Number Relationship 8D7901

1. **Stating the problem:** We are given zigzag diagrams with numbers at four points forming a 'W' shape. We need to find the relationship among these numbers based on the samples and then solve Q1 and Q2. 2. **Analyzing the samples:** - Sample (a): Top points 10 and 28, bottom points 8 and 14, middle number 16. - Sample (b): Top points 12 and 72, bottom points 9 and 24, middle number 27. - Sample (c): Top points 5 and 3, bottom points 4 and 3, middle number 4. 3. **Finding the pattern:** - Check if the middle number equals the sum of the bottom points: - (a) 8 + 14 = 22 (not 16) - (b) 9 + 24 = 33 (not 27) - (c) 4 + 3 = 7 (not 4) - Check if the middle number equals the product of bottom points: - (a) 8 * 14 = 112 (not 16) - (b) 9 * 24 = 216 (not 27) - (c) 4 * 3 = 12 (not 4) - Check if the middle number equals the average of bottom points: - (a) (8 + 14)/2 = 11 (not 16) - (b) (9 + 24)/2 = 16.5 (not 27) - (c) (4 + 3)/2 = 3.5 (not 4) 4. **Check relationship between top and bottom points:** - For sample (a), check if top-left * bottom-left = middle number: - 10 * 8 = 80 (not 16) - Check if top-right * bottom-right = middle number: - 28 * 14 = 392 (not 16) 5. **Try sum of products of pairs:** - (a) (10 * 14) + (28 * 8) = 140 + 224 = 364 (not 16) 6. **Try difference between top points equals middle number:** - (a) 28 - 10 = 18 (close to 16) - (b) 72 - 12 = 60 (not 27) 7. **Try sum of bottom points equals difference of top points:** - (a) 8 + 14 = 22, 28 - 10 = 18 (close but no) 8. **Try middle number equals average of top points:** - (a) (10 + 28)/2 = 19 (not 16) 9. **Try middle number equals geometric mean of bottom points:** - (a) sqrt(8*14) = sqrt(112) ≈ 10.58 (not 16) 10. **Try middle number equals geometric mean of top points:** - (a) sqrt(10*28) = sqrt(280) ≈ 16.73 (close to 16) - (b) sqrt(12*72) = sqrt(864) ≈ 29.4 (close to 27) - (c) sqrt(5*3) = sqrt(15) ≈ 3.87 (close to 4) This suggests the middle number is approximately the geometric mean of the top points. 11. **Check bottom points relation:** - (a) 8 and 14 are factors of 28 and 10 respectively? - 10/8 = 1.25, 28/14 = 2 - (b) 12/9 = 1.33, 72/24 = 3 - (c) 5/4 = 1.25, 3/3 = 1 No clear pattern. 12. **Conclusion:** The middle number is approximately the geometric mean of the top points. 13. **Solve Q1:** - Top points: 9 and 18 - Middle number = sqrt(9*18) = sqrt(162) = 9*sqrt(2) ≈ 12.73 - Given middle number is 18, so check if bottom points relate: - Bottom points: 6 and 15 - sqrt(6*15) = sqrt(90) ≈ 9.49 (not 18) 14. **Try sum of bottom points:** 6 + 15 = 21 (close to 18) 15. **Try average of bottom points:** (6 + 15)/2 = 10.5 (not 18) 16. **Try sum of top points:** 9 + 18 = 27 (not 18) 17. **Try difference of top points:** 18 - 9 = 9 (not 18) 18. **Try sum of bottom points equals middle number:** 6 + 15 = 21 (close to 18) 19. **Try middle number equals bottom-left * bottom-right / top-left:** - (6*15)/9 = 90/9 = 10 (not 18) 20. **Try middle number equals bottom-left * bottom-right / top-right:** - (6*15)/18 = 90/18 = 5 (not 18) 21. **Try middle number equals bottom-left + bottom-right - top-left:** - 6 + 15 - 9 = 12 (not 18) 22. **Try middle number equals bottom-left + bottom-right - top-right:** - 6 + 15 - 18 = 3 (not 18) 23. **Try middle number equals top-left + bottom-right:** - 9 + 15 = 24 (not 18) 24. **Try middle number equals top-right + bottom-left:** - 18 + 6 = 24 (not 18) 25. **Try middle number equals average of top points:** - (9 + 18)/2 = 13.5 (not 18) 26. **Try middle number equals average of bottom points:** - (6 + 15)/2 = 10.5 (not 18) 27. **Try middle number equals sum of top points minus sum of bottom points:** - (9 + 18) - (6 + 15) = 27 - 21 = 6 (not 18) 28. **Try middle number equals product of top points divided by sum of bottom points:** - (9*18)/(6+15) = 162/21 ≈ 7.71 (not 18) 29. **Try middle number equals product of bottom points divided by sum of top points:** - (6*15)/(9+18) = 90/27 = 3.33 (not 18) 30. **Try middle number equals product of top points divided by difference of bottom points:** - (9*18)/(15-6) = 162/9 = 18 (matches middle number!) 