Fractions Using All Digits 1 to 9 only once

Goal:   To express integers as a fraction or a mixed fraction using ALL digits 1 to 9 and whenever possible the digits 0 to 9 only once.

I have posted some of the following results a long time ago. They are buried some where in older posts.

13458/6729   =   2                    13584/6792   =   2
13854/6927   =   2                    14538/7269   =   2
14586/7293   =   2                    14658/7329   =   2
15384/7692   =   2                    15846/7923   =   2
15864/7932   =   2                    18534/9267   =   2
18546/9273   =   2                    18654/9327   =   2

17469/5823   =   3                    17496/5832   =   3

15768/3942   =   4                    17568/4392   =   4
23184/5796   =   4                    31824/7956   =   4

13485/2697   =   5                    13845/2769   =   5
14685/2937   =   5                    14835/2967   =   5
14865/2973   =   5                    16485/3297   =   5
18645/3729   =   5                    31485/6297   =   5
38145/7629   =   5                    46185/9237   =   5
48135/9627   =   5                    48615/9723   =   5

17658/2943   =   6
27918/4653   =   6
34182/5697   =   6

16758/2394   =   7                    18459/2637   =   7
31689/4527   =   7                    36918/5274   =   7
37926/5418   =   7                    41832/5976   =   7
53298/7614   =   7

The two equalities        65821/9403   =   7   =   28651/4093
use all 10 digits 0-9 once in each equality.

76328/9541   =   8                    76248/9531   =   8
76184/9523   =   8                    75368/9421   =   8
75328/9416   =   8                    74816/9352   =   8
74568/9321   =   8                    74528/9316   =   8
73456/9182   =   8                    73248/9156   =   8
71632/8954   =   8                    71624/8953   =   8
71536/8942   =   8                    71456/8932   =   8
67512/8439   =   8                    67352/8419   =   8
67152/8394   =   8                    65432/8179   =   8
65392/8174   =   8                    63528/7941   =   8
63152/7894   =   8                    59368/7421   =   8
59328/7416   =   8                    58912/7364   =   8
58496/7312   =   8                    56984/7123   =   8
54712/6839   =   8                    54328/6791   =   8
54312/6789   =   8                    53928/6741   =   8
51832/6479   =   8                    47368/5921   =   8
47328/5916   =   8                    47136/5892   =   8
46712/5839   =   8                    46328/5791   =   8
46312/5789   =   8                    42968/5371   =   8
41896/5237   =   8                    38152/4769   =   8
37528/4691   =   8                    37512/4689   =   8
36728/4591   =   8                    36712/4589   =   8
25496/3187   =   8                    73264/9158   =   8

57429/6381   =   9
58239/6471   =   9
75249/8361   =   9

each ratio use each of 10 digits 0-9 once:

97524/10836   =   9
95823/10647   =   9
95742/10638   =   9

45792/3816   =   12                    73548/6129   =   12
89532/7461   =   12                    91584/7632   =   12

9 5472/1368   =   13
9 6435/1287   =   14
3 8952/(746/1)   =   15
12 3576/894   =   16
9 5742/(638/1)   =   18
6 13258/947   =   20
18 3645/729   =   23
15 9432/786   =   27
24 9756/813   =   36
27 5148/396   =   40
49 5761/823   =   56
18 36450/729   =   68
65 1892/473   =   69
59 3614/278   =   72
81 3645/729   =   86
75 3648/192   =   94

3 69258/714   =   100
96 2148/537   =   100
96 1752/438   =   100
96 1428/357   =   100
94 1578/263   =   100
91 7524/836   =   100
91 5823/647   =   100
91 5742/638   =   100
82 3546/197   =   100
81 7524/396   =   100
81 5643/297   =   100

94 5761/823   =   101
108 3645/729   =   113
81 36450/729   =   131
180 3645/729   =   185
409 5761/823   =   416
468 5103/729   =   475
486 5103/729   =   493
490 5761/823   =   497
579 4160/832   =   584
597 4160/832   =   602
648 5103/729   =   655
684 5103/729   =   691
759 4160/832   =   764
795 4160/832   =   800
846 5103/729   =   853
864 5103/729   =   871
904 5761/823   =   911
940 5761/823   =   947

Find more examples.

