Enter two fractions, a/b and c/d, using positive integer numerators and denominators. Their mediant is (a+c)/(b+d). It lies strictly between the two fractions when their values differ; when their values are equal, the mediant has that same value. Either fraction may be entered first.
Write the displayed mediant numerator over the displayed mediant denominator to obtain the exact, unreduced fraction. With the default inputs, 1/3 and 1/2 give (1+1)/(3+2) = 2/5 = 0.4.
Use positive integers from 1 to 1,000,000,000 for all four inputs. Decimal results are rounded to eight places, so very close fractions can appear equal. The calculator does not automatically reduce the result. Unavailable results indicate a calculation error, not zero.
Suppose a/b < c/d and both denominators are positive. Multiplying by bd gives ad < bc. The difference between the mediant and a/b is (bc−ad)/[b(b+d)], which is positive. The difference between c/d and the mediant is (bc−ad)/[d(b+d)], which is also positive. Therefore a/b < (a+c)/(b+d) < c/d.
If the input values are equal, then ad = bc and both differences are zero. The mediant equals both inputs.
| First fraction | Second fraction | Mediant as entered | Comparison |
|---|---|---|---|
| 1/3 | 1/2 | 2/5 | 1/3 < 2/5 < 1/2 |
| 2/3 | 3/4 | 5/7 | 2/3 < 5/7 < 3/4 |
| 3/4 | 1/4 | 4/8 = 1/2 | 1/4 < 1/2 < 3/4 |
| 1/2 | 2/4 | 3/6 = 1/2 | All three values are equal |
| 5/2 | 7/3 | 12/5 | 7/3 < 12/5 < 5/2 |
Is the mediant the sum of the fractions?
No. Fraction addition gives a/b + c/d = (ad+bc)/(bd). The mediant adds the numerators and denominators separately. For example, 1/3 + 1/2 = 5/6, while their mediant is 2/5.
Is the mediant the arithmetic mean?
The arithmetic mean gives equal weight to the two fraction values. The mediant weights them by their entered denominators. They agree when the denominators are equal or the fraction values are equal.
Does simplifying the inputs change the mediant?
It can. The mediant depends on the representations entered. For example, 1/2 and 1/3 give 2/5, but replacing 1/2 with 2/4 gives 3/7. Reduce both inputs first if you want the mediant of their lowest-term representations. Reducing the resulting mediant afterward does not change its value.
Can I use improper fractions?
Yes. Positive fractions greater than one follow the same rule. Enter each fraction as a positive integer numerator and denominator.