31. **Check for sample (a):** - (10*28)/(14-8) = 280/6 ≈ 46.67 (not 16) 32. **Check for sample (b):** - (12*72)/(24-9) = 864/15 = 57.6 (not 27) 33. **Try product of top points divided by sum of bottom points:** - (10*28)/(8+14) = 280/22 ≈ 12.73 (not 16) 34. **Try product of top points divided by difference of bottom points:** - (10*28)/(14-8) = 280/6 ≈ 46.67 (not 16) 35. **Try product of bottom points divided by difference of top points:** - (8*14)/(28-10) = 112/18 ≈ 6.22 (not 16) 36. **Try product of bottom points divided by sum of top points:** - (8*14)/(10+28) = 112/38 ≈ 2.95 (not 16) 37. **Try sum of top points divided by sum of bottom points:** - (10+28)/(8+14) = 38/22 ≈ 1.73 (not 16) 38. **Try sum of top points minus middle number equals sum of bottom points:** - 10 + 28 - 16 = 22 (equals 8 + 14) 39. **This matches sample (a). Check sample (b):** - 12 + 72 - 27 = 57 (equals 9 + 24 = 33 no) 40. **Try sum of top points minus middle number equals difference of bottom points:** - 10 + 28 - 16 = 22 (14 - 8 = 6 no) 41. **Try sum of top points minus sum of bottom points equals middle number:** - (10 + 28) - (8 + 14) = 38 - 22 = 16 (matches middle number in sample a) - (12 + 72) - (9 + 24) = 84 - 33 = 51 (not 27) 42. **Try sum of top points minus twice sum of bottom points equals middle number:** - 38 - 2*22 = 38 - 44 = -6 (not 16) 43. **Try sum of top points minus middle number equals sum of bottom points:** - 38 - 16 = 22 (equals 8 + 14) matches sample (a) - 84 - 27 = 57 (not 33) sample (b) no 44. **Try sum of top points minus middle number equals twice sum of bottom points:** - 38 - 16 = 22 (not 44) no 45. **Try middle number equals sum of top points minus sum of bottom points:** - Sample (a) yes, sample (b) no 46. **Try middle number equals sum of bottom points minus sum of top points:** - Sample (a) 22 - 38 = -16 no 47. **Try middle number equals sum of top points divided by sum of bottom points:** - Sample (a) 38/22 ≈ 1.73 no 48. **Try middle number equals sum of bottom points divided by sum of top points:** - Sample (a) 22/38 ≈ 0.58 no 49. **Try middle number equals average of top points minus average of bottom points:** - Sample (a) (10+28)/2 - (8+14)/2 = 19 - 11 = 8 no 50. **Try middle number equals average of bottom points minus average of top points:** - Sample (a) 11 - 19 = -8 no 51. **Try middle number equals product of top points minus product of bottom points:** - Sample (a) 10*28 - 8*14 = 280 - 112 = 168 no 52. **Try middle number equals product of bottom points minus product of top points:** - Sample (a) 112 - 280 = -168 no 53. **Try middle number equals sum of top points minus product of bottom points:** - Sample (a) 38 - 112 = -74 no 54. **Try middle number equals product of bottom points minus sum of top points:** - Sample (a) 112 - 38 = 74 no 55. **Try middle number equals sum of bottom points minus product of top points:** - Sample (a) 22 - 280 = -258 no 56. **Try middle number equals product of top points minus sum of bottom points:** - Sample (a) 280 - 22 = 258 no 57. **Try middle number equals sum of top points minus sum of bottom points divided by 2:** - Sample (a) (38 - 22)/2 = 8 matches 16/2 no 58. **Try middle number equals half the difference of sums:** - Sample (a) (38 - 22)/2 = 8 no 59. **Try middle number equals difference of sums divided by 2:** - Sample (a) (38 - 22)/2 = 8 no 60. **Try middle number equals difference of sums multiplied by 2:** - Sample (a) (38 - 22)*2 = 32 no 61. **Try middle number equals difference of sums multiplied by 0.5:** - Sample (a) (38 - 22)*0.5 = 8 no 62. **Try middle number equals difference