Mathematics 10 — Exercise 11.1 · Q1 (i–ii)

Full question

The table gives the scores in a memory test, in classes 44–48, 49–53, 54–58, 59–63, 64–68, 69–73 and 74–78. Draw a cumulative-frequency graph. Use it to estimate the median, first and third quartiles Q₁ and Q₃, third decile D₃, 90th percentile P₉₀ and interquartile range. Give answers to the nearest whole number.

Step-by-step solution

1. The scores are whole numbers. Convert each interval to continuous boundaries by subtracting 0.5 from its lower limit and adding 0.5 to its upper limit:44–48 becomes 43.5–48.5,49–53 becomes 48.5–53.5, and so on.

2. Cumulative frequency means a running total. Begin with zero at the first lower boundary.

3. Class 87/2–97/2: add frequency 3 to the preceding total.

   $$0+3=3$$

4. Class 97/2–107/2: add frequency 12 to the preceding total.

   $$3+12=15$$

5. Class 107/2–117/2: add frequency 15 to the preceding total.

   $$15+15=30$$

6. Class 117/2–127/2: add frequency 23 to the preceding total.

   $$30+23=53$$

7. Class 127/2–137/2: add frequency 12 to the preceding total.

   $$53+12=65$$

8. Class 137/2–147/2: add frequency 8 to the preceding total.

   $$65+8=73$$

9. Class 147/2–157/2: add frequency 7 to the preceding total.

   $$73+7=80$$

10. Plot each UPPER boundary against its cumulative frequency, and join adjacent points.

11. Median: find the cumulative-frequency position 1/2×80.

    $$\frac{1}{2}\times 80=40$$

12. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 30=-30$$

13. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{117\times 1}{1\times 2}$$

14. Calculate these two products separately.

    $$\frac{117}{2}$$

15. Calculate this operation before combining the remaining terms.

    $$117\times \frac{1}{2}=\frac{117}{2}$$

16. Keep the common denominator and add the signed numerators.

    $$\frac{100+2691}{46}=\frac{2791}{46}$$

17. Calculate this operation before combining the remaining terms.

    $$\frac{50}{23}+\frac{117}{2}=\frac{2791}{46}$$

18. Median lies between cumulative totals 30 and 53. Interpolate along that graph segment: lower boundary + (position−previous total)/class frequency × class width.

    $$\frac{40-30}{23}\times 5+\frac{117}{2}$$

19. Calculate this operation before combining the remaining terms.

    $$40-30=10$$

20. Calculate this operation before combining the remaining terms.

    $$10\times 1=10$$

21. Calculate this operation before combining the remaining terms.

    $$1\times 23=23$$

22. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{10\times 1}{1\times 23}$$

23. Calculate these two products separately.

    $$\frac{10}{23}$$

24. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{1\times 1}{1\times 23}$$

25. Calculate these two products separately.

    $$\frac{1}{23}$$

26. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{23}=\frac{1}{23}$$

27. Calculate this operation before combining the remaining terms.

    $$10\times \frac{1}{23}=\frac{10}{23}$$

28. Calculate this operation before combining the remaining terms.

    $$10\times 5=50$$

29. Calculate this operation before combining the remaining terms.

    $$23\times 1=23$$

30. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{50\times 1}{1\times 23}$$

31. Calculate this operation before combining the remaining terms.

    $$50\times \frac{1}{23}=\frac{50}{23}$$

32. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{10\times 5}{23\times 1}$$

33. Calculate these two products separately.

    $$\frac{50}{23}$$

34. Calculate this operation before combining the remaining terms.

    $$\frac{10}{23}\times 5=\frac{50}{23}$$

35. To add these fractions, first use a common denominator: the least common multiple of their denominators.

    $$\mathrm{LCD}=46$$

36. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{117}{2}=\frac{117\times 23}{2\times 23}$$

37. Calculate the converted numerator; the denominator is now common.

    $$\frac{2691}{46}$$

38. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{50}{23}=\frac{50\times 2}{23\times 2}$$

