Task: Arithmetic-Relationship (source dataset: Auto-Relate)

You are given a table with numerical values in multiple columns. Your task is to analyze the data and identify semantic relationships, expressed in arithmetic relationships, that exist between columns.

These relationships may involve any combination of the following operations: addition (+), subtraction (−), multiplication (*), and division (/). The expressions can involve two or more columns and may be nested (e.g., B = (A + C) / D).

Please only identify arithmetic relationships that hold semantically based on the data, and do not return relationships that are likely spurious, e.g., (A = B * 0), where A is an all 0 column. Please only return minimal atomic relationships, e.g., if A = B + C and D = E + F, then there is no need to return A = B + C + E + F - D.

No explanation, return all the formula(s) in JSON format: {"Arithmetic-Relationship": [<list-of-formulas>]}. If no relationship is found, return {"Arithmetic-Relationship": []}.

Input Table:
| A                                                  | B                    |   C |   D |       E |   F |       G |   H |       I |   J |       K |   L |       M |   N |   O |   P |       Q |   R |       S |   T |       U |    V |       W |
|:---------------------------------------------------|:---------------------|----:|----:|--------:|----:|--------:|----:|--------:|----:|--------:|----:|--------:|----:|----:|----:|--------:|----:|--------:|----:|--------:|-----:|--------:|
| enrolled in at least one Advanced Placement course | Alabama              | 414 |   0 |  0      |   2 | 0.48309 |   4 |  0.9662 | 254 |  61.353 | 150 | 36.2319 |   0 |   0 |   4 |  0.9662 |   0 |   0     |   0 |  0      | 1400 | 100     |
| enrolled in at least one Advanced Placement course | Alaska               |  21 |   6 | 28.5714 |   0 | 0       |   0 |  0      |   1 |   4.762 |  11 | 52.381  |   0 |   0 |   3 | 14.2857 |   0 |   0     |   2 |  9.5238 |  503 | 100     |
| enrolled in at least one Advanced Placement course | Arizona              |  77 |   6 |  7.7922 |   0 | 0       |  18 | 23.3766 |  13 |  16.883 |  34 | 44.1558 |   0 |   0 |   6 |  7.7922 |   2 |   2.597 |   1 |  1.2987 | 1977 | 100     |
| enrolled in at least one Advanced Placement course | Colorado             |  66 |   0 |  0      |   1 | 1.51515 |  22 | 33.3333 |  19 |  28.788 |  21 | 31.8182 |   0 |   0 |   3 |  4.5455 |   1 |   1.515 |   1 |  1.5152 | 1868 |  91.328 |
| enrolled in at least one Advanced Placement course | Connecticut          | 128 |   0 |  0      |   0 | 0       |  35 | 27.3438 |  37 |  28.906 |  51 | 39.8438 |   0 |   0 |   5 |  3.9063 |  13 |  10.156 |   5 |  3.9063 | 1238 | 100     |
| enrolled in at least one Advanced Placement course | Delaware             |   1 |   0 |  0      |   0 | 0       |   0 |  0      |   1 | 100     |   0 |  0      |   0 |   0 |   0 |  0      |   1 | 100     |   0 |  0      |  235 | 100     |
| enrolled in at least one Advanced Placement course | District of Columbia |  40 |   0 |  0      |   0 | 0       |   3 |  7.5    |  37 |  92.5   |   0 |  0      |   0 |   0 |   0 |  0      |   2 |   5     |   1 |  2.5    |  221 | 100     |
