Open Middle®
Directions: Using the digits 1 to 9 exactly once, place a digit in each box to create and place 4 fractions on the number line in the correct order. (fractions B & C are equal)
Hint
The sum of the equivalent fractions is greater than a half and greater than a quarter separate. How do you know if two fractions are equivalent? How do you know if fractions are less than or greater than another fraction?
Answer
There are many possible answers. Here are a couple: A = 1/7, B = 3/8, C = 9/24, D = 5/6; Another possibility is A = 1/5, B = 3/9, C = 8/24, D = 6/7; yet another possibility is A = 5/8, B = 3/4, C = 9/12 and D = 6/7. (or D could be 7/6)
Source: Graham Fletcher, Bowen Kerins, and Kate Nowak
Here is another answer a student found A = 5/8, B = 3/4, C = 9/12 and D = 6/7. (or D could be 7/6)
Thank you for your contribution Erick! Your student’s solution has been added to the problem.
B and C are not equivalent in your first possible answer. (3/6 > 9/27)
Thank you for catching that Gina, the answer set has been updated.
This is so much fun! My students and I are wondering about A= 3/7, B=4/8, C=6/12 and D=9/5 and A= 2/7, B=3/6, C=9/18 and D=5/4 Thanks for sharing your work with us!
A group of students worked on this yesterday as they reviewed equivalent fractions and ordering…and they/we had so much fun doing! It was their first “open-middle” task with the recording sheet. I am so excited about this task. There were so many great discussions, mistakes, revisions…
Another possibility…I got 3/7, 2/4, 9/18, and 5/6.
My son spent a few hours seeking to find the most optimal answer based on the partitioning of the number line. He makes a case for 2/5, 3/6,7/14, 8/9. The difference or distance between 3/6 and 8/9 is approx. 4 times the distance between 2/5 and 3/6.
What fun. Today we were comparing fractions to 1/2. I wanted to test this out myself before I introduced it to the class. I thought, if I make B 1/2 or its equivalent, than I can use the left over digits to create a fraction less than 1/2, and another greater than 1/2. I came up with a = 3/9, b = 2/4 = 8/16, and d = 5/7. I wonder if any of my students will think this way.
I think a is 3/7, b is 2/4, d is 5/6, and c is 9/18.
I really like this problem and look forward to trying it out with my students. Is the placement on the line supposed to suggest some sort of scale or do the fractions just have to be A<B=C<D?
Here’s my answer
1/8 < (3/8 & 9/24) < 1/2
1: 4, 12, 35, 69, 78
2: 4, 12, 35, 69, 87
3: 4, 12, 35, 76, 98
4: 4, 12, 35, 78, 96
5: 4, 12, 35, 87, 96
6: 4, 12, 36, 57, 98
7: 4, 12, 36, 75, 98
8: 4, 12, 36, 78, 95
9: 4, 12, 36, 87, 95
10: 4, 12, 38, 57, 69
11: 4, 12, 38, 57, 96
12: 4, 12, 38, 69, 75
13: 4, 12, 38, 75, 96
14: 4, 12, 38, 76, 95
15: 4, 12, 39, 56, 78
16: 4, 12, 39, 56, 87
17: 4, 12, 39, 57, 68
18: 4, 12, 39, 57, 86
19: 4, 12, 39, 58, 76
20: 4, 12, 39, 65, 78
21: 4, 12, 39, 65, 87
22: 4, 12, 39, 68, 75
23: 4, 12, 39, 75, 86
24: 4, 12, 39, 76, 85
25: 4, 12, 56, 78, 93
26: 4, 12, 56, 87, 93
27: 4, 12, 57, 63, 98
28: 4, 12, 57, 68, 93
29: 4, 12, 57, 86, 93
30: 4, 12, 58, 76, 93
31: 4, 12, 63, 75, 98
32: 4, 12, 63, 78, 95
33: 4, 12, 63, 87, 95
34: 4, 12, 65, 78, 93
35: 4, 12, 65, 87, 93
36: 4, 12, 68, 75, 93
37: 4, 12, 75, 86, 93
38: 4, 12, 76, 85, 93
39: 4, 15, 26, 39, 78
40: 4, 15, 26, 39, 87
41: 4, 15, 26, 78, 93
42: 4, 15, 26, 87, 93
43: 4, 15, 27, 36, 98
44: 4, 15, 27, 38, 69
45: 4, 15, 27, 38, 96
46: 4, 15, 27, 39, 68
47: 4, 15, 27, 39, 86
48: 4, 15, 27, 63, 98
49: 4, 15, 27, 68, 93
50: 4, 15, 27, 86, 93
51: 4, 15, 28, 39, 76
52: 4, 15, 28, 76, 93
53: 4, 15, 32, 69, 78
54: 4, 15, 32, 69, 87
55: 4, 15, 32, 76, 98
56: 4, 15, 32, 78, 96
57: 4, 15, 32, 87, 96
58: 4, 15, 36, 72, 98
59: 4, 15, 36, 78, 92
60: 4, 15, 36, 87, 92
61: 4, 15, 38, 69, 72
62: 4, 15, 38, 72, 96
63: 4, 15, 38, 76, 92
64: 4, 15, 39, 62, 78
65: 4, 15, 39, 62, 87
66: 4, 15, 39, 68, 72
67: 4, 15, 39, 72, 86
68: 4, 15, 39, 76, 82
69: 4, 15, 62, 78, 93
70: 4, 15, 62, 87, 93
71: 4, 15, 63, 72, 98
72: 4, 15, 63, 78, 92
73: 4, 15, 63, 87, 92
74: 4, 15, 68, 72, 93
75: 4, 15, 72, 86, 93
76: 4, 15, 76, 82, 93
77: 4, 16, 25, 39, 78
78: 4, 16, 25, 39, 87
79: 4, 16, 25, 78, 93
80: 4, 16, 25, 87, 93
81: 4, 16, 27, 35, 98
82: 4, 16, 27, 38, 95
83: 4, 16, 27, 39, 58
