Common Denominator Finder

A denominator names the size of the pieces a whole has been cut into — fourths, sixths, twelfths. Numerators can only be added or subtracted directly when they're counting pieces of the same size, which is exactly what rewriting over a common denominator sets up. It's the step that makes adding and subtracting fractions with unlike denominators possible in the first place — without it, 1/4 + 1/6 has no direct way to combine 1 and 1 across denominators that don't match.

Once both fractions share a denominator, the arithmetic collapses to something trivial: add or subtract the numerators, and keep the shared denominator unchanged. 3/12 + 2/12 = 5/12, with no further conversion needed. All of the real work in adding unlike fractions happens at the rewriting stage, not the addition itself.

The same rewriting trick settles a second, separate question: which of two fractions is larger. 5/8 and 7/12 don't compare cleanly at a glance, since a bigger numerator over a bigger denominator could go either way. Rewritten over their LCD of 24, they become 15/24 and 14/24 — now that the pieces are identical in size, the fraction with more of them, 15/24, is unambiguously the larger value.

Every equivalent fraction produced along the way — 3/12, 2/12, 15/24, 14/24 — is a genuine equivalent fraction of the original, just expressed with different-sized pieces. That's the whole trick: change the pieces, not the amount.