39. Calculate the converted numerator; the denominator is now common.

    $$\frac{100}{46}$$

40. Calculate this operation before combining the remaining terms.

    $$2691+100=2791$$

41. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{2791\times 1}{1\times 46}$$

42. Calculate these two products separately.

    $$\frac{2791}{46}$$

43. Calculate this operation before combining the remaining terms.

    $$2791\times \frac{1}{46}=\frac{2791}{46}$$

44. Keep the common denominator and add the signed numerators.

    $$\frac{2691+100}{46}=\frac{2791}{46}$$

45. Calculate this operation before combining the remaining terms.

    $$\frac{117}{2}+\frac{50}{23}=\frac{2791}{46}$$

46. Median ≈ 60.674, rounded to a whole number: 61.

    $$\frac{2791}{46}$$

47. Q1: find the cumulative-frequency position 1/4×80.

    $$\frac{1}{4}\times 80=20$$

48. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 15=-15$$

49. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{107\times 1}{1\times 2}$$

50. Calculate this operation before combining the remaining terms.

    $$107\times \frac{1}{2}=\frac{107}{2}$$

51. Keep the common denominator and add the signed numerators.

    $$\frac{10+321}{6}=\frac{331}{6}$$

52. Calculate this operation before combining the remaining terms.

    $$\frac{5}{3}+\frac{107}{2}=\frac{331}{6}$$

53. Q1 lies between cumulative totals 15 and 30. Interpolate along that graph segment: lower boundary + (position−previous total)/class frequency × class width.

    $$\frac{20-15}{15}\times 5+\frac{107}{2}$$

54. Calculate this operation before combining the remaining terms.

    $$20-15=5$$

55. Calculate this operation before combining the remaining terms.

    $$5\times 1=5$$

56. Calculate this operation before combining the remaining terms.

    $$1\times 15=15$$

57. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{5\times 1}{1\times 15}$$

58. Calculate these two products separately.

    $$\frac{5}{15}$$

59. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=5$$

60. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{5\times 1}{1\times 5}$$

61. Calculate these two products separately.

    $$\frac{5}{5}$$

62. Show the two divisions explicitly.

    $$\frac{\frac{5}{5}}{\frac{5}{5}}$$

63. Write the reduced fraction.

    $$\frac{1}{1}$$

64. Calculate this operation before combining the remaining terms.

    $$5\times \frac{1}{5}=1$$

65. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{15\times 1}{1\times 5}$$

66. Calculate this operation before combining the remaining terms.

    $$15\times \frac{1}{5}=3$$

67. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{1\times 1}{1\times 3}$$

68. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{3}=\frac{1}{3}$$

69. Show the two divisions explicitly.

    $$\frac{\frac{5}{5}}{\frac{15}{5}}$$

70. Evaluate the numerator division.

    $$\frac{5}{5}=1$$

71. Evaluate the denominator division.

    $$\frac{15}{5}=3$$

72. Write the reduced fraction.

    $$\frac{1}{3}$$

73. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{1\times 1}{1\times 15}$$

74. Calculate these two products separately.

    $$\frac{1}{15}$$

75. Calculate this operation before combining the remaining terms.

    $$1\times \frac{1}{15}=\frac{1}{15}$$

76. Calculate this operation before combining the remaining terms.

    $$5\times \frac{1}{15}=\frac{1}{3}$$

77. Calculate this operation before combining the remaining terms.

    $$1\times 5=5$$

78. Calculate this operation before combining the remaining terms.

    $$3\times 1=3$$

79. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{5\times 1}{1\times 3}$$

80. Calculate this operation before combining the remaining terms.

    $$5\times \frac{1}{3}=\frac{5}{3}$$

81. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{1\times 5}{3\times 1}$$

82. Calculate these two products separately.

    $$\frac{5}{3}$$

83. Calculate this operation before combining the remaining terms.

    $$\frac{1}{3}\times 5=\frac{5}{3}$$

84. To add these fractions, first use a common denominator: the least common multiple of their denominators.

    $$\mathrm{LCD}=6$$

85. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{107}{2}=\frac{107\times 3}{2\times 3}$$

