Mediant of Two Fractions Calculator

This calculator performs that exact operation without reducing or otherwise changing the result. Enter four integers representing the two fractions, and it returns the mediant as separate numerator and denominator values plus a readable fraction. It also rejects either input fraction when its denominator is zero.

How to calculate the mediant

Suppose the first fraction is a/b and the second is c/d. Their mediant is (a + c)/(b + d). Enter a as the first numerator, b as the first denominator, c as the second numerator, and d as the second denominator. The calculator adds across the numerators and denominators independently, then reports both sums and a formatted fraction. For example, the mediant of 1/3 and 2/5 is 3/8 because 1 + 2 equals 3 and 3 + 5 equals 8. This operation is different from ordinary fraction addition: adding 1/3 and 2/5 requires a common denominator and produces 11/15, while their mediant is 3/8. Keep that distinction in mind when choosing the operation. The output deliberately preserves the direct sums instead of simplifying them, because those sums are the definition of the mediant and often carry useful information about how the result was constructed. Negative numerators and denominators are accepted as long as each denominator is nonzero and every value is a safe integer.

What the result means

For two positive fractions with positive denominators, the mediant lies strictly between them when the fractions have different values. This makes the construction useful for finding a rational number inside an interval. The familiar Farey sequence and Stern–Brocot tree use mediants to organize rational numbers and generate increasingly precise bounds. The returned numerator and denominator are the literal component sums, not necessarily a fraction in lowest terms. If you enter 1/2 and 3/4, the result is 4/6 even though 4/6 has the same numerical value as 2/3. Preserving 4/6 matters when you are studying the mediant operation itself, tracing a tree path, or checking a hand calculation. Signs deserve care: the between-ness property assumes the conventional positive-denominator setting, while the arithmetic definition still works with signed inputs. Also note that the two input denominators may sum to zero. In that case the tool still returns the formal mediant with a zero denominator, because the stated invalid-input rule applies to either original denominator and the component-wise sum remains well defined as a formal fraction expression.

Validation and reliable API use

Each of the four fields must be supplied as an integer that JavaScript can represent exactly. Decimal values, numeric strings, missing values, infinities, and integers outside the safe range are rejected instead of being rounded or silently coerced. The first and second denominators are checked separately, and either one being zero produces an invalid-input error. After validation, the calculation uses only two additions, so the same request always produces the same JSON result and never depends on a network service, stored state, time, or randomness. API consumers receive the fields numerator, denominator, and fraction. The separate numeric fields are convenient for later computation, while the fraction string is ready for display or logging. Before sending large integer values, remember that even two individually safe integers can have a sum outside the safe range; such an overflow is rejected to protect exactness. Automated clients can call the capability for $0.002 per request, while interactive users can perform the same deterministic calculation in the browser. For reproducible workflows, store the original four inputs alongside the returned mediant so the construction can be audited later.