Several rules for the calculation with fractions are useful:
Cancelling. If both the numerator and the denominator of a fraction are multiplied or divided by the same number, then the fraction does not change its value. For instance, 4/6 = 2/3 and 1/x = x / x2.
Adding fractions. To add or subtract two fractions, you first need to change the two fractions so that they have a common denominator; then you can add or subtract the numerators. For instance, 2/3 + 1/4 = 8/12 + 3/12 = 11/12.
Multiplying fractions. To multiply two fractions, multiply the numerators to get the new numerator, and multiply the denominators to get the new denominator. For instance, 2/3 × 1/4 = (2×1) / (3× 4) = 2 / 12 = 1 / 6.
Dividing fractions. To divide one fraction by another one, flip numerator and denominator of the second one, and then multiply the two fractions. For instance, (2/3) / (4/5) = 2/3 × 5/4 = (2×5) / (3×4) = 10/12 = 5/6.
In abstract algebra, these rules can be proved to hold in any field. Furthermore, if one starts with any integral domain Q, one can always construct a field consisting of all fractions of elements of Q, the field of fractions of Q.