Fraction reduction
Fraction reduction is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
There is a fine line between numerator and denominator. ─── anonymous
A method to "reduce" some reducible fractions is to cross out a digit from the numerator and the denominator. An example is:
16 16──── and then (simply) cross─out the sixes: ──── 6464
resulting in:
1
─── (the result)
4
Naturally, this "method" of reduction must reduce to the proper value (shown as a fraction).
(Of course, this "method" shouldn't be taught to impressionable or gullible minds.) 😇
- Task
Find and show some fractions that can be reduced by the above "method".
- show 2-digit fractions found (like the example shown above)
- show 3-digit fractions
- show 4-digit fractions
- show 5-digit fractions (and higher) (optional)
- show each (above) n-digit fractions separately from other different n-sized fractions, don't mix different "sizes" together
- for each "size" fraction, only show a dozen examples (the 1st twelve found)
- (it's recognized that not every programming solution will have the same generation algorithm)
- for each "size" fraction:
- show a count of how many reducible fractions were found. The example (above) is size 2
- show a count of which digits were crossed out (one line for each different digit)
- for each "size" fraction, show a count of how many were found. The example (above) is size 2
- show each n-digit example (to be shown on one line):
- show each n-digit fraction
- show each reduced n-digit fraction
- show what digit was crossed out for the numerator and the denominator
- Task requirements/restrictions
-
- only proper fractions and their reductions (the result) are to be used (no vulgar fractions)
- only positive fractions are to be used (no negative signs anywhere)
- only base ten integers are to be used for the numerator and denominator
- no zeros (decimal digit) can be used within the numerator or the denominator
- the numerator and denominator should be composed of the same number of digits
- no digit can be repeated in the numerator
- no digit can be repeated in the denominator
- (naturally) there should be a shared decimal digit in the numerator and the denominator
- fractions can be shown as 16/64 (for example)
Show all output here, on this page.
- Related task
Farey sequence. ─── Well, somewhat related, it has fractions.
- References
-
- Wikipedia entry; proper and improper fractions.
REXX
<lang rexx>/*REXX pgm reduces fractions by "crossing out" matching digits in nominator&denominator.*/ parse arg high show . /*obtain optional arguments from the CL*/ if high== | high=="," then high= 4 /*Not specified? Then use the default.*/ if show== | show=="," then show= 12 /* " " " " " " */ say center(' some samples of reduced fractions by crossing out digits ', 79, "═") $.=0 /*placeholder array for counts; init. 0*/
do L=2 to high /*do 2-dig fractions to HIGH-dig fract.*/
say
do n=10**(L-1) to 10**L - 1 /*generate some N digit fractions. */
if pos(0, n)\==0 then iterate /*Does it have a zero? Then skip it.*/
if hasDup(n) then iterate /* " " " " dup? " " " */
do d=n+1 to 10**L - 1 /*only process like-sized #.s*/
if pos(0, d)\==0 then iterate /*Have a zero? Then skip it. */
if verify(d, n, 'M')==0 then iterate /*No digs in common? Skip it*/
if hasDup(d) then iterate /*Any digs are dups? " "*/
q=n/d /*compute quotient just once.*/
do e=1 for L; xo= substr(n, e, 1) /*try crossing out each digit*/
nn= elide(n, xo) /*elide from the numerator. */
dd= elide(d, xo) /* " " " denominator.*/
if nn/dd \== q then iterate /*Not the same quotient? Skip*/
$.L= $.L + 1 /*Eureka! We found one. */
$.L.xo= $.L.xo + 1 /*count the silly reduction. */
if $.L>show then iterate /*Too many found? Don't show.*/
say center(n'/'d " = " nn'/'dd " by crossing out the" xo"'s.", 79)
end /*e*/
end /*d*/
end /*n*/
end /*L*/
say; @with= ' with crossed-out' /* [↓] show counts for any reductions.*/
do k=1 for 9 /*Is this a zero count? Then skip it. */
if $.k==0 then iterate /*Is this a zero count? Then skip it. */
say; say center('There are ' $.k " "k'-digit fractions.', 79, "═")
@for= ' For ' /*literal for SAY indentation (below). */
do #=1 for 9; if $.k.#==0 then iterate
say @for k"-digit fractions, there are " right($.k.#, k-1) @with #"'s."
end /*#*/
end /*k*/
exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ hasDup: procedure; parse arg x; w= length(x); if w<2 then return 0
do i=1 for w-1; if pos(substr(x,i,1), substr(x,i+1)) \== 0 then return 1
end /*i*/; return 0
/*──────────────────────────────────────────────────────────────────────────────────────*/ elide: procedure; parse arg x,c; return space( translate(x, , c), 0)</lang>
- output when using the input of: 5 12
══════════ some samples of reduced fractions by crossing out digits ═══════════
16/64 = 1/4 by crossing out the 6's.
