Lesson Video: Adding Fractions with Unlike Denominators Mathematics • 5th Grade

Video Transcript

In this video, we will learn how to add two proper fractions with different denominators by finding a common denominator. We’ll see how we can use models to add fractions by finding a common denominator and then how to use the least common denominator to add fractions.

We already know that to add fractions, the size of the parts must be the same. And to determine the size of the parts, we look at the denominator. The denominator tells us how many parts make one whole. For example, one-fifth and two-fifths both have a denominator of five. So in a bar model for fifths, each whole is divided into five equal parts. To find the sum of two fractions that both have a denominator of five, we just need to count the number of equal parts. There are three equal parts in total. We say that one-fifth add another two-fifths makes three-fifths.

We run into a real challenge when the denominators are different, such as with the fractions one-third and one-fifth. To find their sum, we need to count the number of equal parts, but a one-third part is larger than a one-fifth part. We can split the bars into smaller equal parts to find a common denominator. If we divide each of the thirds into five equal pieces and divide each of the fifths into three equal pieces, then we have divided both bars into the same-size parts. These parts are fifteenths.

Now we can see that one-third and one-fifth can each be represented as an equivalent number of fifteenths. The first bar shows one-third is equivalent to five-fifteenths. And the second bar shows how one-fifth is equivalent to three-fifteenths. This means that five-fifteenths plus three-fifteenths is equivalent to one-third plus one-fifth. Finally, we add the same-size parts to get eight-fifteenths.

We can also use area models to add fractions with different denominators. The area model is useful for finding a common denominator. Let’s say we wanted to add one-fourth and two-thirds. To use this method, we begin by sketching a rectangular area model to represent one-fourth. We know that the denominator represents the number of equal parts, so we divide the model into four rows. Then we shade one of the four rows.

We sketch a second area model of the same size to represent two-thirds. This time, we will divide the area model into equal columns instead of rows. Since the denominator is three, we will have three columns. Then we shade two of the three equal parts.

Our next step is to create a third area model with the same number of rows as the first and the same number of columns as the second. We have created an area model made of 12 equal parts. So 12 is going to be our common denominator. Now we return to our first two area models. With our new common denominator in mind, we divide each model into the same 12 equal parts. This allows us to find fractions equivalent to one-fourth and two-thirds with a common denominator of 12.

We see three out of 12 equal parts shaded in orange. This means that one-fourth is equivalent to three-twelfths. By counting the number of equal parts shaded in pink, we find that two-thirds is equivalent to eight-twelfths. Finally, we use the third area model to show the number of shaded parts altogether, which reveals our combined total of 11 out of 12 shaded parts.

Using the area model, we have shown that one-fourth plus two-thirds is equivalent to three-twelfths plus eight-twelfths. Once we have equivalent fractions with a common denominator, we just add the numerators together. This leads us to the sum eleven twelfths.

As we saw in the last example, it is possible to find a common denominator by multiplying the number of rows by the number of columns. In other words, multiplying the denominators together gives us a common denominator.

However, this strategy does not always give us the least common denominator. In future examples, we may work with equivalent fractions that have larger numbers as numerators and denominators, which is going to make our calculations more difficult. If we can identify a smaller common multiple of the denominators, that would simplify our calculations.

Let’s consider the sum of five-sixths and three-eighths.

According to the area model method, we would use a common denominator of 48. Creating an area model with 48 equal parts, along with all the necessary shading might take a lot more work than we have time for. Listing multiples of each denominator will give us many common denominators. We want to find out if any of these common denominators are smaller than 48. The first five multiples of six are six, 12, 18, 24, and 30. If we start listing the multiples of eight, we get eight, 16, 24, and 32.

We can define the least common denominator as the smallest multiple the denominators have in common. By comparing the two lists, we see that 24 is the smallest multiple the denominators have in common. This is the least common denominator for five-sixths and three-eights.