84: 4, 16, 27, 39, 85
85: 4, 16, 27, 58, 93
86: 4, 16, 27, 85, 93
87: 4, 16, 28, 39, 57
88: 4, 16, 28, 39, 75
89: 4, 16, 28, 57, 93
90: 4, 16, 28, 75, 93
91: 4, 16, 32, 57, 98
92: 4, 16, 32, 75, 98
93: 4, 16, 32, 78, 95
94: 4, 16, 32, 87, 95
95: 4, 16, 35, 72, 98
96: 4, 16, 35, 78, 92
97: 4, 16, 35, 87, 92
98: 4, 16, 38, 57, 92
99: 4, 16, 38, 72, 95
100: 4, 16, 38, 75, 92
101: 4, 16, 39, 52, 78
102: 4, 16, 39, 52, 87
103: 4, 16, 39, 57, 82
104: 4, 16, 39, 58, 72
105: 4, 16, 39, 72, 85
106: 4, 16, 39, 75, 82
107: 4, 16, 52, 78, 93
108: 4, 16, 52, 87, 93
109: 4, 16, 57, 82, 93
110: 4, 16, 58, 72, 93
111: 4, 16, 72, 85, 93
112: 4, 16, 75, 82, 93
113: 4, 18, 25, 39, 76
114: 4, 18, 25, 76, 93
115: 4, 18, 26, 39, 57
116: 4, 18, 26, 39, 75
117: 4, 18, 26, 57, 93
118: 4, 18, 26, 75, 93
119: 4, 18, 27, 35, 69
120: 4, 18, 27, 35, 96
121: 4, 18, 27, 36, 95
122: 4, 18, 27, 39, 56
123: 4, 18, 27, 39, 65
124: 4, 18, 27, 56, 93
125: 4, 18, 27, 63, 95
126: 4, 18, 27, 65, 93
127: 4, 18, 32, 57, 69
128: 4, 18, 32, 57, 96
129: 4, 18, 32, 69, 75
130: 4, 18, 32, 75, 96
131: 4, 18, 32, 76, 95
132: 4, 18, 35, 69, 72
133: 4, 18, 35, 72, 96
134: 4, 18, 35, 76, 92
135: 4, 18, 36, 57, 92
136: 4, 18, 36, 72, 95
137: 4, 18, 36, 75, 92
138: 4, 18, 39, 52, 76
139: 4, 18, 39, 56, 72
140: 4, 18, 39, 57, 62
141: 4, 18, 39, 62, 75
142: 4, 18, 39, 65, 72
143: 4, 18, 52, 76, 93
144: 4, 18, 56, 72, 93
145: 4, 18, 57, 62, 93
146: 4, 18, 57, 63, 92
147: 4, 18, 62, 75, 93
148: 4, 18, 63, 72, 95
149: 4, 18, 63, 75, 92
150: 4, 18, 65, 72, 93
151: 4, 21, 35, 69, 78
152: 4, 21, 35, 69, 87
153: 4, 21, 35, 76, 98
154: 4, 21, 35, 78, 96
155: 4, 21, 35, 87, 96
156: 4, 21, 36, 57, 98
157: 4, 21, 36, 75, 98
158: 4, 21, 36, 78, 95
159: 4, 21, 36, 87, 95
160: 4, 21, 38, 57, 69
161: 4, 21, 38, 57, 96
162: 4, 21, 38, 69, 75
163: 4, 21, 38, 75, 96
164: 4, 21, 38, 76, 95
165: 4, 21, 39, 56, 78
166: 4, 21, 39, 56, 87
167: 4, 21, 39, 57, 68
168: 4, 21, 39, 57, 86
169: 4, 21, 39, 58, 76
170: 4, 21, 39, 65, 78
171: 4, 21, 39, 65, 87
172: 4, 21, 39, 68, 75
173: 4, 21, 39, 75, 86
174: 4, 21, 39, 76, 85
175: 4, 21, 56, 78, 93
176: 4, 21, 56, 87, 93
177: 4, 21, 57, 63, 98
178: 4, 21, 57, 68, 93
179: 4, 21, 57, 86, 93
180: 4, 21, 58, 76, 93
181: 4, 21, 63, 75, 98
182: 4, 21, 63, 78, 95
183: 4, 21, 63, 87, 95
184: 4, 21, 65, 78, 93
185: 4, 21, 65, 87, 93
186: 4, 21, 68, 75, 93
187: 4, 21, 75, 86, 93
188: 4, 21, 76, 85, 93
189: 4, 25, 36, 78, 91
190: 4, 25, 36, 87, 91
191: 4, 25, 38, 76, 91
192: 4, 25, 39, 76, 81
193: 4, 25, 63, 78, 91
194: 4, 25, 63, 87, 91
195: 4, 25, 76, 81, 93
196: 4, 26, 35, 78, 91
197: 4, 26, 35, 87, 91
198: 4, 26, 38, 57, 91
199: 4, 26, 38, 75, 91
200: 4, 26, 39, 51, 78
201: 4, 26, 39, 51, 87
202: 4, 26, 39, 57, 81
203: 4, 26, 39, 75, 81
204: 4, 26, 51, 78, 93
205: 4, 26, 51, 87, 93
206: 4, 26, 57, 81, 93
207: 4, 26, 75, 81, 93
208: 4, 27, 35, 68, 91
209: 4, 27, 35, 69, 81
210: 4, 27, 35, 81, 96
211: 4, 27, 35, 86, 91
212: 4, 27, 36, 51, 98
213: 4, 27, 36, 58, 91
214: 4, 27, 36, 81, 95
215: 4, 27, 36, 85, 91
216: 4, 27, 38, 51, 69
217: 4, 27, 38, 51, 96
218: 4, 27, 38, 56, 91
219: 4, 27, 38, 65, 91
220: 4, 27, 39, 51, 68
221: 4, 27, 39, 51, 86
222: 4, 27, 39, 56, 81
223: 4, 27, 39, 65, 81
224: 4, 27, 51, 63, 98
225: 4, 27, 51, 68, 93
226: 4, 27, 51, 86, 93
227: 4, 27, 56, 81, 93
228: 4, 27, 58, 63, 91
229: 4, 27, 63, 81, 95
230: 4, 27, 63, 85, 91
231: 4, 27, 65, 81, 93
232: 4, 28, 35, 76, 91
233: 4, 28, 36, 57, 91
234: 4, 28, 36, 75, 91
235: 4, 28, 39, 51, 76
236: 4, 28, 51, 76, 93
237: 4, 28, 57, 63, 91
238: 4, 28, 63, 75, 91
239: 4, 32, 51, 69, 78
240: 4, 32, 51, 69, 87
241: 4, 32, 51, 76, 98
242: 4, 32, 51, 78, 96
243: 4, 32, 51, 87, 96
244: 4, 32, 56, 78, 91
245: 4, 32, 56, 87, 91
246: 4, 32, 57, 68, 91
247: 4, 32, 57, 69, 81
248: 4, 32, 57, 81, 96
249: 4, 32, 57, 86, 91
250: 4, 32, 58, 76, 91
251: 4, 32, 65, 78, 91
252: 4, 32, 65, 87, 91
253: 4, 32, 68, 75, 91
254: 4, 32, 69, 75, 81
255: 4, 32, 75, 81, 96
256: 4, 32, 75, 86, 91
257: 4, 32, 76, 81, 95
258: 4, 32, 76, 85, 91
259: 4, 35, 62, 78, 91
260: 4, 35, 62, 87, 91
261: 4, 35, 68, 72, 91
262: 4, 35, 69, 72, 81
263: 4, 35, 72, 81, 96
264: 4, 35, 72, 86, 91
265: 4, 35, 76, 81, 92
266: 4, 35, 76, 82, 91
267: 4, 36, 51, 72, 98
268: 4, 36, 51, 78, 92
269: 4, 36, 51, 87, 92
270: 4, 36, 52, 78, 91
271: 4, 36, 52, 87, 91