David a/k/a @InfinitelyManic found:

26843 / 1579 = 17
28543 / 1679 = 17
29546 / 1738 = 17
36958 / 2174 = 17
45713 / 2689 = 17
45781 / 2693 = 17
54689 / 3217 = 17
59126 / 3478 = 17
64957 / 3821 = 17
65297 / 3841 = 17
67184 / 3952 = 17
67218 / 3954 = 17
76823 / 4519 = 17
76891 / 4523 = 17
78132 / 4596 = 17
78523 / 4619 = 17
78591 / 4623 = 17
81532 / 4796 = 17
83572 / 4916 = 17
83657 / 4921 = 17
89437 / 5261 = 17
89471 / 5263 = 17
89641 / 5273 = 17
91426 / 5378 = 17
92837 / 5461 = 17
92871 / 5463 = 17
93126 / 5478 = 17

math grad - Interest: Number theory
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4 Responses to Fractions Using All Digits 1 to 9 only once

1. David a/k/a @InfinitelyManic says:

The mixed fractions are really interesting.

#Python3 Command Line Interface

>>> from itertools import permutations
>>> list5 = list(permutations(‘123456789’,5))
>>> list4 = list(permutations(‘123456789’,4))
>>> n = [int(“”.join(n)) for n in list5]
>>> d = [int(“”.join(n)) for n in list4]
>>> from fractions import Fraction
>>> [ print(n[i],”/”,d[j],”=”,Fraction(n[i],d[j])) for i in range(0,len(n)) for j in range(0,len(d)) if set(str(n[i])).isdisjoint(set(str(d[j]))) is True and Fraction(n[i],d[j]).denominator == 1 and Fraction(n[i],d[j]).denominator == 17]

#### some results for 5 digit numerator & 4 digit denominator; where fraction reduces to 17 ######

26843 / 1579 = 17
28543 / 1679 = 17
29546 / 1738 = 17
36958 / 2174 = 17
45713 / 2689 = 17
45781 / 2693 = 17
54689 / 3217 = 17
59126 / 3478 = 17
64957 / 3821 = 17
65297 / 3841 = 17
67184 / 3952 = 17
67218 / 3954 = 17
76823 / 4519 = 17
76891 / 4523 = 17
78132 / 4596 = 17
78523 / 4619 = 17
78591 / 4623 = 17
81532 / 4796 = 17
83572 / 4916 = 17
83657 / 4921 = 17
89437 / 5261 = 17
89471 / 5263 = 17
89641 / 5273 = 17
91426 / 5378 = 17
92837 / 5461 = 17
92871 / 5463 = 17
93126 / 5478 = 17