19/95 = 1/5 by crossing out the 9's.
26/65 = 2/5 by crossing out the 6's.
49/98 = 4/8 by crossing out the 9's.
132/231 = 12/21 by crossing out the 3's.
134/536 = 14/56 by crossing out the 3's.
134/938 = 14/98 by crossing out the 3's.
136/238 = 16/28 by crossing out the 3's.
138/345 = 18/45 by crossing out the 3's.
139/695 = 13/65 by crossing out the 9's.
143/341 = 13/31 by crossing out the 4's.
146/365 = 14/35 by crossing out the 6's.
149/298 = 14/28 by crossing out the 9's.
149/596 = 14/56 by crossing out the 9's.
149/894 = 14/84 by crossing out the 9's.
154/253 = 14/23 by crossing out the 5's.
1234/4936 = 124/496 by crossing out the 3's.
1239/6195 = 123/615 by crossing out the 9's.
1246/3649 = 126/369 by crossing out the 4's.
1249/2498 = 124/248 by crossing out the 9's.
1259/6295 = 125/625 by crossing out the 9's.
1279/6395 = 127/635 by crossing out the 9's.
1283/5132 = 128/512 by crossing out the 3's.
1297/2594 = 127/254 by crossing out the 9's.
1297/3891 = 127/381 by crossing out the 9's.
1298/2596 = 128/256 by crossing out the 9's.
1298/3894 = 128/384 by crossing out the 9's.
1298/5192 = 128/512 by crossing out the 9's.
12349/24698 = 1234/2468 by crossing out the 9's.
12356/67958 = 1236/6798 by crossing out the 5's.
12358/14362 = 1258/1462 by crossing out the 3's.
12358/15364 = 1258/1564 by crossing out the 3's.
12358/17368 = 1258/1768 by crossing out the 3's.
12358/19372 = 1258/1972 by crossing out the 3's.
12358/21376 = 1258/2176 by crossing out the 3's.
12358/25384 = 1258/2584 by crossing out the 3's.
12359/61795 = 1235/6175 by crossing out the 9's.
12364/32596 = 1364/3596 by crossing out the 2's.
12379/61895 = 1237/6185 by crossing out the 9's.
12386/32654 = 1386/3654 by crossing out the 2's.
═══════════════════════There are 4 2-digit fractions.════════════════════════
For 2-digit fractions, there are 2 with crossed-out 6's.
For 2-digit fractions, there are 2 with crossed-out 9's.
══════════════════════There are 122 3-digit fractions.═══════════════════════
For 3-digit fractions, there are 9 with crossed-out 3's.
For 3-digit fractions, there are 1 with crossed-out 4's.
For 3-digit fractions, there are 6 with crossed-out 5's.
For 3-digit fractions, there are 15 with crossed-out 6's.
For 3-digit fractions, there are 16 with crossed-out 7's.
For 3-digit fractions, there are 15 with crossed-out 8's.
For 3-digit fractions, there are 60 with crossed-out 9's.
══════════════════════There are 660 4-digit fractions.═══════════════════════
For 4-digit fractions, there are 14 with crossed-out 1's.
For 4-digit fractions, there are 25 with crossed-out 2's.
For 4-digit fractions, there are 92 with crossed-out 3's.
For 4-digit fractions, there are 14 with crossed-out 4's.
For 4-digit fractions, there are 29 with crossed-out 5's.
For 4-digit fractions, there are 63 with crossed-out 6's.
For 4-digit fractions, there are 16 with crossed-out 7's.
For 4-digit fractions, there are 17 with crossed-out 8's.
For 4-digit fractions, there are 390 with crossed-out 9's.
══════════════════════There are 5087 5-digit fractions.══════════════════════
For 5-digit fractions, there are 75 with crossed-out 1's.
For 5-digit fractions, there are 40 with crossed-out 2's.
For 5-digit fractions, there are 376 with crossed-out 3's.
For 5-digit fractions, there are 78 with crossed-out 4's.
For 5-digit fractions, there are 209 with crossed-out 5's.
For 5-digit fractions, there are 379 with crossed-out 6's.
For 5-digit fractions, there are 591 with crossed-out 7's.
For 5-digit fractions, there are 351 with crossed-out 8's.
For 5-digit fractions, there are 2988 with crossed-out 9's.