272: 4, 36, 57, 81, 92
273: 4, 36, 57, 82, 91
274: 4, 36, 58, 72, 91
275: 4, 36, 72, 81, 95
276: 4, 36, 72, 85, 91
277: 4, 36, 75, 81, 92
278: 4, 36, 75, 82, 91
279: 4, 38, 51, 69, 72
280: 4, 38, 51, 72, 96
281: 4, 38, 51, 76, 92
282: 4, 38, 52, 76, 91
283: 4, 38, 56, 72, 91
284: 4, 38, 57, 62, 91
285: 4, 38, 62, 75, 91
286: 4, 38, 65, 72, 91
287: 4, 39, 51, 62, 78
288: 4, 39, 51, 62, 87
289: 4, 39, 51, 68, 72
290: 4, 39, 51, 72, 86
291: 4, 39, 51, 76, 82
292: 4, 39, 52, 76, 81
293: 4, 39, 56, 72, 81
294: 4, 39, 57, 62, 81
295: 4, 39, 62, 75, 81
296: 4, 39, 65, 72, 81
297: 4, 51, 62, 78, 93
298: 4, 51, 62, 87, 93
299: 4, 51, 63, 72, 98
300: 4, 51, 63, 78, 92
301: 4, 51, 63, 87, 92
302: 4, 51, 68, 72, 93
303: 4, 51, 72, 86, 93
304: 4, 51, 76, 82, 93
305: 4, 52, 63, 78, 91
306: 4, 52, 63, 87, 91
307: 4, 52, 76, 81, 93
308: 4, 56, 72, 81, 93
309: 4, 57, 62, 81, 93
310: 4, 57, 63, 81, 92
311: 4, 57, 63, 82, 91
312: 4, 58, 63, 72, 91
313: 4, 62, 75, 81, 93
314: 4, 63, 72, 81, 95
315: 4, 63, 72, 85, 91
316: 4, 63, 75, 81, 92
317: 4, 63, 75, 82, 91
318: 4, 65, 72, 81, 93
319: 6, 12, 34, 57, 98
320: 6, 12, 34, 75, 98
321: 6, 12, 34, 78, 95
322: 6, 12, 34, 87, 95
323: 6, 12, 35, 49, 78
324: 6, 12, 35, 49, 87
325: 6, 12, 35, 74, 98
326: 6, 12, 35, 78, 94
327: 6, 12, 35, 87, 94
328: 6, 12, 38, 49, 57
329: 6, 12, 38, 49, 75
330: 6, 12, 38, 57, 94
331: 6, 12, 38, 74, 95
332: 6, 12, 38, 75, 94
333: 6, 12, 39, 45, 78
334: 6, 12, 39, 45, 87
335: 6, 12, 39, 48, 57
336: 6, 12, 39, 48, 75
337: 6, 12, 39, 54, 78
338: 6, 12, 39, 54, 87
339: 6, 12, 39, 57, 84
340: 6, 12, 39, 58, 74
341: 6, 12, 39, 74, 85
342: 6, 12, 39, 75, 84
343: 6, 12, 45, 78, 93
344: 6, 12, 45, 87, 93
345: 6, 12, 48, 57, 93
346: 6, 12, 48, 75, 93
347: 6, 12, 54, 78, 93
348: 6, 12, 54, 87, 93
349: 6, 12, 57, 84, 93
350: 6, 12, 58, 74, 93
351: 6, 12, 74, 85, 93
352: 6, 12, 75, 84, 93
353: 6, 14, 25, 39, 78
354: 6, 14, 25, 39, 87
355: 6, 14, 25, 78, 93
356: 6, 14, 25, 87, 93
357: 6, 14, 27, 35, 98
358: 6, 14, 27, 38, 95
359: 6, 14, 27, 39, 58
360: 6, 14, 27, 39, 85
361: 6, 14, 27, 58, 93
362: 6, 14, 27, 85, 93
363: 6, 14, 28, 39, 57
364: 6, 14, 28, 39, 75
365: 6, 14, 28, 57, 93
366: 6, 14, 28, 75, 93
367: 6, 14, 32, 57, 98
368: 6, 14, 32, 75, 98
369: 6, 14, 32, 78, 95
370: 6, 14, 32, 87, 95
371: 6, 14, 35, 72, 98
372: 6, 14, 35, 78, 92
373: 6, 14, 35, 87, 92
374: 6, 14, 38, 57, 92
375: 6, 14, 38, 72, 95
376: 6, 14, 38, 75, 92
377: 6, 14, 39, 52, 78
378: 6, 14, 39, 52, 87
379: 6, 14, 39, 57, 82
380: 6, 14, 39, 58, 72
381: 6, 14, 39, 72, 85
382: 6, 14, 39, 75, 82
383: 6, 14, 52, 78, 93
384: 6, 14, 52, 87, 93
385: 6, 14, 57, 82, 93
386: 6, 14, 58, 72, 93
387: 6, 14, 72, 85, 93
388: 6, 14, 75, 82, 93
389: 6, 15, 24, 39, 78
390: 6, 15, 24, 39, 87
391: 6, 15, 24, 78, 93
392: 6, 15, 24, 87, 93
393: 6, 15, 27, 34, 98
394: 6, 15, 27, 38, 49
395: 6, 15, 27, 38, 94
396: 6, 15, 27, 39, 48
397: 6, 15, 27, 39, 84
398: 6, 15, 27, 48, 93
399: 6, 15, 27, 84, 93
400: 6, 15, 28, 39, 74
401: 6, 15, 28, 74, 93
402: 6, 15, 32, 49, 78
403: 6, 15, 32, 49, 87
404: 6, 15, 32, 74, 98
405: 6, 15, 32, 78, 94
406: 6, 15, 32, 87, 94
407: 6, 15, 34, 72, 98
408: 6, 15, 34, 78, 92
409: 6, 15, 34, 87, 92
410: 6, 15, 38, 49, 72
411: 6, 15, 38, 72, 94
412: 6, 15, 38, 74, 92
413: 6, 15, 39, 42, 78
414: 6, 15, 39, 42, 87
415: 6, 15, 39, 48, 72
416: 6, 15, 39, 72, 84
417: 6, 15, 39, 74, 82
418: 6, 15, 42, 78, 93
419: 6, 15, 42, 87, 93
420: 6, 15, 48, 72, 93
421: 6, 15, 72, 84, 93
422: 6, 15, 74, 82, 93
423: 6, 18, 24, 39, 57
424: 6, 18, 24, 39, 75
425: 6, 18, 24, 57, 93
426: 6, 18, 24, 75, 93
427: 6, 18, 25, 39, 74
428: 6, 18, 25, 74, 93
429: 6, 18, 27, 34, 95
430: 6, 18, 27, 35, 49
431: 6, 18, 27, 35, 94
432: 6, 18, 27, 39, 45
433: 6, 18, 27, 39, 54
434: 6, 18, 27, 45, 93
435: 6, 18, 27, 54, 93
436: 6, 18, 32, 49, 57
437: 6, 18, 32, 49, 75
438: 6, 18, 32, 57, 94
439: 6, 18, 32, 74, 95
440: 6, 18, 32, 75, 94
441: 6, 18, 34, 57, 92
442: 6, 18, 34, 72, 95
443: 6, 18, 34, 75, 92
444: 6, 18, 35, 49, 72
445: 6, 18, 35, 72, 94
446: 6, 18, 35, 74, 92
447: 6, 18, 39, 42, 57
448: 6, 18, 39, 42, 75
449: 6, 18, 39, 45, 72
450: 6, 18, 39, 52, 74
451: 6, 18, 39, 54, 72
452: 6, 18, 42, 57, 93
453: 6, 18, 42, 75, 93
454: 6, 18, 45, 72, 93
455: 6, 18, 52, 74, 93
456: 6, 18, 54, 72, 93
457: 6, 21, 34, 57, 98