2. David a/k/a @InfinitelyManic says:

..that last bit of code should say numerator == 17

3. David a/k/a @InfinitelyManic says:

### 1-9 where n/d <= 100 followed by some 0-9 where n/d < =100 ….script is still running….
13458 / 6729 = 2
13584 / 6792 = 2
13854 / 6927 = 2
14538 / 7269 = 2
14586 / 7293 = 2
14658 / 7329 = 2
15384 / 7692 = 2
15846 / 7923 = 2
15864 / 7932 = 2
18534 / 9267 = 2
18546 / 9273 = 2
18654 / 9327 = 2
17469 / 5823 = 3
17496 / 5832 = 3
15768 / 3942 = 4
17568 / 4392 = 4
23184 / 5796 = 4
31824 / 7956 = 4
13485 / 2697 = 5
13845 / 2769 = 5
14685 / 2937 = 5
14835 / 2967 = 5
14865 / 2973 = 5
16485 / 3297 = 5
18645 / 3729 = 5
31485 / 6297 = 5
38145 / 7629 = 5
46185 / 9237 = 5
48135 / 9627 = 5
48615 / 9723 = 5
17658 / 2943 = 6
27918 / 4653 = 6
34182 / 5697 = 6
16758 / 2394 = 7
18459 / 2637 = 7
31689 / 4527 = 7
36918 / 5274 = 7
37926 / 5418 = 7
41832 / 5976 = 7
53298 / 7614 = 7
25496 / 3187 = 8
36712 / 4589 = 8
36728 / 4591 = 8
37512 / 4689 = 8
37528 / 4691 = 8
38152 / 4769 = 8
41896 / 5237 = 8
42968 / 5371 = 8
46312 / 5789 = 8
46328 / 5791 = 8
46712 / 5839 = 8
47136 / 5892 = 8
47328 / 5916 = 8
47368 / 5921 = 8
51832 / 6479 = 8
53928 / 6741 = 8
54312 / 6789 = 8
54328 / 6791 = 8
54712 / 6839 = 8
56984 / 7123 = 8
58496 / 7312 = 8
58912 / 7364 = 8
59328 / 7416 = 8
59368 / 7421 = 8
63152 / 7894 = 8
63528 / 7941 = 8
65392 / 8174 = 8
65432 / 8179 = 8
67152 / 8394 = 8
67352 / 8419 = 8
67512 / 8439 = 8
71456 / 8932 = 8
71536 / 8942 = 8
71624 / 8953 = 8
71632 / 8954 = 8
73248 / 9156 = 8
73264 / 9158 = 8
73456 / 9182 = 8
74528 / 9316 = 8
74568 / 9321 = 8
74816 / 9352 = 8
75328 / 9416 = 8
75368 / 9421 = 8
76184 / 9523 = 8
76248 / 9531 = 8
76328 / 9541 = 8
57429 / 6381 = 9
58239 / 6471 = 9
75249 / 8361 = 9
45792 / 3816 = 12
73548 / 6129 = 12
89532 / 7461 = 12
91584 / 7632 = 12
67392 / 5184 = 13
81549 / 6273 = 13
94653 / 7281 = 13
25746 / 1839 = 14
27384 / 1956 = 14
41538 / 2967 = 14
46158 / 3297 = 14
51492 / 3678 = 14
54768 / 3912 = 14
61572 / 4398 = 14
65982 / 4713 = 14
27945 / 1863 = 15
92745 / 6183 = 15
45936 / 2871 = 16
73296 / 4581 = 16
98352 / 6147 = 16
26843 / 1579 = 17
28543 / 1679 = 17
29546 / 1738 = 17
36958 / 2174 = 17
45713 / 2689 = 17
45781 / 2693 = 17
54689 / 3217 = 17
59126 / 3478 = 17
64957 / 3821 = 17
65297 / 3841 = 17
67184 / 3952 = 17
67218 / 3954 = 17
76823 / 4519 = 17
76891 / 4523 = 17
78132 / 4596 = 17
78523 / 4619 = 17
78591 / 4623 = 17
81532 / 4796 = 17
83572 / 4916 = 17
83657 / 4921 = 17
89437 / 5261 = 17
89471 / 5263 = 17
89641 / 5273 = 17
91426 / 5378 = 17
92837 / 5461 = 17
92871 / 5463 = 17
93126 / 5478 = 17
28674 / 1593 = 18
51984 / 2736 = 19
81567 / 4293 = 19
51678 / 2349 = 22
36294 / 1578 = 23
81627 / 3549 = 23
81972 / 3564 = 23
39528 / 1647 = 24
46872 / 1953 = 24
42978 / 1653 = 26
56498 / 2173 = 26
61854 / 2379 = 26
67314 / 2589 = 26
67418 / 2593 = 26
76518 / 2943 = 26
82654 / 3179 = 26
89726 / 3451 = 26
92846 / 3571 = 26
39852 / 1476 = 27
49572 / 1836 = 27
69741 / 2583 = 27
96714 / 3582 = 27
75348 / 2691 = 28
37584 / 1296 = 29
73689 / 2541 = 29
75168 / 2349 = 32
48265 / 1379 = 35
63945 / 1827 = 35
64295 / 1837 = 35
74865 / 2139 = 35
93485 / 2671 = 35
65934 / 1782 = 37
65892 / 1734 = 38
74328 / 1956 = 38
93654 / 2178 = 43
58476 / 1329 = 44
59268 / 1347 = 44
67892 / 1543 = 44
69432 / 1578 = 44
95348 / 2167 = 44
58374 / 1269 = 46
95472 / 1836 = 52
65879 / 1243 = 53
75896 / 1432 = 53
84376 / 1592 = 53
92538 / 1746 = 53
73986 / 1254 = 59
79546 / 1283 = 62
94736 / 1528 = 62
83754 / 1269 = 66
98736 / 1452 = 68