86. Calculate the converted numerator; the denominator is now common.

    $$\frac{321}{6}$$

87. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{5}{3}=\frac{5\times 2}{3\times 2}$$

88. Calculate the converted numerator; the denominator is now common.

    $$\frac{10}{6}$$

89. Calculate this operation before combining the remaining terms.

    $$321+10=331$$

90. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{331\times 1}{1\times 6}$$

91. Calculate these two products separately.

    $$\frac{331}{6}$$

92. Calculate this operation before combining the remaining terms.

    $$331\times \frac{1}{6}=\frac{331}{6}$$

93. Keep the common denominator and add the signed numerators.

    $$\frac{321+10}{6}=\frac{331}{6}$$

94. Calculate this operation before combining the remaining terms.

    $$\frac{107}{2}+\frac{5}{3}=\frac{331}{6}$$

95. Q1 ≈ 55.167, rounded to a whole number: 55.

    $$\frac{331}{6}$$

96. Q3: find the cumulative-frequency position 3/4×80.

    $$\frac{3}{4}\times 80=60$$

97. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 53=-53$$

98. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{127\times 1}{1\times 2}$$

99. Calculate these two products separately.

    $$\frac{127}{2}$$

100. Calculate this operation before combining the remaining terms.

     $$127\times \frac{1}{2}=\frac{127}{2}$$

101. Keep the common denominator and add the signed numerators.

     $$\frac{35+762}{12}=\frac{797}{12}$$

102. Calculate this operation before combining the remaining terms.

     $$\frac{35}{12}+\frac{127}{2}=\frac{797}{12}$$

103. Q3 lies between cumulative totals 53 and 65. Interpolate along that graph segment: lower boundary + (position−previous total)/class frequency × class width.

     $$\frac{60-53}{12}\times 5+\frac{127}{2}$$

104. Calculate this operation before combining the remaining terms.

     $$60-53=7$$

105. Calculate this operation before combining the remaining terms.

     $$7\times 1=7$$

106. Calculate this operation before combining the remaining terms.

     $$1\times 12=12$$

107. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{7\times 1}{1\times 12}$$

108. Calculate these two products separately.

     $$\frac{7}{12}$$

109. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{1\times 1}{1\times 12}$$

110. Calculate these two products separately.

     $$\frac{1}{12}$$

111. Calculate this operation before combining the remaining terms.

     $$1\times \frac{1}{12}=\frac{1}{12}$$

112. Calculate this operation before combining the remaining terms.

     $$7\times \frac{1}{12}=\frac{7}{12}$$

113. Calculate this operation before combining the remaining terms.

     $$7\times 5=35$$

114. Calculate this operation before combining the remaining terms.

     $$12\times 1=12$$

115. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{35\times 1}{1\times 12}$$

116. Calculate this operation before combining the remaining terms.

     $$35\times \frac{1}{12}=\frac{35}{12}$$

117. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{7\times 5}{12\times 1}$$

118. Calculate these two products separately.

     $$\frac{35}{12}$$

119. Calculate this operation before combining the remaining terms.

     $$\frac{7}{12}\times 5=\frac{35}{12}$$

120. To add these fractions, first use a common denominator: the least common multiple of their denominators.

     $$\mathrm{LCD}=12$$

121. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

     $$\frac{127}{2}=\frac{127\times 6}{2\times 6}$$

122. Calculate the converted numerator; the denominator is now common.

     $$\frac{762}{12}$$

123. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

     $$\frac{35}{12}=\frac{35\times 1}{12\times 1}$$

124. Calculate the converted numerator; the denominator is now common.

     $$\frac{35}{12}$$

125. Calculate this operation before combining the remaining terms.

     $$762+35=797$$

126. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{797\times 1}{1\times 12}$$

