How to find a fraction between two fractions

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If a and b are different, the average (a + b) ÷ 2 always falls between the two and is another fraction. For 2/3 and 3/4 that average is 17/24. To calculate it and check it with your own values, open fraction between two fractions. Here you see the method, the case where there is no answer and the reason the in-between fractions never run out.

The midpoint gives a single answer, while the common denominator lets you choose among several.

The result is rational because adding fractions and dividing by 2 keeps the form of a quotient of integers. It also lands inside the interval: the distance from a to the midpoint is (b − a) ÷ 2, positive if a is less than b, and the distance from the midpoint to b is the same. If you give the endpoints the other way around, the sum does not change and the result is the same.

When the numerators are consecutive

With a common denominator there is not always room between the numerators. Between 3/7 and 4/7 the numerators 3 and 4 are consecutive, and no integer fits. The way out is to scale up: multiply the numerator and denominator of both fractions by 2 and you get 6/14 and 8/14, with 7/14 in the middle. If you need several, multiply by a larger factor and you will have more free numerators.

This method returns an interior fraction, but it does not tell you which one is the central one. For the central one, use the midpoint.

With negative endpoints or endpoints of different signs

The method does not change with negatives. Between −1/2 and 1/4, eighths give −4/8 and 2/8; the sum is −2/8 and its half is −1/8. The check is −4/8 < −1/8 < 2/8. The midpoint sits to the left of zero even though the upper endpoint is positive, because −1/2 weighs more than 1/4.

If both endpoints are negative, order them carefully: between −3/8 and −1/8 the lesser is −3/8, because it is farther from zero. If they are opposites, like −2/5 and 2/5, the sum is 0 and the midpoint is 0.

Why there are infinitely many fractions between two values

After finding a midpoint m between a and b, find the midpoint between a and m. It is different from a and from m, and it is still inside the original interval. You can repeat this as many times as you like, so you never reach a last in-between fraction.

Between 0 and 1 there is 1/2, between 0 and 1/2 there is half of that half, and you can keep halving without end. That is why no fraction has a “next one” among the rationals, unlike the integers, where after 2 comes 3. The English Wikipedia article on rational numbers describes this property by saying that the rationals form a dense subset of the real numbers.

There is one case without in-between fractions: endpoints that are equal. Between 3/5 and 6/10 there is nothing, because they are the same number written two ways and there is no value strictly between a number and itself. That is why the first step of the method is to check that the endpoints are different.

Midpoint and mediant are not the same

The mediant of p/q and r/s is (p + r)/(q + s), and it also falls inside the interval when the denominators are positive. Between 1/5 and 1/3 the mediant is (1 + 1)/(5 + 3) = 2/8, which simplified is 1/4, while the midpoint is (3/15 + 5/15) ÷ 2 = 4/15. Both fall inside, but 4/15 is 1/15 from each endpoint, and the mediant is 1/20 from the left one and 1/12 from the right one.

Choose the midpoint if you want equal distance. The mediant is useful for locating simple fractions; the Farey sequence uses it to build ordered lists.

Sources and tools

Source: the English Wikipedia article Rational number, which says the rational numbers form a dense subset of the real numbers (consulted October 17, 2026). To see the positions, use the number line; to order more than two values, order fractions.