A worked example: 3/4 + 5/4
Adding 3/4 and 5/4 doesn't need a search for the smallest shared denominator — multiplying the two denominators together always works. Here both denominators are 4, so the shared denominator is 4 × 4 = 16, and the numerators become 3 × 4 = 12 and 5 × 4 = 20, giving 32/16 before simplifying. Reducing 32/16 by its greatest common factor of 16 leaves 2/1, which is just the whole number 2 — confirmed by the calculator's decimal output of exactly 2.
The formula: cross-multiply denominators, then simplify
For addition and subtraction, the fraction calculator computes the new numerator as numerator1 × denominator2 ± numerator2 × denominator1, and the new denominator as denominator1 × denominator2. That result is then reduced using the greatest common factor of the numerator and denominator, so 32/16 collapses to 2/1 in one step rather than requiring a separate simplification pass.
Multiplying and dividing: 5/6 × 3/4 and 2/3 ÷ 5/6
Multiplication skips the common-denominator step entirely: numerators multiply straight across and so do denominators. 5/6 × 3/4 gives (5 × 3)/(6 × 4) = 15/24, which simplifies to 5/8 — matching the calculator's decimal output of 0.625. Division instead multiplies by the reciprocal of the second fraction: 2/3 ÷ 5/6 becomes (2 × 6)/(3 × 5) = 12/15, which simplifies to 4/5, or 0.8 as a decimal.
Why simplifying to lowest terms matters
A subtraction like 5/6 − 1/6 illustrates the same pattern: the raw cross-multiplied result is 24/36, which is correct but not in its simplest form. Dividing both numbers by their greatest common factor of 12 reduces it to 2/3, or 0.66666667 as a decimal. Without that final reduction, equivalent fractions like 24/36 and 2/3 would look like different answers even though they represent the exact same value — which is why the calculator always returns the reduced form.