Adding Fractions: Using a Common Denominator

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Add fractions with a common denominator. (Click for larger view) Image by Mike DeHaan

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 fractions: Explanation one. Image by Mike DeHaan

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)

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