127. Calculate this operation before combining the remaining terms.

     $$797\times \frac{1}{12}=\frac{797}{12}$$

128. Keep the common denominator and add the signed numerators.

     $$\frac{762+35}{12}=\frac{797}{12}$$

129. Calculate this operation before combining the remaining terms.

     $$\frac{127}{2}+\frac{35}{12}=\frac{797}{12}$$

130. Q3 ≈ 66.417, rounded to a whole number: 66.

     $$\frac{797}{12}$$

131. D3: find the cumulative-frequency position 3/10×80.

     $$\frac{3}{10}\times 80=24$$

132. Keep the common denominator and add the signed numerators.

     $$\frac{6+107}{2}=\frac{113}{2}$$

133. Calculate this operation before combining the remaining terms.

     $$3+\frac{107}{2}=\frac{113}{2}$$

134. D3 lies between cumulative totals 15 and 30. Interpolate along that graph segment: lower boundary + (position−previous total)/class frequency × class width.

     $$\frac{24-15}{15}\times 5+\frac{107}{2}$$

135. Calculate this operation before combining the remaining terms.

     $$24-15=9$$

136. Calculate this operation before combining the remaining terms.

     $$9\times 1=9$$

137. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{9\times 1}{1\times 15}$$

138. Calculate these two products separately.

     $$\frac{9}{15}$$

139. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

     $$\mathrm{GCD}=3$$

140. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{9\times 1}{1\times 3}$$

141. Calculate these two products separately.

     $$\frac{9}{3}$$

142. Show the two divisions explicitly.

     $$\frac{\frac{9}{3}}{\frac{3}{3}}$$

143. Evaluate the denominator division.

     $$\frac{3}{3}=1$$

144. Calculate this operation before combining the remaining terms.

     $$9\times \frac{1}{3}=3$$

145. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{15\times 1}{1\times 3}$$

146. Calculate these two products separately.

     $$\frac{15}{3}$$

147. Show the two divisions explicitly.

     $$\frac{\frac{15}{3}}{\frac{3}{3}}$$

148. Write the reduced fraction.

     $$5\times \frac{1}{1}$$

149. Calculate this operation before combining the remaining terms.

     $$15\times \frac{1}{3}=5$$

150. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{3\times 1}{1\times 5}$$

151. Calculate this operation before combining the remaining terms.

     $$3\times \frac{1}{5}=\frac{3}{5}$$

152. Show the two divisions explicitly.

     $$\frac{\frac{9}{3}}{\frac{15}{3}}$$

153. Evaluate the numerator division.

     $$\frac{9}{3}=3$$

154. Evaluate the denominator division.

     $$\frac{15}{3}=5$$

155. Write the reduced fraction.

     $$\frac{3}{5}$$

156. Calculate this operation before combining the remaining terms.

     $$9\times \frac{1}{15}=\frac{3}{5}$$

157. Calculate this operation before combining the remaining terms.

     $$3\times 5=15$$

158. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{3\times 5}{5\times 1}$$

159. Calculate these two products separately.

     $$\frac{15}{5}$$

160. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

     $$\mathrm{GCD}=5$$

161. Calculate this operation before combining the remaining terms.

     $$\frac{1}{1}=1$$

162. Show the two divisions explicitly.

     $$\frac{\frac{15}{5}}{\frac{5}{5}}$$

163. Evaluate the numerator division.

     $$\frac{15}{5}=3$$

164. Evaluate the denominator division.

     $$\frac{5}{5}=1$$

165. Write the reduced fraction.

     $$3\times \frac{1}{1}$$

166. Calculate this operation before combining the remaining terms.

     $$\frac{3}{5}\times 5=3$$

167. To add these fractions, first use a common denominator: the least common multiple of their denominators.

     $$\mathrm{LCD}=2$$

168. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

     $$\frac{107}{2}=\frac{107\times 1}{2\times 1}$$

169. Calculate the converted numerator; the denominator is now common.

     $$\frac{107}{2}$$

170. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

     $$3\times \frac{1}{1}=\frac{3\times 2}{1\times 2}$$

171. Calculate the converted numerator; the denominator is now common.

     $$\frac{6}{2}$$

172. Calculate this operation before combining the remaining terms.

     $$107+6=113$$

173. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{113\times 1}{1\times 2}$$