######### 0-9
90762 / 45381 = 2
90372 / 45186 = 2
90276 / 45138 = 2
98760 / 12345 = 8
98532 / 14076 = 7
98456 / 12307 = 8
97062 / 48531 = 2
97032 / 48516 = 2
97026 / 48513 = 2
97524 / 10836 = 9
97302 / 48651 = 2
97230 / 48615 = 2
96702 / 48351 = 2
96584 / 12073 = 8
96270 / 48135 = 2
96174 / 32058 = 3
95823 / 10647 = 9
95742 / 10638 = 9
94068 / 23517 = 4
94860 / 23715 = 4
93702 / 46851 = 2
93270 / 18654 = 5
92730 / 18546 = 5
92670 / 18534 = 5
92370 / 46185 = 2
91746 / 30582 = 3
87632 / 10954 = 8
87624 / 10953 = 8
87536 / 10942 = 8
87456 / 10932 = 8

4. David a/k/a @InfinitelyManic says:

Some more 0-9 There could be duplicates
58614 / 29307 = 2
59126 / 3478 = 17
59268 / 1347 = 44
60948 / 15237 = 4
61458 / 30729 = 2
61572 / 4398 = 14
61584 / 30792 = 2
61749 / 20583 = 3
61854 / 2379 = 26
61854 / 30927 = 2
62970 / 31485 = 2
63945 / 1827 = 35
64158 / 32079 = 2
64295 / 1837 = 35
64957 / 3821 = 17
65297 / 3841 = 17
65418 / 32709 = 2
65814 / 32907 = 2
65879 / 1243 = 53
65892 / 1734 = 38
65934 / 1782 = 37
65982 / 4713 = 14
67184 / 3952 = 17
67218 / 3954 = 17
67290 / 13458 = 5
67314 / 2589 = 26
67392 / 5184 = 13
67418 / 2593 = 26
67892 / 1543 = 44
67920 / 13584 = 5
68940 / 17235 = 4
69174 / 23058 = 3
69270 / 13854 = 5
69408 / 17352 = 4
69432 / 1578 = 44
69702 / 34851 = 2
69741 / 2583 = 27
70296 / 35148 = 2
70962 / 35481 = 2
72690 / 14538 = 5
72930 / 14586 = 5
73290 / 14658 = 5
73296 / 4581 = 16
73548 / 6129 = 12
73689 / 2541 = 29
73986 / 1254 = 59
74328 / 1956 = 38
74865 / 2139 = 35
75168 / 2349 = 32
75348 / 2691 = 28
75896 / 1432 = 53
76290 / 38145 = 2
76518 / 2943 = 26
76823 / 4519 = 17
76891 / 4523 = 17
76902 / 38451 = 2
76920 / 15384 = 5
78132 / 4596 = 17
78523 / 4619 = 17
78591 / 4623 = 17
79230 / 15846 = 5
79320 / 15864 = 5
79546 / 1283 = 62
81532 / 4796 = 17
81549 / 6273 = 13
81567 / 4293 = 19
81576 / 20394 = 4
81627 / 3549 = 23
81756 / 20439 = 4
81972 / 3564 = 23
82654 / 3179 = 26
83572 / 4916 = 17
83657 / 4921 = 17
83672 / 10459 = 8
83752 / 10469 = 8
83754 / 1269 = 66
84296 / 10537 = 8
84376 / 1592 = 53
84632 / 10579 = 8
84736 / 10592 = 8
85392 / 10674 = 8
85432 / 10679 = 8
85936 / 10742 = 8
86352 / 10794 = 8
86940 / 21735 = 4