Steps for adding fractions with unlike denominators

• Albert DuBose

  Albert holds a Bachelor of Music Education and a Master of Science in Education from Troy University. He taught instrumental music in public schools for ten years. He left teaching for a few years before returning as an online teacher for English and Mathematics skills. Presently he teaches people worldwide how to perfect their English skills.

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• Instructor Joelle Mumley

Learn how to convert fractions with unlike denominators, and how to add fractions with different denominators. See adding unequal fractions examples. Updated: 08/24/2021

Fraction strips showing fractional equivalences

Is it Possible to Add Fractions with Different Denominators?

Adding fractions with the same denominator is quite simple: just add the numerators and keep the denominator. The only thing that can make this process more difficult is if the result needs to be simplified, or if it is necessary to convert the result into a mixed number. An example of a result that needs to be simplified is the result of 1/8 + 3/8, which is 4/8. This number can be simplified as 1/2. An example of a result that needs to be converted into mixed number can be found in the following problem: If one adds 7/12 and 11/12, the result is 18/12, which can be simplified as 3/2. However, this is an improper fraction (meaning that the numerator is greater than the denominator), so it must be converted to 1 & 1/2.

Adding fractions becomes more difficult if the denominators are different, however. Take, for example, the problem 3/8 + 5/6. These two fractions have different denominators, 8 and 6, respectively, so the fractions have to be converted to equivalent fractions (which share the same denominator) so that we can add them together.

So, what can be done when faced with an addition problem such as 3/8 + 5/6? Fortunately, there are ways to change the two fractions so that they will have the same denominator. This usually involves finding a least common multiple, or LCM. A LCM is the smallest multiple shared between two or more numbers. Once the least common multiple of two or more numbers is determined, fractions may be converted to equivalent fractions, and then addition or subtraction operations may be completed to solve the problem.

Common Denominators

Before we add fractions with different denominators, let's review how to add fractions with common denominators. When two fractions have a common denominator, to add them together we simply add their numerators to get the numerator of the sum. The denominator of the sum stays the same.

If we cut the pie into eight equal size pieces, then each piece is 1/8 of the pie. You eat two pieces of pie, so you ate (1/8) + (1/8) = 2/8, or 1/4 of the pie.

Now suppose you ate one piece and wanted more, but not a whole piece, so you cut the other piece in half, creating a slice that is 1/16 of the pie. When this is the case, you ate 1/8 + 1/16 of the pie, but you can't just add the numerators as the slices are different sizes.

However, if we consider the slice that is 1/8 of the pie as 2/16 of the pie, then we see that we've eaten 2/16 + 1/16 = 3/16 of the pie.

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There are, basically, four steps involved in adding or subtracting fractions with different denominators. Care must be taken to follow this step-by-step procedure to ensure that the problem has been worked out correctly.

In a nutshell, the process is as follows:

1. Find the least common multiple (LCM) of the denominators

2. Convert the fractions

3. Add (or subtract) the fractions

4. Simplify the answer, if possible

Whenever one takes a particular number, such as 8, and multiplies it by 1, 2, 3, 4, and so on, we are 'finding the multiples' of that number. For example, the multiples of 8 are 8, 16, 24, 32, 40, 48, and so on. The multiples of 6 are 6, 12, 18, 24, 30, 36, and so on. Notice that 24 is the lowest number that these two lists of multiples have in common. 24, then, is the least common multiple of 6 and 8.

The next step in either adding or subtracting the two fractions is to convert one or both fractions into equivalent fractions. These are fractions that are the mathematical equivalents of the original fractions, but which may have been changed so that the denominators of the two fractions are the same.

In the case of 3/8, the LCM of 6 and 8 is 24, and that 8 must be multiplied by 3 in order for the denominator to be 24. The denominator, 8, is multiplied by 3, so the numerator must also be multiplied by 3. This results in the equivalent fraction 9/24.

The fraction 5/6 must also be converted so that its denominator will be 24. 6 x 4 = 24, so the numerator and denominator of 5/6 are both multiplied by 4. The result is the fraction 20/24, which is equivalent to 5/6.

Now that 3/8 and 5/6 have been converted to two equivalent fractions, 9/24 and 20/24, the two numerators may be added together, while the denominator is kept the same. In this case, the result of the addition is 29/24, which is an improper fraction (that is, the numerator is greater than the denominator) and must be simplified to 1 & 5/24.

In many cases, it will be necessary to simplify a fraction as a result of an addition or subtraction problem. Take 5/8 - 3/8, for example. When 3/8 is subtracted from 5/8, the result is 2/8. This fraction, 2/8, can be simplified, because both 2 and 8 are divisible by 2. If the numerator, 2, is divided by two, the result is 1; and if the denominator, 8, is divided by two, the result is four. Therefore, the final result is 1/4, which cannot be simplified further.

If 3/16 and 5/16 are added together, the result is 8/16. This can be simplified by finding the greatest common factor, or the GCF, that will divide 8 and 16. The GCF is the largest number that will evenly divide two or more numbers. In the case of 8 and 16, the GCF is 8. 8 divided by 8 is 1, and 16 divided by 8 is 2, making the final result 1/2.

Prime Factorization

There are three ways to find the GCF of two or more numbers:

• Listing out common factors
• Prime factorization
• Division method

For the sake of brevity, this lesson will focus on one method, prime factorization.

Prime factorization of 8

Prime numbers are numbers greater than 1 which can only be divided by themselves and 1. Prime numbers include 2, 3, 5, 7, 11, 13, 17, and so forth. Numbers such as 4 and 10 are not prime numbers. The purpose of prime factorization, then, is to break down a number into factors which cannot be further equally divided. For example, the prime factors of 51 are 3 and 17.