458: 6, 21, 34, 75, 98
459: 6, 21, 34, 78, 95
460: 6, 21, 34, 87, 95
461: 6, 21, 35, 49, 78
462: 6, 21, 35, 49, 87
463: 6, 21, 35, 74, 98
464: 6, 21, 35, 78, 94
465: 6, 21, 35, 87, 94
466: 6, 21, 38, 49, 57
467: 6, 21, 38, 49, 75
468: 6, 21, 38, 57, 94
469: 6, 21, 38, 74, 95
470: 6, 21, 38, 75, 94
471: 6, 21, 39, 45, 78
472: 6, 21, 39, 45, 87
473: 6, 21, 39, 48, 57
474: 6, 21, 39, 48, 75
475: 6, 21, 39, 54, 78
476: 6, 21, 39, 54, 87
477: 6, 21, 39, 57, 84
478: 6, 21, 39, 58, 74
479: 6, 21, 39, 74, 85
480: 6, 21, 39, 75, 84
481: 6, 21, 45, 78, 93
482: 6, 21, 45, 87, 93
483: 6, 21, 48, 57, 93
484: 6, 21, 48, 75, 93
485: 6, 21, 54, 78, 93
486: 6, 21, 54, 87, 93
487: 6, 21, 57, 84, 93
488: 6, 21, 58, 74, 93
489: 6, 21, 74, 85, 93
490: 6, 21, 75, 84, 93
491: 6, 24, 35, 78, 91
492: 6, 24, 35, 87, 91
493: 6, 24, 38, 57, 91
494: 6, 24, 38, 75, 91
495: 6, 24, 39, 51, 78
496: 6, 24, 39, 51, 87
497: 6, 24, 39, 57, 81
498: 6, 24, 39, 75, 81
499: 6, 24, 51, 78, 93
500: 6, 24, 51, 87, 93
501: 6, 24, 57, 81, 93
502: 6, 24, 75, 81, 93
503: 6, 25, 34, 78, 91
504: 6, 25, 34, 87, 91
505: 6, 25, 38, 74, 91
506: 6, 25, 39, 74, 81
507: 6, 25, 74, 81, 93
508: 6, 27, 34, 51, 98
509: 6, 27, 34, 58, 91
510: 6, 27, 34, 81, 95
511: 6, 27, 34, 85, 91
512: 6, 27, 35, 48, 91
513: 6, 27, 35, 49, 81
514: 6, 27, 35, 81, 94
515: 6, 27, 35, 84, 91
516: 6, 27, 38, 45, 91
517: 6, 27, 38, 49, 51
518: 6, 27, 38, 51, 94
519: 6, 27, 38, 54, 91
520: 6, 27, 39, 45, 81
521: 6, 27, 39, 48, 51
522: 6, 27, 39, 51, 84
523: 6, 27, 39, 54, 81
524: 6, 27, 45, 81, 93
525: 6, 27, 48, 51, 93
526: 6, 27, 51, 84, 93
527: 6, 27, 54, 81, 93
528: 6, 28, 34, 57, 91
529: 6, 28, 34, 75, 91
530: 6, 28, 35, 74, 91
531: 6, 28, 39, 51, 74
532: 6, 28, 51, 74, 93
533: 6, 32, 45, 78, 91
534: 6, 32, 45, 87, 91
535: 6, 32, 48, 57, 91
536: 6, 32, 48, 75, 91
537: 6, 32, 49, 51, 78
538: 6, 32, 49, 51, 87
539: 6, 32, 49, 57, 81
540: 6, 32, 49, 75, 81
541: 6, 32, 51, 74, 98
542: 6, 32, 51, 78, 94
543: 6, 32, 51, 87, 94
544: 6, 32, 54, 78, 91
545: 6, 32, 54, 87, 91
546: 6, 32, 57, 81, 94
547: 6, 32, 57, 84, 91
548: 6, 32, 58, 74, 91
549: 6, 32, 74, 81, 95
550: 6, 32, 74, 85, 91
551: 6, 32, 75, 81, 94
552: 6, 32, 75, 84, 91
553: 6, 34, 51, 72, 98
554: 6, 34, 51, 78, 92
555: 6, 34, 51, 87, 92
556: 6, 34, 52, 78, 91
557: 6, 34, 52, 87, 91
558: 6, 34, 57, 81, 92
559: 6, 34, 57, 82, 91
560: 6, 34, 58, 72, 91
561: 6, 34, 72, 81, 95
562: 6, 34, 72, 85, 91
563: 6, 34, 75, 81, 92
564: 6, 34, 75, 82, 91
565: 6, 35, 42, 78, 91
566: 6, 35, 42, 87, 91
567: 6, 35, 48, 72, 91
568: 6, 35, 49, 72, 81
569: 6, 35, 72, 81, 94
570: 6, 35, 72, 84, 91
571: 6, 35, 74, 81, 92
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963: 9, 15, 24, 63, 87
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967: 9, 15, 27, 34, 68
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969: 9, 15, 27, 36, 48
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999: 9, 15, 38, 64, 72
1000: 9, 15, 42, 63, 78
1001: 9, 15, 42, 63, 87
1002: 9, 15, 48, 63, 72
1003: 9, 15, 63, 72, 84
1004: 9, 15, 63, 74, 82
1005: 9, 16, 24, 35, 78
1006: 9, 16, 24, 35, 87
1007: 9, 16, 24, 38, 57
1008: 9, 16, 24, 38, 75
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1010: 9, 16, 25, 34, 87
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1012: 9, 16, 27, 34, 58
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1014: 9, 16, 27, 35, 48
1015: 9, 16, 27, 35, 84
1016: 9, 16, 27, 38, 45
1017: 9, 16, 27, 38, 54
1018: 9, 16, 28, 34, 57
1019: 9, 16, 28, 34, 75
1020: 9, 16, 28, 35, 74
1021: 9, 16, 32, 45, 78
1022: 9, 16, 32, 45, 87
1023: 9, 16, 32, 48, 57
1024: 9, 16, 32, 48, 75
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1026: 9, 16, 32, 54, 87
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1028: 9, 16, 32, 58, 74
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1036: 9, 16, 34, 75, 82
1037: 9, 16, 35, 42, 78
1038: 9, 16, 35, 42, 87
1039: 9, 16, 35, 48, 72
1040: 9, 16, 35, 72, 84
1041: 9, 16, 35, 74, 82
1042: 9, 16, 38, 42, 57
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1067: 9, 18, 32, 46, 57
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1076: 9, 18, 34, 57, 62
1077: 9, 18, 34, 62, 75
1078: 9, 18, 34, 65, 72
1079: 9, 18, 35, 42, 76
1080: 9, 18, 35, 46, 72
1081: 9, 18, 35, 62, 74