174. Calculate these two products separately.

     $$\frac{113}{2}$$

175. Calculate this operation before combining the remaining terms.

     $$113\times \frac{1}{2}=\frac{113}{2}$$

176. Keep the common denominator and add the signed numerators.

     $$\frac{107+6}{2}=\frac{113}{2}$$

177. Calculate this operation before combining the remaining terms.

     $$\frac{107}{2}+3=\frac{113}{2}$$

178. D3 ≈ 56.500, rounded to a whole number: 57.

     $$\frac{113}{2}$$

179. P90: find the cumulative-frequency position 9/10×80.

     $$\frac{9}{10}\times 80=72$$

180. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times 65=-65$$

181. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{137\times 1}{1\times 2}$$

182. Calculate these two products separately.

     $$\frac{137}{2}$$

183. Calculate this operation before combining the remaining terms.

     $$137\times \frac{1}{2}=\frac{137}{2}$$

184. Keep the common denominator and add the signed numerators.

     $$\frac{35+548}{8}=\frac{583}{8}$$

185. Calculate this operation before combining the remaining terms.

     $$\frac{35}{8}+\frac{137}{2}=\frac{583}{8}$$

186. P90 lies between cumulative totals 65 and 73. Interpolate along that graph segment: lower boundary + (position−previous total)/class frequency × class width.

     $$\frac{72-65}{8}\times 5+\frac{137}{2}$$

187. Calculate this operation before combining the remaining terms.

     $$72-65=7$$

188. Calculate this operation before combining the remaining terms.

     $$1\times 8=8$$

189. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{7\times 1}{1\times 8}$$

190. Calculate these two products separately.

     $$\frac{7}{8}$$

191. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{1\times 1}{1\times 8}$$

192. Calculate these two products separately.

     $$\frac{1}{8}$$

193. Calculate this operation before combining the remaining terms.

     $$1\times \frac{1}{8}=\frac{1}{8}$$

194. Calculate this operation before combining the remaining terms.

     $$7\times \frac{1}{8}=\frac{7}{8}$$

195. Calculate this operation before combining the remaining terms.

     $$8\times 1=8$$

196. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{35\times 1}{1\times 8}$$

197. Calculate this operation before combining the remaining terms.

     $$35\times \frac{1}{8}=\frac{35}{8}$$

198. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{7\times 5}{8\times 1}$$

199. Calculate these two products separately.

     $$\frac{35}{8}$$

200. Calculate this operation before combining the remaining terms.

     $$\frac{7}{8}\times 5=\frac{35}{8}$$

201. To add these fractions, first use a common denominator: the least common multiple of their denominators.

     $$\mathrm{LCD}=8$$

202. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

     $$\frac{137}{2}=\frac{137\times 4}{2\times 4}$$

203. Calculate the converted numerator; the denominator is now common.

     $$\frac{548}{8}$$

204. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

     $$\frac{35}{8}=\frac{35\times 1}{8\times 1}$$

205. Calculate the converted numerator; the denominator is now common.

     $$\frac{35}{8}$$

206. Calculate this operation before combining the remaining terms.

     $$548+35=583$$

207. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{583\times 1}{1\times 8}$$

208. Calculate these two products separately.

     $$\frac{583}{8}$$

209. Calculate this operation before combining the remaining terms.

     $$583\times \frac{1}{8}=\frac{583}{8}$$

210. Keep the common denominator and add the signed numerators.

     $$\frac{548+35}{8}=\frac{583}{8}$$

211. Calculate this operation before combining the remaining terms.

     $$\frac{137}{2}+\frac{35}{8}=\frac{583}{8}$$

212. P90 ≈ 72.875, rounded to a whole number: 73.

     $$\frac{583}{8}$$

213. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{\left(-1\right)\times 331}{1\times 6}$$