| enrolled in at least one Advanced Placement course | Florida              | 203 |   1 |  0.4926 |   1 | 0.49261 |  13 |  6.4039 |  87 |  42.857 |  89 | 43.8424 |   0 |   0 |  12 |  5.9113 |  14 |   6.897 |   5 |  2.4631 | 3952 | 100     |
| enrolled in at least one Advanced Placement course | Georgia              | 172 |   0 |  0      |   0 | 0       |   4 |  2.3256 | 145 |  84.302 |  20 | 11.6279 |   0 |   0 |   3 |  1.7442 |   0 |   0     |   0 |  0      | 2407 | 100     |
| enrolled in at least one Advanced Placement course | Hawaii               |   0 |   0 |  0      |   0 | 0       |   0 |  0      |   0 |   0     |   0 |  0      |   0 |   0 |   0 |  0      |   0 |   0     |   0 |  0      |  290 | 100     |
| enrolled in at least one Advanced Placement course | Idaho                | 298 |   4 |  1.3423 |   0 | 0       | 111 | 37.2483 |   3 |   1.007 | 172 | 57.7181 |   0 |   0 |   8 |  2.6846 |  25 |   8.389 |  88 | 29.5302 |  720 | 100     |
| enrolled in at least one Advanced Placement course | Illinois             | 529 |   6 |  1.1342 |   5 | 0.94518 |  72 | 13.6106 |  83 |  15.69  | 341 | 64.4612 |   0 |   0 |  22 |  4.1588 |  16 |   3.025 |  18 |  3.4026 | 4081 |  99.706 |
| enrolled in at least one Advanced Placement course | Indiana              | 659 |   1 |  0.1517 |   1 | 0.15175 |  36 |  5.4628 | 115 |  17.451 | 448 | 67.9818 |   0 |   0 |  58 |  8.8012 |  16 |   2.428 |  15 |  2.2762 | 1879 | 100     |
| enrolled in at least one Advanced Placement course | Iowa                 | 507 |   1 |  0.1972 |   1 | 0.19724 |  34 |  6.7061 | 129 |  25.444 | 301 | 59.3688 |   0 |   0 |  41 |  8.0868 |  10 |   1.972 |  18 |  3.5503 | 1365 | 100     |
| enrolled in at least one Advanced Placement course | Kentucky             |  45 |   0 |  0      |   0 | 0       |   1 |  2.2222 |  19 |  42.222 |  22 | 48.8889 |   0 |   0 |   3 |  6.6667 |   3 |   6.667 |   0 |  0      | 1407 | 100     |
| enrolled in at least one Advanced Placement course | Louisiana            | 315 |   0 |  0      |   3 | 0.95238 |  18 |  5.7143 | 200 |  63.492 |  82 | 26.0317 |   0 |   0 |  12 |  3.8095 |   8 |   2.54  |  39 | 12.381  | 1367 | 100     |
| enrolled in at least one Advanced Placement course | Maine                | 131 |   1 |  0.7634 |   1 | 0.76336 |   1 |  0.7634 |   6 |   4.58  | 121 | 92.3664 |   0 |   0 |   1 |  0.7634 |  10 |   7.634 |   3 |  2.2901 |  589 | 100     |
| enrolled in at least one Advanced Placement course | Maryland             |  75 |   0 |  0      |   1 | 1.33333 |   4 |  5.3333 |  41 |  54.667 |  24 | 32      |   0 |   0 |   5 |  6.6667 |   4 |   5.333 |   1 |  1.3333 | 1434 |  86.053 |
| enrolled in at least one Advanced Placement course | Massachusetts        |  19 |   0 |  0      |   0 | 0       |   4 | 21.0526 |  10 |  52.632 |   5 | 26.3158 |   0 |   0 |   0 |  0      |   1 |   5.263 |   0 |  0      | 1873 | 100     |
| enrolled in at least one Advanced Placement course | Michigan             | 415 |   4 |  0.9639 |   1 | 0.24096 |  11 |  2.6506 |  88 |  21.205 | 297 | 71.5663 |   0 |   0 |  14 |  3.3735 |   4 |   0.964 |   3 |  0.7229 | 3616 |  99.972 |
