Adding Fractions

Expressed in symbols, the rule for adding fraction is as follows:

Let’s break this down to see everything that is expressed in this rule. The numerator of the sum is a · d + b · c. You can remember the numerator without having to memorize this particular formula by remembering the pattern of cross-multiplying. To create the numerator, you multiply each numerator by the opposing denominator, forming a “cross” pattern.

To get the denominator of the sum, you just multiply the two denominators ( b and d ) together.

Example

Work out each of the following sums of fractions.

Solution

(a) To work out the sum of these two fractions, you can use the rule for adding fractions:

You can also do this fraction addition in a slightly different and more efficient way:

The feature of this sum that allows you to calculate more efficiently is the fact that the two denominators (“3” and “12”) are related by the fact that “12” is a multiple of “3” because 3 × 4 = 12. You can put both fractions in the sum over the same denominator (a denominator of “12”) by multiplying the numerator of by “4” and the denominator of by “4.” This will put both fractions over the denominator of “12” and allow you to add them by just adding the numerators and putting the result over a denominator of “12.”

Note that Algebra Solver can easily do all kinds of problems with fractions that you enter. Click here for a sample screenshot.

(b) Although it does not appear to be an example of fraction addition, you can add

using the rules of fraction addition. The key is to re-write 2 · x as a fraction by putting it over a denominator of “1”. Doing this:

The result of this fraction addition can be simplified by combining the two square roots:

as the operations of squaring and taking a square root “reverse” or cancel each other leaving just x - 1. This simplification means that the sum of the two fractions could simplified to:

It is possible to simplify this fraction further, by taking the x inside the square root that still remains in the denominator. Taking a quantity inside a square root always involves squaring the quantity:

Although it is certainly theoretically possible to perform this last “simplification,” it probably would not make much of a difference to the simplicity of the sum of the two fractions, so it is largely a matter of taste as to whether or not you do it.

Note the use of FOILing to expand the term ( x + 1) that appears in the numerator when the fractions are added. One thing to note here is that although the same term (i.e. the term ( x + 1)) appears in both the numerator there is no special significance of this – it is just a coincidence – and no easy simplifications or cancellations that occur as a result.