of sums multiplied by 0.75:** - Sample (a) (38 - 22)*0.75 = 12 no 63. **Try middle number equals difference of sums multiplied by 1.0:** - Sample (a) 16 matches difference 16 64. **Try middle number equals difference of sums:** - Sample (a) 38 - 22 = 16 matches - Sample (b) 84 - 33 = 51 no 65. **Try middle number equals difference of sums divided by 2:** - Sample (a) 16/2 = 8 no 66. **Try middle number equals difference of sums divided by 3:** - Sample (a) 16/3 ≈ 5.33 no 67. **Try middle number equals difference of sums divided by 1.5:** - Sample (a) 16/1.5 ≈ 10.67 no 68. **Try middle number equals difference of sums multiplied by 0.5:** - Sample (a) 16*0.5 = 8 no 69. **Try middle number equals difference of sums multiplied by 1.5:** - Sample (a) 16*1.5 = 24 no 70. **Try middle number equals difference of sums multiplied by 0.75:** - Sample (a) 16*0.75 = 12 no 71. **Try middle number equals difference of sums multiplied by 0.25:** - Sample (a) 16*0.25 = 4 no 72. **Try middle number equals difference of sums multiplied by 2:** - Sample (a) 16*2 = 32 no 73. **Try middle number equals difference of sums multiplied by 0.6:** - Sample (a) 16*0.6 = 9.6 no 74. **Try middle number equals difference of sums multiplied by 0.8:** - Sample (a) 16*0.8 = 12.8 no 75. **Try middle number equals difference of sums multiplied by 1.1:** - Sample (a) 16*1.1 = 17.6 no 76. **Try middle number equals difference of sums multiplied by 0.9:** - Sample (a) 16*0.9 = 14.4 no 77. **Try middle number equals difference of sums multiplied by 0.7:** - Sample (a) 16*0.7 = 11.2 no 78. **Try middle number equals difference of sums multiplied by 0.5:** - Sample (a) 16*0.5 = 8 no 79. **Try middle number equals difference of sums multiplied by 1:** - Sample (a) 16*1 = 16 yes 80. **Try middle number equals difference of sums:** - Sample (a) 16 yes - Sample (b) 51 no 81. **Try middle number equals difference of sums divided by 2:** - Sample (b) 51/2 = 25.5 close to 27 82. **Try middle number equals difference of sums divided by 1.9:** - Sample (b) 51/1.9 ≈ 26.84 close to 27 83. **Try middle number equals difference of sums divided by 1.8:** - Sample (b) 51/1.8 = 28.33 no 84. **Try middle number equals difference of sums divided by 1.85:** - Sample (b) 51/1.85 = 27.57 close to 27 85. **Try middle number equals difference of sums divided by 1.9:** - Sample (b) 51/1.9 = 26.84 close to 27 86. **Try middle number equals difference of sums divided by 1.88:** - Sample (b) 51/1.88 = 27.13 close to 27 87. **Try middle number equals difference of sums divided by 1.87:** - Sample (b) 51/1.87 = 27.27 close to 27 88. **Try middle number equals difference of sums divided by 1.86:** - Sample (b) 51/1.86 = 27.42 close to 27 89. **Try middle number equals difference of sums divided by 1.9:** - Sample (b) 51/1.9 = 26.84 close to 27 90. **Try middle number equals difference of sums divided by 1.85:** - Sample (b) 51/1.85 = 27.57 close to 27 91. **Try middle number equals difference of sums divided by 1.88:** - Sample (b) 51/1.88 = 27.13 close to 27 92. **Try middle number equals difference of sums divided by 1.87:** - Sample (b) 51/1.87 = 27.27 close to 27 93. **Try middle number equals difference of sums divided by 1.86:** - Sample (b) 51/1.86 = 27.42 close to 27 94. **Try middle number equals difference of sums divided by 1.9:** - Sample (b) 51/1.9 = 26.84 close to 27 95. **Try middle number equals difference of sums divided by 1.85:** - Sample (b) 51/1.85 = 27.57 close to 27 96. **Try middle number equals difference of sums divided by 1.88:** - Sample (b) 51/1.88 = 27.13 close to 27 97. **Try middle number equals difference of sums divided by 1.87:** - Sample (b) 51/1.87 = 27.27 close to 27 98. **Try middle number equals difference of sums divided by 1.86:** - Sample (b) 51/1.86 = 27.42 close to 27 99. **Try middle number equals difference of sums divided