214. Calculate these two products separately.

     $$-\frac{331}{6}$$

215. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times \frac{331}{6}=-\frac{331}{6}$$

216. Interquartile range is Q3−Q1.

     $$\frac{797}{12}-\frac{331}{6}$$

217. To add these fractions, first use a common denominator: the least common multiple of their denominators.

     $$\mathrm{LCD}=12$$

218. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

     $$\frac{797}{12}=\frac{797\times 1}{12\times 1}$$

219. Calculate the converted numerator; the denominator is now common.

     $$\frac{797}{12}$$

220. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{\left(-331\right)\times 1}{1\times 6}$$

221. Calculate this operation before combining the remaining terms.

     $$\left(-331\right)\times \frac{1}{6}=-\frac{331}{6}$$

222. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

     $$-\frac{331}{6}=\frac{\left(-331\right)\times 2}{6\times 2}$$

223. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{\left(-662\right)\times 1}{1\times 12}$$

224. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

     $$\mathrm{GCD}=2$$

225. Show the two divisions explicitly.

     $$\frac{-\frac{662}{2}}{\frac{12}{2}}$$

226. Evaluate the numerator division.

     $$-\frac{662}{2}=-331$$

227. Evaluate the denominator division.

     $$\frac{12}{2}=6$$

228. Calculate this operation before combining the remaining terms.

     $$\left(-662\right)\times \frac{1}{12}=-\frac{331}{6}$$

229. Calculate the converted numerator; the denominator is now common.

     $$-\frac{662}{12}$$

230. Calculate this operation before combining the remaining terms.

     $$\left(-1\right)\times 662=-662$$

231. Calculate this operation before combining the remaining terms.

     $$797-662=135$$

232. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{135\times 1}{1\times 12}$$

233. Calculate these two products separately.

     $$\frac{135}{12}$$

234. Calculate this operation before combining the remaining terms.

     $$135\times \frac{1}{12}=\frac{45}{4}$$

235. Keep the common denominator and add the signed numerators.

     $$\frac{797-662}{12}=\frac{135}{12}$$

236. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

     $$\mathrm{GCD}=3$$

237. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{135\times 1}{1\times 3}$$

238. Calculate these two products separately.

     $$\frac{135}{3}$$

239. Show the two divisions explicitly.

     $$\frac{\frac{135}{3}}{\frac{3}{3}}$$

240. Write the reduced fraction.

     $$45\times \frac{1}{1}$$

241. Calculate this operation before combining the remaining terms.

     $$135\times \frac{1}{3}=45$$

242. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{12\times 1}{1\times 3}$$

243. Calculate these two products separately.

     $$\frac{12}{3}$$

244. Show the two divisions explicitly.

     $$\frac{\frac{12}{3}}{\frac{3}{3}}$$

245. Write the reduced fraction.

     $$4\times \frac{1}{1}$$

246. Calculate this operation before combining the remaining terms.

     $$12\times \frac{1}{3}=4$$

247. For multiplication of fractions, multiply the numerators together and the denominators together.

     $$\frac{45\times 1}{1\times 4}$$

248. Calculate this operation before combining the remaining terms.

     $$45\times \frac{1}{4}=\frac{45}{4}$$

249. Show the two divisions explicitly.

     $$\frac{\frac{135}{3}}{\frac{12}{3}}$$

250. Evaluate the numerator division.

     $$\frac{135}{3}=45$$

251. Evaluate the denominator division.

     $$\frac{12}{3}=4$$

252. Write the reduced fraction.

     $$\frac{45}{4}$$

253. Calculate this operation before combining the remaining terms.

     $$\frac{797}{12}-\frac{331}{6}=\frac{45}{4}$$

Answer

Median ≈ 61; Q1 ≈ 55; Q3 ≈ 66; D3 ≈ 57; P90 ≈ 73; IQR ≈ 11