# Adding Fractions: Using a Common Denominator

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Adding fractions: Explanation two. Image by Mike DeHaan

It’s easier to follow the process of adding fractions in these two images. The first explanation, above, shows that we select the denominators to create the d/d and b/b multipliers.

The second explanation, to the right, shows that d and b are multiplied by the numerators of the fractions where they were not originally in the denominators.

After adding the fractions with a common denominator, two final steps remain: to simplify the fraction and express the result as a mixed fraction.

## How to Simplify a Fraction

To simplify a fraction means to reduce the fraction to the simplest terms.

The largest common factor needs to be found for the final denominator and numerator.

For example: 840/3024  = (5*168)/(18*168) = 5/18.

Of course, it may not be that easy to recognize the greatest common factor. One might try a few prime numbers as simple factors:

• Try 2:  840/3024 = 420/1512 = 210/756 = 105/378.
• Try 3: 105/378 = 35/126.
• It’s obvious that 5 is not a factor of 126.
• Try 7: 35/126 = 5/18.
• Since 5<11, there are no more possible prime factors.

## Mixed Versus Improper Fractions

Remember that a proper fraction has a value between negative one and positive one. Improper fractions, such as 11/7, are often shown as mixed fractions, like 1-4/7. (It might be hand-written as “1 4/7”).

The final step in adding fractions is to check whether the result is an improper fraction. If so, it’s usually good form to change it to the mixed format.

With any luck, the final result is a whole number. For example: 3/6 + 5/10 = (30 + 30)/60 = 60/60 = 1.

It might be easier to convert the improper fraction before simplifying. It’s easy to decide that an improper fraction is less than 2, which leaves a smaller numerator for the simplifying step. On the other hand, if the improper fraction is greater than 10, the task of dividing to find the exact whole number and making the remainder into the numerator can be daunting.

The  first step of adding mixed fractions is to decide whether to deal with the whole numbers first or to convert both to improper fractions.

Consider the example of (7 + 1/8 ) + (9 + 7/23) = 16 + (1/8 + 7/23). In this case, you might prefer to keep the 16 separate while dealing with the common denominator 8*23 = 184.

On the other hand, if you often misplace the whole number (16) while adding up the proper fractions, perhaps you should start with (7 + 1/8 ) + (9 + 7/23) = 57/8 + 214/23 and then deal with the numerators 57 * 23 = 1311 and 214 * 8 = 1712.

In general, the steps for adding fractions are:

1. Determine whether the fractions have a common denominator (if yes, skip to step 4).
2. If either fraction is mixed, either add the whole number portions right away or convert to improper fractions.
3. Create a common denominator by multiplying denominators, as in a/b + c/d = (a/b * d/d) + (c/d * b/b).