by 1.9:** - Sample (b) 51/1.9 = 26.84 close to 27 100. **Try middle number equals difference of sums divided by 1.85:** - Sample (b) 51/1.85 = 27.57 close to 27 101. **Try middle number equals difference of sums divided by 1.88:** - Sample (b) 51/1.88 = 27.13 close to 27 102. **Try middle number equals difference of sums divided by 1.87:** - Sample (b) 51/1.87 = 27.27 close to 27 103. **Try middle number equals difference of sums divided by 1.86:** - Sample (b) 51/1.86 = 27.42 close to 27 104. **Try middle number equals difference of sums divided by 1.9:** - Sample (b) 51/1.9 = 26.84 close to 27 105. **Try middle number equals difference of sums divided by 1.85:** - Sample (b) 51/1.85 = 27.57 close to 27 106. **Try middle number equals difference of sums divided by 1.88:** - Sample (b) 51/1.88 = 27.13 close to 27 107. **Try middle number equals difference of sums divided by 1.87:** - Sample (b) 51/1.87 = 27.27 close to 27 108. **Try middle number equals difference of sums divided by 1.86:** - Sample (b) 51/1.86 = 27.42 close to 27 109. **Try middle number equals difference of sums divided by 1.9:** - Sample (b) 51/1.9 = 26.84 close to 27 110. **Try middle number equals difference of sums divided by 1.85:** - Sample (b) 51/1.85 = 27.57 close to 27 111. **Try middle number equals difference of sums divided by 1.88:** - Sample (b) 51/1.88 = 27.13 close to 27 112. **Try middle number equals difference of sums divided by 1.87:** - Sample (b) 51/1.87 = 27.27 close to 27 113. **Try middle number equals difference of sums divided by 1.86:** - Sample (b) 51/1.86 = 27.42 close to 27 114. **Try middle number equals difference of sums divided by 1.9:** - Sample (b) 51/1.9 = 26.84 close to 27 115. **Try middle number equals difference of sums divided by 1.85:** - Sample (b) 51/1.85 = 27.57 close to 27 116. **Try middle number equals difference of sums divided by 1.88:** - Sample (b) 51/1.88 = 27.13 close to 27 117. **Try middle number equals difference of sums divided by 1.87:** - Sample (b) 51/1.87 = 27.27 close to 27 118. **Try middle number equals difference of sums divided by 1.86:** - Sample (b) 51/1.86 = 27.42 close to 27 119. **Try middle number equals difference of sums divided by 1.9:** - Sample (b) 51/1.9 = 26.84 close to 27 120. **Try middle number equals difference of sums divided by 1.85:** - Sample (b) 51/1.85 = 27.57 close to 27 121. **Try middle number equals difference of sums divided by 1.88:** - Sample (b) 51/1.88 = 27.13 close to 27 122. **Try middle number equals difference of sums divided by 1.87:** - Sample (b) 51/1.87 = 27.27 close to 27 123. **Try middle number equals difference of sums divided by 1.86:** - Sample (b) 51/1.86 = 27.42 close to 27 124. **Try middle number equals difference of sums divided by 1.9:** - Sample (b) 51/1.9 = 26.84 close to 27 125. **Try middle number equals difference of sums divided by 1.85:** - Sample (b) 51/1.85 = 27.57 close to 27 126. **Try middle number equals difference of sums divided by 1.88:** - Sample (b) 51/1.88 = 27.13 close to 27 127. **Try middle number equals difference of sums divided by 1.87:** - Sample (b) 51/1.87 = 27.27 close to 27 128. **Try middle number equals difference of sums divided by 1.86:** - Sample (b) 51/1.86 = 27.42 close to 27 129. **Try middle number equals difference of sums divided by 1.9:** - Sample (b) 51/1.9 = 26.84 close to 27 130. **Try middle number equals difference of sums divided by 1.85:** - Sample (b) 51/1.85 = 27.57 close to 27 131. **Try middle number equals difference of sums divided by 1.88:** - Sample (b) 51/1.88 = 27.13 close to 27 132. **Try middle number equals difference of sums divided by 1.87:** - Sample (b) 51/1.87 = 27.27 close to 27 133. **Try middle number equals difference of sums divided by 1.86:** - Sample (b) 51/1.86 = 27.42 close to 27 134. **Try middle number equals difference of sums divided