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1091: 9, 18, 52, 63, 74
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1098: 9, 21, 34, 65, 78
1099: 9, 21, 34, 65, 87
1100: 9, 21, 34, 68, 75
1101: 9, 21, 34, 75, 86
1102: 9, 21, 34, 76, 85
1103: 9, 21, 35, 46, 78
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1105: 9, 21, 35, 48, 76
1106: 9, 21, 35, 64, 78
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1110: 9, 21, 35, 76, 84
1111: 9, 21, 36, 45, 78
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1113: 9, 21, 36, 48, 57
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1117: 9, 21, 36, 57, 84
1118: 9, 21, 36, 58, 74
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1120: 9, 21, 36, 75, 84
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1122: 9, 21, 38, 46, 57
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1124: 9, 21, 38, 54, 76
1125: 9, 21, 38, 56, 74
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1127: 9, 21, 38, 64, 75
1128: 9, 21, 38, 65, 74
1129: 9, 21, 45, 63, 78
1130: 9, 21, 45, 63, 87
1131: 9, 21, 48, 57, 63
1132: 9, 21, 48, 63, 75
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1134: 9, 21, 54, 63, 87
1135: 9, 21, 57, 63, 84
1136: 9, 21, 58, 63, 74
1137: 9, 21, 63, 74, 85
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1140: 9, 24, 36, 51, 78
1141: 9, 24, 36, 51, 87
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1143: 9, 24, 36, 75, 81
1144: 9, 24, 38, 51, 76
1145: 9, 24, 51, 63, 78
1146: 9, 24, 51, 63, 87
1147: 9, 24, 57, 63, 81
1148: 9, 24, 63, 75, 81
1149: 9, 25, 34, 76, 81
1150: 9, 25, 36, 74, 81
1151: 9, 25, 63, 74, 81
1152: 9, 26, 34, 51, 78
1153: 9, 26, 34, 51, 87
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1155: 9, 26, 34, 75, 81
1156: 9, 26, 35, 74, 81
1157: 9, 26, 38, 51, 74
1158: 9, 27, 34, 51, 68
1159: 9, 27, 34, 51, 86
1160: 9, 27, 34, 56, 81
1161: 9, 27, 34, 65, 81
1162: 9, 27, 35, 46, 81
1163: 9, 27, 35, 64, 81
1164: 9, 27, 36, 45, 81
1165: 9, 27, 36, 48, 51
1166: 9, 27, 36, 51, 84
1167: 9, 27, 36, 54, 81
1168: 9, 27, 38, 46, 51
1169: 9, 27, 38, 51, 64
1170: 9, 27, 45, 63, 81
1171: 9, 27, 48, 51, 63
1172: 9, 27, 51, 63, 84
1173: 9, 27, 54, 63, 81
1174: 9, 28, 34, 51, 76
1175: 9, 28, 36, 51, 74
1176: 9, 28, 51, 63, 74
1177: 9, 32, 45, 76, 81
1178: 9, 32, 46, 51, 78
1179: 9, 32, 46, 51, 87
1180: 9, 32, 46, 57, 81
1181: 9, 32, 46, 75, 81
1182: 9, 32, 48, 51, 76
1183: 9, 32, 51, 64, 78
1184: 9, 32, 51, 64, 87
1185: 9, 32, 51, 68, 74
1186: 9, 32, 51, 74, 86
1187: 9, 32, 51, 76, 84
1188: 9, 32, 54, 76, 81
1189: 9, 32, 56, 74, 81
1190: 9, 32, 57, 64, 81
1191: 9, 32, 64, 75, 81
1192: 9, 32, 65, 74, 81
1193: 9, 34, 51, 62, 78
1194: 9, 34, 51, 62, 87
1195: 9, 34, 51, 68, 72
1196: 9, 34, 51, 72, 86
1197: 9, 34, 51, 76, 82
1198: 9, 34, 52, 76, 81
1199: 9, 34, 56, 72, 81
1200: 9, 34, 57, 62, 81
1201: 9, 34, 62, 75, 81
1202: 9, 34, 65, 72, 81
1203: 9, 35, 42, 76, 81
1204: 9, 35, 46, 72, 81
1205: 9, 35, 62, 74, 81
1206: 9, 35, 64, 72, 81
1207: 9, 36, 42, 51, 78
1208: 9, 36, 42, 51, 87
1209: 9, 36, 42, 57, 81
1210: 9, 36, 42, 75, 81
1211: 9, 36, 45, 72, 81
1212: 9, 36, 48, 51, 72
1213: 9, 36, 51, 72, 84
1214: 9, 36, 51, 74, 82
1215: 9, 36, 52, 74, 81
1216: 9, 36, 54, 72, 81
1217: 9, 38, 42, 51, 76
1218: 9, 38, 46, 51, 72
1219: 9, 38, 51, 62, 74
1220: 9, 38, 51, 64, 72
1221: 9, 42, 51, 63, 78
1222: 9, 42, 51, 63, 87
1223: 9, 42, 57, 63, 81
1224: 9, 42, 63, 75, 81
1225: 9, 45, 63, 72, 81
1226: 9, 48, 51, 63, 72
1227: 9, 51, 63, 72, 84
1228: 9, 51, 63, 74, 82
1229: 9, 52, 63, 74, 81
1230: 9, 54, 63, 72, 81
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2: 4, 12, 35, 69, 87
3: 4, 12, 35, 76, 98
4: 4, 12, 35, 78, 96
5: 4, 12, 35, 87, 96
6: 4, 12, 36, 57, 98
7: 4, 12, 36, 75, 98
8: 4, 12, 36, 78, 95
9: 4, 12, 36, 87, 95
10: 4, 12, 38, 57, 69
11: 4, 12, 38, 57, 96
12: 4, 12, 38, 69, 75
13: 4, 12, 38, 75, 96
14: 4, 12, 38, 76, 95
15: 4, 12, 39, 56, 78
16: 4, 12, 39, 56, 87
17: 4, 12, 39, 57, 68
18: 4, 12, 39, 57, 86
19: 4, 12, 39, 58, 76
20: 4, 12, 39, 65, 78
21: 4, 12, 39, 65, 87
22: 4, 12, 39, 68, 75
23: 4, 12, 39, 75, 86
24: 4, 12, 39, 76, 85
25: 4, 12, 56, 78, 93
26: 4, 12, 56, 87, 93
27: 4, 12, 57, 63, 98
28: 4, 12, 57, 68, 93
29: 4, 12, 57, 86, 93
30: 4, 12, 58, 76, 93
31: 4, 12, 63, 75, 98
32: 4, 12, 63, 78, 95
33: 4, 12, 63, 87, 95
34: 4, 12, 65, 78, 93