| enrolled in at least one Advanced Placement course | Minnesota            | 123 |   1 |  0.813  |   1 | 0.81301 |   3 |  2.439  |  16 |  13.008 |  94 | 76.4228 |   0 |   0 |   8 |  6.5041 |   5 |   4.065 |   2 |  1.626  | 2170 |  99.954 |
| enrolled in at least one Advanced Placement course | Mississippi          | 121 |   0 |  0      |   0 | 0       |   1 |  0.8264 |  97 |  80.165 |  23 | 19.0083 |   0 |   0 |   0 |  0      |   0 |   0     |   0 |  0      |  978 | 100     |
| enrolled in at least one Advanced Placement course | Missouri             | 103 |   0 |  0      |   0 | 0       |  11 | 10.6796 |  24 |  23.301 |  66 | 64.0777 |   0 |   0 |   2 |  1.9417 |   2 |   1.942 |   6 |  5.8252 | 2372 | 100     |
| enrolled in at least one Advanced Placement course | Montana              | 577 | 497 | 86.1352 |   0 | 0       |   1 |  0.1733 |   0 |   0     |  76 | 13.1716 |   0 |   0 |   3 |  0.5199 |   1 |   0.173 |  90 | 15.5979 |  825 | 100     |
| enrolled in at least one Advanced Placement course | Nebraska             | 186 |   4 |  2.1505 |   0 | 0       |  15 |  8.0645 |  33 |  17.742 | 124 | 66.6667 |   0 |   0 |  10 |  5.3763 |   2 |   1.075 |   5 |  2.6882 | 1064 | 100     |
| enrolled in at least one Advanced Placement course | Nevada               |   9 |   1 | 11.1111 |   0 | 0       |   2 | 22.2222 |   0 |   0     |   6 | 66.6667 |   0 |   0 |   0 |  0      |   0 |   0     |   0 |  0      |  658 | 100     |
| enrolled in at least one Advanced Placement course | New Hampshire        |  64 |   0 |  0      |   0 | 0       |   3 |  4.6875 |   4 |   6.25  |  56 | 87.5    |   0 |   0 |   1 |  1.5625 |   7 |  10.938 |   0 |  0      |  483 | 100     |
| enrolled in at least one Advanced Placement course | New Jersey           |  69 |   0 |  0      |   0 | 0       |  26 | 37.6812 |  37 |  53.623 |   6 |  8.6957 |   0 |   0 |   0 |  0      |   0 |   0     |   0 |  0      | 2577 | 100     |
| enrolled in at least one Advanced Placement course | New Mexico           |  27 |   8 | 29.6296 |   1 | 3.7037  |   7 | 25.9259 |   0 |   0     |  11 | 40.7407 |   0 |   0 |   0 |  0      |   0 |   0     |   8 | 29.6296 |  880 | 100     |
| enrolled in at least one Advanced Placement course | New York             |  94 |   0 |  0      |   0 | 0       |  11 | 11.7021 |  45 |  47.872 |  35 | 37.234  |   0 |   0 |   3 |  3.1915 |   5 |   5.319 |   2 |  2.1277 | 4916 | 100     |
| enrolled in at least one Advanced Placement course | North Carolina       |  39 |   0 |  0      |   0 | 0       |   5 | 12.8205 |  32 |  82.051 |   2 |  5.1282 |   0 |   0 |   0 |  0      |   0 |   0     |   5 | 12.8205 | 2618 | 100     |
| enrolled in at least one Advanced Placement course | North Dakota         |  64 |   5 |  7.8125 |   0 | 0       |   5 |  7.8125 |   1 |   1.563 |  53 | 82.8125 |   0 |   0 |   0 |  0      |   4 |   6.25  |   0 |  0      |  481 | 100     |
