A fraction is a value in two parts; each part, the numerator or denominator, is an integer. The numerator is the fraction's top number, while the denominator is its bottom number. Lower-order fractional math like addition and subtraction requires that the denominators of the involved fractions be the same value. When finding a fraction that comes between two others, you ignore normal fractional math in favor of a simpler method.
Write Out the Starting Fractions
Obtain two fractions for example purposes. For this example, let the fractions be 1/2 and 3/4.
Add the Numerators Together
Sum the fractions' numerators. In this example, 1 + 3 = 4.
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Add the Denominators Together
Sum the fraction's denominators. In this example, 2 + 4 = 6.
Write a New Fraction
Write a new fraction with the numerators' sum as the new numerator and the denominators' sum as the new denominator. In this example, the new fraction is 4/6.
Simplify the Fraction
Simplify the fraction by eliminating the greatest common factor shared by the numerator and denominator. To do this, list the factors of each number and factor out the largest shared number.
In this case, the factors of 4 are 1, 2 and 4, and the factors of 6 are 1, 2, 3 and 6. Both numbers have 1 and 2 as factors, with 2 being the greatest factor.
Eliminating 2 from both the numerator and denominator results in (4 ÷ 2) / (6 ÷ 2), which becomes 2/3.
TL;DR (Too Long; Didn't Read)
To check your answer, write the fractions with common denominators and compare the numerators. The example fractions of 1/2, 2/3 and 3/4 with common denominators become 6/12, 8/12 and 9/12. The numerator 8 is between 6 and 9, so the fraction you created – 8/12, or 2/3 when simplified – is between the two fractions you started with.