35: 4, 12, 65, 87, 93
36: 4, 12, 68, 75, 93
37: 4, 12, 75, 86, 93
38: 4, 12, 76, 85, 93
39: 4, 15, 26, 39, 78
40: 4, 15, 26, 39, 87
41: 4, 15, 26, 78, 93
42: 4, 15, 26, 87, 93
43: 4, 15, 27, 36, 98
44: 4, 15, 27, 38, 69
45: 4, 15, 27, 38, 96
46: 4, 15, 27, 39, 68
47: 4, 15, 27, 39, 86
48: 4, 15, 27, 63, 98
49: 4, 15, 27, 68, 93
50: 4, 15, 27, 86, 93
51: 4, 15, 28, 39, 76
52: 4, 15, 28, 76, 93
53: 4, 15, 32, 69, 78
54: 4, 15, 32, 69, 87
55: 4, 15, 32, 76, 98
56: 4, 15, 32, 78, 96
57: 4, 15, 32, 87, 96
58: 4, 15, 36, 72, 98
59: 4, 15, 36, 78, 92
60: 4, 15, 36, 87, 92
61: 4, 15, 38, 69, 72
62: 4, 15, 38, 72, 96
63: 4, 15, 38, 76, 92
64: 4, 15, 39, 62, 78
65: 4, 15, 39, 62, 87
66: 4, 15, 39, 68, 72
67: 4, 15, 39, 72, 86
68: 4, 15, 39, 76, 82
69: 4, 15, 62, 78, 93
70: 4, 15, 62, 87, 93
71: 4, 15, 63, 72, 98
72: 4, 15, 63, 78, 92
73: 4, 15, 63, 87, 92
74: 4, 15, 68, 72, 93
75: 4, 15, 72, 86, 93
76: 4, 15, 76, 82, 93
77: 4, 16, 25, 39, 78
78: 4, 16, 25, 39, 87
79: 4, 16, 25, 78, 93
80: 4, 16, 25, 87, 93
81: 4, 16, 27, 35, 98
82: 4, 16, 27, 38, 95
83: 4, 16, 27, 39, 58
84: 4, 16, 27, 39, 85
85: 4, 16, 27, 58, 93
86: 4, 16, 27, 85, 93
87: 4, 16, 28, 39, 57
88: 4, 16, 28, 39, 75
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90: 4, 16, 28, 75, 93
91: 4, 16, 32, 57, 98
92: 4, 16, 32, 75, 98
93: 4, 16, 32, 78, 95
94: 4, 16, 32, 87, 95
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96: 4, 16, 35, 78, 92
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98: 4, 16, 38, 57, 92
99: 4, 16, 38, 72, 95
100: 4, 16, 38, 75, 92
101: 4, 16, 39, 52, 78
102: 4, 16, 39, 52, 87
103: 4, 16, 39, 57, 82
104: 4, 16, 39, 58, 72
105: 4, 16, 39, 72, 85
106: 4, 16, 39, 75, 82
107: 4, 16, 52, 78, 93
108: 4, 16, 52, 87, 93
109: 4, 16, 57, 82, 93
110: 4, 16, 58, 72, 93
111: 4, 16, 72, 85, 93
112: 4, 16, 75, 82, 93
113: 4, 18, 25, 39, 76
114: 4, 18, 25, 76, 93
115: 4, 18, 26, 39, 57
116: 4, 18, 26, 39, 75
117: 4, 18, 26, 57, 93
118: 4, 18, 26, 75, 93
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120: 4, 18, 27, 35, 96
121: 4, 18, 27, 36, 95
122: 4, 18, 27, 39, 56
123: 4, 18, 27, 39, 65
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125: 4, 18, 27, 63, 95
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130: 4, 18, 32, 75, 96
131: 4, 18, 32, 76, 95
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136: 4, 18, 36, 72, 95
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139: 4, 18, 39, 56, 72
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141: 4, 18, 39, 62, 75
142: 4, 18, 39, 65, 72
143: 4, 18, 52, 76, 93
144: 4, 18, 56, 72, 93
145: 4, 18, 57, 62, 93
146: 4, 18, 57, 63, 92
147: 4, 18, 62, 75, 93
148: 4, 18, 63, 72, 95
149: 4, 18, 63, 75, 92
150: 4, 18, 65, 72, 93
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152: 4, 21, 35, 69, 87
153: 4, 21, 35, 76, 98
154: 4, 21, 35, 78, 96
155: 4, 21, 35, 87, 96
156: 4, 21, 36, 57, 98
157: 4, 21, 36, 75, 98
158: 4, 21, 36, 78, 95
159: 4, 21, 36, 87, 95
160: 4, 21, 38, 57, 69
161: 4, 21, 38, 57, 96
162: 4, 21, 38, 69, 75
163: 4, 21, 38, 75, 96
164: 4, 21, 38, 76, 95
165: 4, 21, 39, 56, 78
166: 4, 21, 39, 56, 87
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169: 4, 21, 39, 58, 76
170: 4, 21, 39, 65, 78
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172: 4, 21, 39, 68, 75
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174: 4, 21, 39, 76, 85
175: 4, 21, 56, 78, 93
176: 4, 21, 56, 87, 93
177: 4, 21, 57, 63, 98
178: 4, 21, 57, 68, 93
179: 4, 21, 57, 86, 93
180: 4, 21, 58, 76, 93
181: 4, 21, 63, 75, 98
182: 4, 21, 63, 78, 95
183: 4, 21, 63, 87, 95
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187: 4, 21, 75, 86, 93
188: 4, 21, 76, 85, 93
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190: 4, 25, 36, 87, 91
191: 4, 25, 38, 76, 91
192: 4, 25, 39, 76, 81
193: 4, 25, 63, 78, 91
194: 4, 25, 63, 87, 91
195: 4, 25, 76, 81, 93
196: 4, 26, 35, 78, 91
197: 4, 26, 35, 87, 91
198: 4, 26, 38, 57, 91
199: 4, 26, 38, 75, 91
200: 4, 26, 39, 51, 78
201: 4, 26, 39, 51, 87
202: 4, 26, 39, 57, 81
203: 4, 26, 39, 75, 81
204: 4, 26, 51, 78, 93
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207: 4, 26, 75, 81, 93
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209: 4, 27, 35, 69, 81
210: 4, 27, 35, 81, 96
211: 4, 27, 35, 86, 91
212: 4, 27, 36, 51, 98
