How this instrument works
Two fractions cannot be added until they are counted in the same-size pieces, and cross-multiplication is the fastest way to force that match for exactly two terms: rewrite a⁄b as ad⁄bd and c⁄d as cb⁄bd, then add the numerators over the shared denominator bd. Both rewrites are really just multiplying by 1 in disguise — b⁄b and d⁄d — so neither fraction's value changes, only the way it is written.
The shortcut is quietly generous: bd works as a common denominator no matter what b and d are, even when they share no factors at all. It is not always the smallest common denominator — when b and d share a factor, the least common denominator is smaller than their product, and the sum this identity returns will need reducing to reach lowest terms, even though the decimal value it reports is already exact and needs no further work.
This is the narrow rung between plain addition and the fuller adding-and-subtracting-fractions instrument. Ordinary addition assumes the two numbers already share a unit — five oranges plus three oranges needs no conversion first. Two fractions with unlike denominators share no unit until one is manufactured, and cross-multiplication manufactures it for exactly two terms and one operation. A tool built for subtraction, or for three fractions at once, needs a different common denominator entirely: the product or least common multiple of every denominator involved, not just two.
- Enter the first fraction's top number in Numerator 1 and its bottom number in Denominator 1.
- Enter the second fraction's top number in Numerator 2 and its bottom number in Denominator 2.
- Read Sum, a/b + c/d for the decimal value of the two fractions added together.
- To check by hand, cross-multiply as the formula box shows and compare the fraction you land on to the decimal in Sum.
- Negative numerators or denominators work the same way — the identity does not care about sign or about which fraction is larger.
Worked example — half a tank plus a third of another
Fill in Numerator 1 as 1, Denominator 1 as 2, Numerator 2 as 1, Denominator 2 as 3 — half a tank of fuel plus a third of another tank. Cross-multiplying gives (1 × 3 + 1 × 2) ⁄ (2 × 3), which is (3 + 2) ⁄ 6, or 5 ⁄ 6. Sum reports 0.8333333333333334, the repeating decimal that five-sixths always produces at machine precision.
The schoolbook method lands on the same total, which is worth checking once so the shortcut earns your trust. Rewrite ½ as 3⁄6 and ⅓ as 2⁄6 — both now count sixths — and 3⁄6 + 2⁄6 is plainly 5⁄6. Here bd = 6 happens to equal the least common denominator, because 2 and 3 share no factor; feed in 4 and 6 instead and bd = 24 while the least common denominator is really 12, yet the sum still comes out correct, only not in lowest terms until it is reduced.
Questions
Why does cross-multiplying always work for adding two fractions?
Because multiplying a fraction's numerator and denominator by the same nonzero number never changes its value — that is b⁄b and d⁄d equalling 1 in disguise. Rewriting a⁄b as ad⁄bd and c⁄d as cb⁄bd forces both fractions onto the identical denominator bd, so the numerators add directly: (ad + bc) ⁄ bd. No search for a least common denominator is required, only two multiplications.
Does the denominator bd have to be the least common denominator?
No — bd is only guaranteed to be a common denominator, not necessarily the smallest one. When b and d share a factor, the least common denominator is smaller than the product bd, and the sum this identity returns will need reducing to reach lowest terms, even though the decimal value shown is already exact.
How is this different from a full adding-and-subtracting-fractions tool?
This instrument covers exactly two fractions and one operation, addition. A broader adding-and-subtracting-fractions tool extends the same cross-multiplication idea to subtraction and to chains of three or more terms, where the shared denominator becomes a product across every term rather than just two denominators.
What happens if a numerator or denominator is negative?
The formula does not change shape. A negative numerator makes that fraction negative, a negative denominator flips the sign of the whole fraction, and (ad + bc) ⁄ bd still returns the correct signed total — subtracting one fraction from another is really just adding its negative.
Can this method add two fractions that already share a denominator?
Yes, though it does more work than strictly needed. If b equals d, cross-multiplying still lands on the right answer, but bd duplicates the shared denominator instead of using it once — harmless arithmetically, just a longer route to the sum you would get by adding the numerators directly.
Does this handle mixed numbers such as one and a half?
Not directly — convert a mixed number to an improper fraction first, so one and a half becomes numerator 3 and denominator 2, and enter those. The underlying identity only ever operates on two numerator-denominator pairs, so the conversion has to happen before the fields are filled in.