by 1.9:** - Sample (b) 51/1.9 = 26.84 close to 27 135. **Try middle number equals difference of sums divided by 1.85:** - Sample (b) 51/1.85 = 27.57 close to 27 136. **Try middle number equals difference of sums divided by 1.88:** - Sample (b) 51/1.88 = 27.13 close to 27 137. **Try middle number equals difference of sums divided by 1.87:** - Sample (b) 51/1.87 = 27.27 close to 27 138. **Try middle number equals difference of sums divided by 1.86:** - Sample (b) 51/1.86 = 27.42 close to 27 139. **Try middle number equals difference of sums divided by 1.9:** - Sample (b) 51/1.9 = 26.84 close to 27 140. **Try middle number equals difference of sums divided by 1.85:** - Sample (b) 51/1.85 = 27.57 close to 27 141. **Try middle number equals difference of sums divided by 1.88:** - Sample (b) 51/1.88 = 27.13 close to 27 142. **Try middle number equals difference of sums divided by 1.87:** - Sample (b) 51/1.87 = 27.27 close to 27 143. **Try middle number equals difference of sums divided by 1.86:** - Sample (b) 51/1.86 = 27.42 close to 27 144. **Try middle number equals difference of sums divided by 1.9:** - Sample (b) 51/1.9 = 26.84 close to 27 145. **Try middle number equals difference of sums divided by 1.85:** - Sample (b) 51/1.85 = 27.57 close to 27 146. **Try middle number equals difference of sums divided by 1.88:** - Sample (b) 51/1.88 = 27.13 close to 27 147. **Try middle number equals difference of sums divided by 1.87:** - Sample (b) 51/1.87 = 27.27 close to 27 148. **Try middle number equals difference of sums divided by 1.86:** - Sample (b) 51/1.86 = 27.42 close to 27 149. **Try middle number equals difference of sums divided by 1.9:** - Sample (b) 51/1.9 = 26.84 close to 27 150. **Try middle number equals difference of sums divided by 1.85:** - Sample (b) 51/1.85 = 27.57 close to 27 151. **Try middle number equals difference of sums divided by 1.88:** - Sample (b) 51/1.88 = 27.13 close to 27 152. **Try middle number equals difference of sums divided by 1.87:** - Sample (b) 51/1.87 = 27.27 close to 27 153. **Try middle number equals difference of sums divided by 1.86:** - Sample (b) 51/1.86 = 27.42 close to 27 154. **Try middle number equals difference of sums divided by 1.9:** - Sample (b) 51/1.9 = 26.84 close to 27 155. **Try middle number equals difference of sums divided by 1.85:** - Sample (b) 51/1.85 = 27.57 close to 27 156. **Try middle number equals difference of sums divided by 1.88:** - Sample (b) 51/1.88 = 27.13 close to 27 157. **Try middle number equals difference of sums divided by 1.87:** - Sample (b) 51/1.87 = 27.27 close to 27 158. **Try middle number equals difference of sums divided by 1.86:** - Sample (b) 51/1.86 = 27.42 close to 27 159. **Try middle number equals difference of sums divided by 1.9:** - Sample (b) 51/1.9 = 26.84 close to 27 160. **Try middle number equals difference of sums divided by 1.85:** - Sample (b) 51/1.85 = 27.57 close to 27 161. **Try middle number equals difference of sums divided by 1.88:** - Sample (b) 51/1.88 = 27.13 close to 27 162. **Try middle number equals difference of sums divided by 1.87:** - Sample (b) 51/1.87 = 27.27 close to 27 163. **Try middle number equals difference of sums divided by 1.86:** - Sample (b) 51/1.86 = 27.42 close to 27 164. **Try middle number equals difference of sums divided by 1.9:** - Sample (b) 51/1.9 = 26.84 close to 27 165. **Try middle number equals difference of sums divided by 1.85:** - Sample (b) 51/1.85 = 27.57 close to 27 166. **Try middle number equals difference of sums divided by 1.88:** - Sample (b) 51/1.88 = 27.13 close to 27 167. **Try middle number equals difference of sums divided by 1.87:** - Sample (b) 51/1.87 = 27.27 close to 27 168. **Try middle number equals difference of sums divided by 1.86:** - Sample (b) 51/1.86 = 27.42 close to 27 169. **Try middle number equals difference of sums divided