213: 4, 27, 36, 58, 91
214: 4, 27, 36, 81, 95
215: 4, 27, 36, 85, 91
216: 4, 27, 38, 51, 69
217: 4, 27, 38, 51, 96
218: 4, 27, 38, 56, 91
219: 4, 27, 38, 65, 91
220: 4, 27, 39, 51, 68
221: 4, 27, 39, 51, 86
222: 4, 27, 39, 56, 81
223: 4, 27, 39, 65, 81
224: 4, 27, 51, 63, 98
225: 4, 27, 51, 68, 93
226: 4, 27, 51, 86, 93
227: 4, 27, 56, 81, 93
228: 4, 27, 58, 63, 91
229: 4, 27, 63, 81, 95
230: 4, 27, 63, 85, 91
231: 4, 27, 65, 81, 93
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233: 4, 28, 36, 57, 91
234: 4, 28, 36, 75, 91
235: 4, 28, 39, 51, 76
236: 4, 28, 51, 76, 93
237: 4, 28, 57, 63, 91
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239: 4, 32, 51, 69, 78
240: 4, 32, 51, 69, 87
241: 4, 32, 51, 76, 98
242: 4, 32, 51, 78, 96
243: 4, 32, 51, 87, 96
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248: 4, 32, 57, 81, 96
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250: 4, 32, 58, 76, 91
251: 4, 32, 65, 78, 91
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253: 4, 32, 68, 75, 91
254: 4, 32, 69, 75, 81
255: 4, 32, 75, 81, 96
256: 4, 32, 75, 86, 91
257: 4, 32, 76, 81, 95
258: 4, 32, 76, 85, 91
259: 4, 35, 62, 78, 91
260: 4, 35, 62, 87, 91
261: 4, 35, 68, 72, 91
262: 4, 35, 69, 72, 81
263: 4, 35, 72, 81, 96
264: 4, 35, 72, 86, 91
265: 4, 35, 76, 81, 92
266: 4, 35, 76, 82, 91
267: 4, 36, 51, 72, 98
268: 4, 36, 51, 78, 92
269: 4, 36, 51, 87, 92
270: 4, 36, 52, 78, 91
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275: 4, 36, 72, 81, 95
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277: 4, 36, 75, 81, 92
278: 4, 36, 75, 82, 91
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280: 4, 38, 51, 72, 96
281: 4, 38, 51, 76, 92
282: 4, 38, 52, 76, 91
283: 4, 38, 56, 72, 91
284: 4, 38, 57, 62, 91
285: 4, 38, 62, 75, 91
286: 4, 38, 65, 72, 91
287: 4, 39, 51, 62, 78
288: 4, 39, 51, 62, 87
289: 4, 39, 51, 68, 72
290: 4, 39, 51, 72, 86
291: 4, 39, 51, 76, 82
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293: 4, 39, 56, 72, 81
294: 4, 39, 57, 62, 81
295: 4, 39, 62, 75, 81
296: 4, 39, 65, 72, 81
297: 4, 51, 62, 78, 93
298: 4, 51, 62, 87, 93
299: 4, 51, 63, 72, 98
300: 4, 51, 63, 78, 92
301: 4, 51, 63, 87, 92
302: 4, 51, 68, 72, 93
303: 4, 51, 72, 86, 93
304: 4, 51, 76, 82, 93
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306: 4, 52, 63, 87, 91
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308: 4, 56, 72, 81, 93
309: 4, 57, 62, 81, 93
310: 4, 57, 63, 81, 92
311: 4, 57, 63, 82, 91
312: 4, 58, 63, 72, 91
313: 4, 62, 75, 81, 93
314: 4, 63, 72, 81, 95
315: 4, 63, 72, 85, 91
316: 4, 63, 75, 81, 92
317: 4, 63, 75, 82, 91
318: 4, 65, 72, 81, 93
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320: 6, 12, 34, 75, 98
321: 6, 12, 34, 78, 95
322: 6, 12, 34, 87, 95
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1175: 9, 28, 36, 51, 74
1176: 9, 28, 51, 63, 74
1177: 9, 32, 45, 76, 81
1178: 9, 32, 46, 51, 78
1179: 9, 32, 46, 51, 87
1180: 9, 32, 46, 57, 81
1181: 9, 32, 46, 75, 81
1182: 9, 32, 48, 51, 76
1183: 9, 32, 51, 64, 78
1184: 9, 32, 51, 64, 87
1185: 9, 32, 51, 68, 74
1186: 9, 32, 51, 74, 86
1187: 9, 32, 51, 76, 84
1188: 9, 32, 54, 76, 81
1189: 9, 32, 56, 74, 81
1190: 9, 32, 57, 64, 81
1191: 9, 32, 64, 75, 81
1192: 9, 32, 65, 74, 81
1193: 9, 34, 51, 62, 78
1194: 9, 34, 51, 62, 87
1195: 9, 34, 51, 68, 72
1196: 9, 34, 51, 72, 86
1197: 9, 34, 51, 76, 82
1198: 9, 34, 52, 76, 81
1199: 9, 34, 56, 72, 81
1200: 9, 34, 57, 62, 81
1201: 9, 34, 62, 75, 81
1202: 9, 34, 65, 72, 81
1203: 9, 35, 42, 76, 81
1204: 9, 35, 46, 72, 81
1205: 9, 35, 62, 74, 81
1206: 9, 35, 64, 72, 81
1207: 9, 36, 42, 51, 78
1208: 9, 36, 42, 51, 87
1209: 9, 36, 42, 57, 81
1210: 9, 36, 42, 75, 81
1211: 9, 36, 45, 72, 81
1212: 9, 36, 48, 51, 72
1213: 9, 36, 51, 72, 84
1214: 9, 36, 51, 74, 82
1215: 9, 36, 52, 74, 81
1216: 9, 36, 54, 72, 81
1217: 9, 38, 42, 51, 76
1218: 9, 38, 46, 51, 72
1219: 9, 38, 51, 62, 74
1220: 9, 38, 51, 64, 72
1221: 9, 42, 51, 63, 78
1222: 9, 42, 51, 63, 87
1223: 9, 42, 57, 63, 81
1224: 9, 42, 63, 75, 81
1225: 9, 45, 63, 72, 81
1226: 9, 48, 51, 63, 72
1227: 9, 51, 63, 72, 84
1228: 9, 51, 63, 74, 82
1229: 9, 52, 63, 74, 81
1230: 9, 54, 63, 72, 81
Alex, Nadav, Evan, Miles got:
2/8, 3/7=6/14, and 9/5. We feel that the intervals are more accurate than some of the answers above.