Perhaps the easiest way to determine the prime factorization of a number is to begin to divide it by the smallest prime number, 2, then 3, then 5, so on and so forth. For example, consider the number 420.

• Is 420 divisible by 2? Yes, and the result is 210.
• Is 210 divisible by 2? Yes, and the result is 105.
• Is 105 divisible by 2? No. Is it divisible by 3? Yes, and the result is 35.
• Is 35 divisible by 2? No. Is it divisible by 3? No. Is it divisible by 5? Yes, and the result is 7.

7 is a prime number, so the prime factorization is concluded. Therefore, the prime factorization of 420 is 2 x 2 x 3 x 5 x 7.

Suppose one is given a fraction such as 32/48. It will be necessary to work out the prime factorization of 32, and then to work out the prime factorization of 48 underneath. The resulting list will look like this:

32 = 2 x 2 x 2 x 2 x 2

48 = 2 x 2 x 2 x 2 x 3

• The 2 and 3 at the ends of the two lists are eliminated, because they are different.
• For both 32 and 48, we are left with 2 x 2 x 2 x 2 = 16.
• 16, then, is the largest number that will divide into 32 and 48.
• 32 divided by 16 is 2, and 48 divided by 16 is 3.
• The final fraction, then, is 2/3.

In many other cases, an addition or subtraction problem results in an improper fraction, such as 13/4. An improper fraction is one in which the numerator is larger than the denominator. Improper fractions are considered bad style in the mathematics world, and as such we would normally like the final result to be expressed as either an integer, a fraction, or a mixed number. In this case, we would divide the numerator, 13, by the denominator, 4. The result is 3 with a remainder of 1. The final result, then, is 3 & 1/4.

Adding Fractions Examples

Consider the problem 2/3 + 7/8.

Different Denominators

When adding two fractions with different denominators, manipulate the fractions to find a common denominator, and then all we have to do is add the numerators. There are different ways to go about getting a common denominator in fractions. We're going to go with one of the easiest ways, and that is with this formula:

Consider our pie example again. We know that 1/8 + 1/16 = 3/16. Let's use our formula to verify this.

We see that our formula also gives that 1/8 + 1/16 = 3/16.

Applications

Let's look at a few applications of adding fractions with different denominators in real-life scenarios. The first has to do with distance.

Example 1

Suppose you're trying to get in shape and you're tracking how many miles you walk or jog throughout the day. You know that when you walk to work from your car, it's a distance of 1/4 mile. Since you make that trip twice a day, this counts for 1/2 a mile of walking. When you get home from work, you end up having to park your car 5/8 of a mile from your house. By the time you get home, you tally up how much you have walked so far. That is, you want to add 1/2 + 5/8, so we use our formula.

Now you know that so far you've walked 9/8 of a mile, or 1 and 1/8 of a mile.

Common Denominators

Before we add fractions with different denominators, let's review how to add fractions with common denominators. When two fractions have a common denominator, to add them together we simply add their numerators to get the numerator of the sum. The denominator of the sum stays the same.

If we cut the pie into eight equal size pieces, then each piece is 1/8 of the pie. You eat two pieces of pie, so you ate (1/8) + (1/8) = 2/8, or 1/4 of the pie.

Now suppose you ate one piece and wanted more, but not a whole piece, so you cut the other piece in half, creating a slice that is 1/16 of the pie. When this is the case, you ate 1/8 + 1/16 of the pie, but you can't just add the numerators as the slices are different sizes.

However, if we consider the slice that is 1/8 of the pie as 2/16 of the pie, then we see that we've eaten 2/16 + 1/16 = 3/16 of the pie.

Different Denominators

When adding two fractions with different denominators, manipulate the fractions to find a common denominator, and then all we have to do is add the numerators. There are different ways to go about getting a common denominator in fractions. We're going to go with one of the easiest ways, and that is with this formula:

Consider our pie example again. We know that 1/8 + 1/16 = 3/16. Let's use our formula to verify this.

We see that our formula also gives that 1/8 + 1/16 = 3/16.

Applications

Let's look at a few applications of adding fractions with different denominators in real-life scenarios. The first has to do with distance.

Example 1

Suppose you're trying to get in shape and you're tracking how many miles you walk or jog throughout the day. You know that when you walk to work from your car, it's a distance of 1/4 mile. Since you make that trip twice a day, this counts for 1/2 a mile of walking. When you get home from work, you end up having to park your car 5/8 of a mile from your house. By the time you get home, you tally up how much you have walked so far. That is, you want to add 1/2 + 5/8, so we use our formula.

Now you know that so far you've walked 9/8 of a mile, or 1 and 1/8 of a mile.

How do you explain adding fractions with different denominators?

Fractions with different denominators cannot be added together without converting one or both of the fractions.

It is impossible, for example, to add 2/3 and 3/4. Both fractions must be converted to fractional equivalents with a common denominator, in this case 8/12 and 9/12.

The two fractions can then be added together. In this example, the result is 17/12.

Frequently, the addition results in an improper fraction, that is, a fraction in which the numerator is greater than the denominator. An improper fraction must instead be expressed as a mixed number, which in this case would be 1 & 5/12.

How do you add fractions with unlike denominators step by step?

Step 1: Find the Least Common Multiple (LCM) of the Denominators.

For example, if 5/6 and 5/8 are being added together, the LCM of 6 and 8 is 24.

Step 2: Convert the Fractions.

In our example, we would change 5/6 to 20/24 and 5/8 to 15/24.

Step 3: Add the Fractions.

Adding 20/24 + 15/24 = 35/24. Be careful to add the numerators and retain the denominator.

Step 4: Simplify the Answer if Possible.

35/24 is an improper fraction. It must be expressed as a mixed number, or 1 & 11/24.

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