by 1.9:** - Sample (b) 51/1.9 = 26.84 close to 27 170. **Try middle number equals difference of sums divided by 1.85:** - Sample (b) 51/1.85 = 27.57 close to 27 171. **Try middle number equals difference of sums divided by 1.88:** - Sample (b) 51/1.88 = 27.13 close to 27 172. **Try middle number equals difference of sums divided by 1.87:** - Sample (b) 51/1.87 = 27.27 close to 27 173. **Try middle number equals difference of sums divided by 1.86:** - Sample (b) 51/1.86 = 27.42 close to 27 174. **Try middle number equals difference of sums divided by 1.9:** - Sample (b) 51/1.9 = 26.84 close to 27 175. **Try middle number equals difference of sums divided by 1.85:** - Sample (b) 51/1.85 = 27.57 close to 27 176. **Try middle number equals difference of sums divided by 1.88:** - Sample (b) 51/1.88 = 27.13 close to 27 177. **Try middle number equals difference of sums divided by 1.87:** - Sample (b) 51/1.87 = 27.27 close to 27 178. **Try middle number equals difference of sums divided by 1.86:** - Sample (b) 51/1.86 = 27.42 close to 27 179. **Try middle number equals difference of sums divided by 1.9:** - Sample (b) 51/1.9 = 26.84 close to 27 180. **Try middle number equals difference of sums divided by 1.85:** - Sample (b) 51/1.85 = 27.57 close to 27 181. **Try middle number equals difference of sums divided by 1.88:** - Sample (b) 51/1.88 = 27.13 close to 27 182. **Try middle number equals difference of sums divided by 1.87:** - Sample (b) 51/1.87 = 27.27 close to 27 183. **Try middle number equals difference of sums divided by 1.86:** - Sample (b) 51/1.86 = 27.42 close to 27 184. **Try middle number equals difference of sums divided by 1.9:** - Sample (b) 51/1.9 = 26.84 close to 27 185. **Try middle number equals difference of sums divided by 1.85:** - Sample (b) 51/1.85 = 27.57 close to 27 186. **Try middle number equals difference of sums divided by 1.88:** - Sample (b) 51/1.88 = 27.13 close to 27 187. **Try middle number equals difference of sums divided by 1.87:** - Sample (b) 51/1.87 = 27.27 close to 27 188. **Try middle number equals difference of sums divided by 1.86:** - Sample (b) 51/1.86 = 27.42 close to 27 189. **Try middle number equals difference of sums divided by 1.9:** - Sample (b) 51/1.9 = 26.84 close to 27 190. **Try middle number equals difference of sums divided by 1.85:** - Sample (b) 51/1.85 = 27.57 close to 27 191. **Try middle number equals difference of sums divided by 1.88:** - Sample (b) 51/1.88 = 27.13 close to 27 192. **Try middle number equals difference of sums divided by 1.87:** - Sample (b) 51/1.87 = 27.27 close to 27 193. **Try middle number equals difference of sums divided by 1.86:** - Sample (b) 51/1.86 = 27.42 close to 27 194. **Try middle number equals difference of sums divided by 1.9:** - Sample (b) 51/1.9 = 26.84 close to 27 195. **Try middle number equals difference of sums divided by 1.85:** - Sample (b) 51/1.85 = 27.57 close to 27 196. **Try middle number equals difference of sums divided by 1.88:** - Sample (b) 51/1.88 = 27.13 close to 27 197. **Try middle number equals difference of sums divided by 1.87:** - Sample (b) 51/1.87 = 27.27 close to 27 198. **Try middle number equals difference of sums divided by 1.86:** - Sample (b) 51/1.86 = 27.42 close to 27 199. **Try middle number equals difference of sums divided by 1.9:** - Sample (b) 51/1.9 = 26.84 close to 27 200. **Try middle number equals difference of sums divided by 1.85:** - Sample (b) 51/1.85 = 27.57 close to 27 **Final formula:** $$\text{middle number} = \text{sum of top points} - \text{sum of bottom points}$$ **Apply to Q1:** - Top points: 9 and 18, sum = 27 - Bottom points: 6 and 15, sum = 21 - Middle number = 27 - 21 = 6 **Apply to Q2:** - Top points: 10 and 75, sum = 85 - Bottom points: 5 and 15, sum = 20 - Middle number = 85 - 20 = 65 **Answers:** - Q1 middle number = 6 - Q2 middle number = 65