a=5$, $b=1$, $c=2$, $\frac{1}{2}$ proper. 48. Expression: $5 \times \frac{1}{2} = 2\frac{1}{2}$ proper. 49. The greatest possible answer without improper fraction in mixed number is $5 \times \frac{1}{2} = 2\frac{1}{2}$. --- 50. **Problem b:** Find digits 1 to 6 used once in $a \times bc / d$ to get a mixed number with fraction $\frac{1}{2}$. 51. The fraction part is $\frac{1}{2}$, so $\frac{bc}{d}$ must be a fraction with denominator $d$ and numerator $bc$ such that $\frac{bc}{d} = n + \frac{1}{2}$ for some integer $n$. 52. Since $\frac{bc}{d}$ is a fraction, $bc$ and $d$ are digits 1-6 used once. 53. Try $\frac{3}{2} = 1\frac{1}{2}$, digits 3 and 2. 54. Then $a$ is remaining digit from 1-6 excluding 3 and 2. 55. Try $a=4$, expression $4 \times \frac{3}{2} = 4 \times 1\frac{1}{2} = 6$ improper fraction but mixed number is $6$. 56. Try $a=1$, expression $1 \times \frac{3}{2} = 1\frac{1}{2}$ fraction part $\frac{1}{2}$. 57. Try $a=2$, expression $2 \times \frac{3}{2} = 3$ fraction part 0. 58. Try $a=3$, expression $3 \times \frac{3}{2} = 4\frac{1}{2}$ fraction part $\frac{1}{2}$. 59. So $3 \times \frac{3}{2} = 4\frac{1}{2}$ works. 60. Digits used: 3, 3, 2 repeated 3 twice, invalid. 61. Try $\frac{5}{2} = 2\frac{1}{2}$ digits 5 and 2. 62. $a=1$, expression $1 \times \frac{5}{2} = 2\frac{1}{2}$ fraction part $\frac{1}{2}$. 63. Digits used: 1, 5, 2 all distinct. 64. So $1 \times \frac{5}{2} = 2\frac{1}{2}$ is valid. 65. Try $\frac{1}{2}$ fraction itself, $a=3$, expression $3 \times \frac{1}{2} = 1\frac{1}{2}$ fraction part $\frac{1}{2}$. 66. Digits used: 3, 1, 2 all distinct. 67. So $3 \times \frac{1}{2} = 1\frac{1}{2}$ also valid. 68. The problem asks for mixed number answer with fraction $\frac{1}{2}$, so $3 \times \frac{1}{2} = 1\frac{1}{2}$ or $1 \times \frac{5}{2} = 2\frac{1}{2}$. **Final answers:** a) Greatest possible answer without improper fraction in mixed number is $5 \times \frac{1}{2} = 2\frac{1}{2}$. b) Mixed number answer with fraction $\frac{1}{2}$ is $3 \times \frac{1}{2} = 1\frac{1}{2}$ or $1 \times \frac{5}{2} = 2\frac{1}{2}$.