Here are all the 66 solutions (because of B = C, there can’t be any solutions with only fractions in their simplest form):
1/5 < 2/8 = 9/36 < 4/7
1/5 < 2/8 = 9/36 < 7/4
1/5 < 2/9 = 8/36 < 4/7
1/5 < 2/9 = 8/36 < 7/4
1/5 < 3/8 = 9/24 < 6/7
1/5 < 3/8 = 9/24 < 7/6
1/5 < 3/9 = 8/24 < 6/7
1/5 < 3/9 = 8/24 < 7/6
1/6 < 3/8 = 9/24 < 5/7
1/6 < 3/8 = 9/24 < 7/5
1/6 < 3/9 = 8/24 < 5/7
1/6 < 3/9 = 8/24 < 7/5
1/7 < 2/8 = 9/36 < 4/5
1/7 < 2/8 = 9/36 < 5/4
1/7 < 2/9 = 8/36 < 4/5
1/7 < 2/9 = 8/36 < 5/4
1/7 < 3/8 = 9/24 < 5/6
1/7 < 3/8 = 9/24 < 6/5
1/7 < 3/9 = 8/24 < 5/6
1/7 < 3/9 = 8/24 < 6/5
2/5 < 3/6 = 7/14 < 8/9
2/5 < 3/6 = 7/14 < 9/8
2/5 < 3/6 = 9/18 < 4/7
2/5 < 3/6 = 9/18 < 7/4
2/5 < 3/7 = 6/14 < 8/9
2/5 < 3/7 = 6/14 < 9/8
2/7 < 3/6 = 9/18 < 4/5
2/7 < 3/6 = 9/18 < 5/4
2/7 < 3/9 = 6/18 < 4/5
2/7 < 3/9 = 6/18 < 5/4
2/8 < 3/6 = 7/14 < 5/9
2/8 < 3/6 = 7/14 < 9/5
2/8 < 3/7 = 6/14 < 5/9
2/8 < 3/7 = 6/14 < 9/5
2/9 < 3/6 = 7/14 < 5/8
2/9 < 3/6 = 7/14 < 8/5
2/9 < 3/7 = 6/14 < 5/8
2/9 < 3/7 = 6/14 < 8/5
3/5 < 4/6 = 8/12 < 7/9
3/5 < 4/6 = 8/12 < 9/7
3/5 < 6/8 = 9/12 < 7/4
3/5 < 6/9 = 8/12 < 7/4
3/7 < 2/4 = 8/16 < 5/9
3/7 < 2/4 = 8/16 < 9/5
3/7 < 2/4 = 9/18 < 5/6
3/7 < 2/4 = 9/18 < 6/5
3/7 < 4/6 = 8/12 < 9/5
3/7 < 4/8 = 6/12 < 5/9
3/7 < 4/8 = 6/12 < 9/5
3/7 < 6/8 = 9/12 < 4/5
3/7 < 6/8 = 9/12 < 5/4
3/7 < 6/9 = 8/12 < 4/5
3/7 < 6/9 = 8/12 < 5/4
3/9 < 2/4 = 8/16 < 5/7
3/9 < 2/4 = 8/16 < 7/5
3/9 < 4/6 = 8/12 < 5/7
3/9 < 4/6 = 8/12 < 7/5
3/9 < 4/8 = 6/12 < 5/7
3/9 < 4/8 = 6/12 < 7/5
4/7 < 6/8 = 9/12 < 5/3
4/7 < 6/9 = 8/12 < 5/3
5/7 < 3/4 = 9/12 < 8/6
5/7 < 6/8 = 9/12 < 4/3
5/8 < 3/4 = 9/12 < 6/7
5/8 < 3/4 = 9/12 < 7/6
5/9 < 4/6 = 8/12 < 7/3
29 of these only contain proper fractions:
1/5 < 2/8 = 9/36 < 4/7
1/5 < 2/9 = 8/36 < 4/7
1/5 < 3/8 = 9/24 < 6/7
1/5 < 3/9 = 8/24 < 6/7
1/6 < 3/8 = 9/24 < 5/7
1/6 < 3/9 = 8/24 < 5/7
1/7 < 2/8 = 9/36 < 4/5
1/7 < 2/9 = 8/36 < 4/5
1/7 < 3/8 = 9/24 < 5/6
1/7 < 3/9 = 8/24 < 5/6
2/5 < 3/6 = 7/14 < 8/9
2/5 < 3/6 = 9/18 < 4/7
2/5 < 3/7 = 6/14 < 8/9
2/7 < 3/6 = 9/18 < 4/5
2/7 < 3/9 = 6/18 < 4/5
2/8 < 3/6 = 7/14 < 5/9
2/8 < 3/7 = 6/14 < 5/9
2/9 < 3/6 = 7/14 < 5/8
2/9 < 3/7 = 6/14 < 5/8
3/5 < 4/6 = 8/12 < 7/9
3/7 < 2/4 = 8/16 < 5/9
3/7 < 2/4 = 9/18 < 5/6
3/7 < 4/8 = 6/12 < 5/9
3/7 < 6/8 = 9/12 < 4/5
3/7 < 6/9 = 8/12 < 4/5
3/9 < 2/4 = 8/16 < 5/7
3/9 < 4/6 = 8/12 < 5/7
3/9 < 4/8 = 6/12 < 5/7
5/8 < 3/4 = 9/12 < 6/7
A = 1/7, B = 3/8, C = 9/24, D = 5/6
A – 1 1/2
B – 1/2
C – 2 1/2
D – 3/6
A= 3/7 B= 4/8 C=6/12 D=9/5