Enter the two bounds, choose midpoint or mediant, and select “Calculate”. Because infinitely many fractions fit between two different ones, either rule gives you a valid answer. If you want to understand why there is always another one, read the guide on a fraction between two fractions.
Worked example: a fraction in between
Write a fraction greater than 1/3 and less than 1/2.
- Add the bounds over denominator 6: 2/6 + 3/6 = 5/6.
- Divide by 2: 5/6 ÷ 2 = 5/12.
- With the mediant, add numerators and denominators: (1 + 1)/(3 + 2) = 2/5.
- To check, rewrite everything over 60: 20/60, 24/60, 25/60, and 30/60.
Result: the midpoint is 5/12 and the mediant is 2/5, both between 1/3 and 1/2.
Check: over 60, 25 is 5 units from each bound, and 24 is 4 from the smaller one and 6 from the larger.
Which method to choose
The midpoint is the right choice when you want a value equally far from both bounds, or the average of two data points. The mediant is the right choice when you want a fraction with a small denominator, and it is the rule that gives the first new term between two neighbors in the Farey sequence. The English Wikipedia article on the Farey sequence states that the fraction between two neighbors is their mediant, found by adding numerators and denominators separately (accessed October 17, 2026).
| Bounds | Midpoint | Mediant |
|---|---|---|
| 3/5 and 5/8 | 49/80 | 8/13 |
| 1/10 and 1/5 | 3/20 | 2/15 |
| 1 and 2 | 3/2 | 3/2 |
| −1/2 and 1/4 | −1/8 | 0 |
| −3/4 and −2/3 | −17/24 | −5/7 |
With bounds of opposite signs the result can be 0, which is still strictly between them, and with consecutive integers the two methods agree.
Infinitely many fractions between two values
The calculator shows a single fraction, but it is not the only one. If you take the midpoint again between the smaller bound and the result, you get 3/8, which is even closer to the bound. You can repeat the process as often as you like: there is always another value, and this property is called the density of the rational numbers. To see where the values fall, open fractions on a number line.
Which values the calculator accepts
Denominators must be nonzero. The bounds must be two different values, because between two equivalent fractions there is nothing strictly in the middle. The result is always an exact fraction, with no decimal rounding.
Related pages
If you only want to know which of two fractions is larger, use compare fractions. To sort more than two values, there is order fractions. The mean of three or more values is calculated with average fractions. If you need to do it by hand with a common denominator, the guide on how to find a fraction between two fractions explains the reasoning and includes exercises with answers.
Frequently asked questions
Why does the mediant always land between the two fractions?
With positive denominators, a/b < c/d is equivalent to a·d < b·c. From that you get a/b < (a + c)/(b + d) < c/d. With 3/7 and 4/5: 3·5 = 15 is less than 7·4 = 28, and the mediant 7/12 satisfies 3/7 < 7/12 < 4/5. That is why the mediant of two different fractions never equals either bound.
Does it work with negative fractions?
Yes, as long as the sign sits in the numerator. Between −3/4 and 1/2, the mediant is (−3 + 1)/(4 + 2) = −1/3. If you wrote 3/(−4), the sum would give 4/(−2) = −2, which falls outside the interval. The calculator moves the sign to the numerator before calculating.
Does the mediant change if I do not simplify the fractions?
Yes, because it depends on how the fractions are written. With 6/9 and 4/5 it would give 5/7, but the calculator simplifies first: 6/9 becomes 2/3, and the mediant of 2/3 and 4/5 is 3/4. Both lie between the bounds. The midpoint does not change with the way you write the fractions.