| enrolled in at least one Advanced Placement course | Ohio                 | 227 |   0 |  0      |   1 | 0.44053 |  16 |  7.0485 | 103 |  45.374 |  87 | 38.326  |   0 |   0 |  20 |  8.8106 |   8 |   3.524 |   5 |  2.2026 | 3631 | 100     |
| enrolled in at least one Advanced Placement course | Oregon               |  46 |   2 |  4.3478 |   0 | 0       |   5 | 10.8696 |   0 |   0     |  36 | 78.2609 |   0 |   0 |   3 |  6.5217 |   3 |   6.522 |   4 |  8.6957 | 1283 | 100     |
| enrolled in at least one Advanced Placement course | Rhode Island         |   6 |   0 |  0      |   0 | 0       |   3 | 50      |   2 |  33.333 |   1 | 16.6667 |   0 |   0 |   0 |  0      |   1 |  16.667 |   0 |  0      |  308 | 100     |
| enrolled in at least one Advanced Placement course | South Carolina       | 141 |   1 |  0.7092 |   0 | 0       |   2 |  1.4184 |  78 |  55.319 |  53 | 37.5887 |   0 |   0 |   7 |  4.9645 |   0 |   0     |   0 |  0      | 1236 | 100     |
| enrolled in at least one Advanced Placement course | South Dakota         |  52 |  23 | 44.2308 |   0 | 0       |   2 |  3.8462 |   1 |   1.923 |  25 | 48.0769 |   0 |   0 |   1 |  1.9231 |   0 |   0     |   0 |  0      |  688 | 100     |
| enrolled in at least one Advanced Placement course | Tennessee            | 129 |   0 |  0      |   0 | 0       |   1 |  0.7752 |  45 |  34.884 |  82 | 63.5659 |   0 |   0 |   1 |  0.7752 |   0 |   0     |   1 |  0.7752 | 1818 | 100     |
| enrolled in at least one Advanced Placement course | Texas                | 241 |   1 |  0.4149 |   2 | 0.82988 | 128 | 53.112  |  59 |  24.481 |  39 | 16.1826 |   0 |   0 |  12 |  4.9793 |  14 |   5.809 |  11 |  4.5643 | 8616 | 100     |
| enrolled in at least one Advanced Placement course | Utah                 | 104 |   0 |  0      |   1 | 0.96154 |  23 | 22.1154 |   5 |   4.808 |  70 | 67.3077 |   0 |   0 |   5 |  4.8077 |   4 |   3.846 |   2 |  1.9231 | 1009 | 100     |
| enrolled in at least one Advanced Placement course | Vermont              |  77 |   0 |  0      |   1 | 1.2987  |   1 |  1.2987 |   2 |   2.597 |  70 | 90.9091 |   0 |   0 |   3 |  3.8961 |  20 |  25.974 |   0 |  0      |  306 | 100     |
| enrolled in at least one Advanced Placement course | Virginia             |  63 |   0 |  0      |   1 | 1.5873  |   6 |  9.5238 |  28 |  44.444 |  26 | 41.2698 |   0 |   0 |   2 |  3.1746 |   7 |  11.111 |   2 |  3.1746 | 1971 | 100     |
| enrolled in at least one Advanced Placement course | West Virginia        | 111 |   0 |  0      |   1 | 0.9009  |   0 |  0      |   9 |   8.108 | 101 | 90.991  |   0 |   0 |   0 |  0      |   0 |   0     |   0 |  0      |  720 | 100     |
| enrolled in at least one Advanced Placement course | Wyoming              |  19 |   0 |  0      |   0 | 0       |   2 | 10.5263 |   2 |  10.526 |  15 | 78.9474 |   0 |   0 |   0 |  0      |   0 |   0     |   0 |  0      |  365 | 100     |



labeled_formulas: O=C*N
O=D*N
O=E*N
O=F*N
O=G*N
O=H*N
O=I*N
O=J*N
O=(C+D)*N
O=(C+E)*N
O=(C+F)*N
O=(C+G)*N
O=(C+H)*N
O=(C+I)*N
O=(C+J)*N
O=(C+K)*N
O=(C+L)*N
O=(C+M)*N
output: {"Arithmetic-Relationship": []}
source: $MMTU_HOME/resources/Relationship-finding/AR/table_181764___xlsx_7a8fb0a59ab74708da41708e12c7d22d63e0ef38.xlsx___line_428