Adding fractions can be a simple task in basic math, and breaking it down to six steps makes the job even easier.

A couple of things to keep in mind as you go through the steps: we’ll never divide by zero – every denominator is non-zero. Also, all examples use positive numbers, though a general fraction like “a/b” could be negative.

## Using Common Denominators

It’s easy to add fractions that have a common denominator. The rule is: a/b + c/b = (a+c)/b.

For example: 4/23 + 5/23 = (4+5)/23 = 9/23.

If the fractions share a common denominator, then the only remaining tasks is to simplify the fraction and express the result as a mixed fraction or whole number. These steps might or might not be possible, depending on the value, as discussed later.

As the above image shows, fractions cannot be added if the denominators are not equal. For example: 1/3 + 2/5 is not 3/3 or 3/5 or 3/15. As Boromir might have advised a math student in The Lord of the Rings, “*One does not simply add fractions with different denominators*.”

Instead, each fraction must be expressed as an equivalent fraction, one that has the same value as the original, but with a common denominator.

The example in the picture shows the process. First, the common denominator should be 15, and the equivalent fractions are 5/15 for 1/3 and 6/15 for 2/5. Second, add 5/15 + 6/15 = (5+6)/15 = 11/15.

## Adding Two Fractions: Make a Common Denominator

The fastest path to a common denominator is to multiply the denominators together. However, that’s not the easiest route. Adding a step to find the least common denominator (LCM) would make the calculations easier, but only if the LCM is less than the product of the denominators. Because the final fraction is simplified towards the end of the process, this article will bypass looking for a least common denominator.

Remember that x/x = 1, and multiplying by 1 leaves the original value unchanged.

Here is the process for adding two fractions with different denominators: a/b and c/d. We need to use a common denominator b*d.

d/d = b/b = 1 are the multipliers, so that the common denominator will be b*d.

Let’s color-code the process to make it easier to follow.

- a/b + c/d = (a/b * d/d) + (c/d * b/b) =…
- …= (a*d)/(b*d) + (c*b)/(d*b) =…
- …= (a*d + c*b)/(b*d)

**Click to Read Page Two: Adding Fractions – the Process**

Dewi Williams says

I find that when teaching this kind of topic it helps to make it as visual as possible at the beginning. What exactly does a quarter of something look like? A third? Two thirds.

Mr. Calculator says

In measurements, fractions appear whenever units are not small enough to express quantities in integers. For example, five quarter-dollars will buy you exactly as mush as a dollar and a quarter. One and a half dollar stands for exactly the same quantity as three half-dollars or six quarter-dollars.

Fractions are unavoidable and sooner or later we all have to learn to work with fractions. The mathematical usage of the word fraction has a very clear everyday connotation as a part of a bigger object. It would be unthinkable nowadays to just introduce fractions as a pair of numbers and postulate their basic properties. Still, to express fractions one needs a pair of numbers with a meaning and intuition attached to them.

When multiplying fractions, the numerators (top numbers) are multiplied together and the denominators (bottom numbers) are multiplied together. To divide fractions, rewrite the problem as multiplying by the reciprocal (multiplicative inverse) of the divisor. To add fractions that have the same, or a common, denominator, simply add the numerators, and use the common denominator. However, fractions cannot be added until they are written with a common denominator. The figure below shows why adding fractions with different denominators is incorrect.