Now we are ready to create equivalent fractions with our new common denominator. To proceed without a model, we must think of the number we multiply six by to get 24. We know that six multiplied by four equals 24. So we multiply the numerator by four as well. The result is the equivalent fraction twenty twenty-fourths. Then we do the same for three-eights. But in this case, we multiply the denominator eight by three to get 24. So we also multiply the numerator by three to get our equivalent fraction. And we find that three-eighths is equivalent to nine twenty-fourths.

Now that we have rewritten our fractions with the least common denominator, we can add the numerators together to find the sum as a fraction over 24. It is important that we only add the numerators, never the denominators. The common denominator represents how the whole is divided into parts of equal size that are allowed to be added together. The parts are 29 in total. So five-sixths plus three-eighths is equivalent to twenty twenty-fourths plus nine twenty-fourths, which equals twenty-nine twenty-fourths.

Here we recognize that the sum is greater than one. In this case, we must determine the whole number part and the remaining fraction part to write our answer as a mixed number. The whole number part, one, is equivalent to twenty-four twenty-fourths. So five of the 29 parts remain. Finally, we write our mixed number answer as one and five twenty-fourths. We have shown this is the sum of five-sixths and three-eighths by using the least common denominator.

We will use this technique in our next example.

Mia is designing a flag. She colors four-sixths of the flag yellow and two-ninths of the flag blue. She leaves the rest of the flag white. What fraction of the flag does Mia color altogether?

Let’s quickly sketch what is happening. Mia is designing a flag. She colors four-sixths yellow and two-ninths blue. We need to find the fraction of the flag that is colored altogether. This means we need to add the fractions.

We know that to add two fractions, we count the number of equal parts. However, in this case, we do not have equal parts. Fractions with equal parts have the same denominator, but these fractions have different denominators: six and nine. So we need to find a common denominator, which is a common multiple of six and nine.

The product of six and nine in this case is certainly a multiple of six and a multiple of nine. But this method does not always lead to the least common denominator. 54 is quite a large number. And because six and nine both have three as a factor, it is likely not the least common denominator.

To find the least common denominator, we will begin by listing the multiples of both six and nine then select the smallest multiple they have in common. The first six multiples of six are six, 12, 18, 24, 30, and 36. The smallest multiple of six that is also a multiple of nine will be our least common denominator. The first four multiples of nine are nine, 18, 27, and 36. Which multiples do they have in common? 18 and 36. Since we want the least common denominator, we select 18.

Now we are ready to create equivalent fractions with our new common denominator. To find the fraction equivalent to four-sixths, we must think of the number we multiply by six to get 18. We know that six multiplied by three equals 18. So we multiply the numerator by three as well. The result is the equivalent fraction twelve eighteenths. Then we do the same for two-ninths. But in this case, we multiply the denominator nine by two to get 18. So we also multiply the numerator by two to find our equivalent fraction. And we find that two-ninths equals four-eighteenths.

Now that we have rewritten our fractions with a common denominator, we can add the parts of equal size together. Twelve eighteenths plus four-eighteenths equals sixteen eighteenths altogether. This means four-sixths plus two-ninths is equivalent to twelve eighteenths plus four-eighteenths, which equals sixteen eighteenths, which is the fraction of the flag Mia colored.

We should check whether our answer can be simplified. In this case, it can, as the numerator and denominator are both divisible by two. So the simplified answer is eight-ninths.

In our final example, we will see that when adding more than two fractions together, we will still need a common denominator.

Calculate one-sixth plus five-eighths plus one-half. Give your answer as a mixed number.

To find the sum of these three fractions, we first need to find the least common denominator. In this case, that will be the smallest common multiple of six, eight, and two. We begin by listing a few multiples of six, which includes six, 12, 18, 24, and 30. When we do the same for eight and two, we find the least common multiple is 24.

We note that sometimes we need to create longer lists of multiples for smaller denominators before we can find something in common. Now we need to convert the fractions to equivalent fractions with denominator 24. We multiply the numerator and denominator of one-sixth by four. This shows that one-sixth is equivalent to four twenty-fourths. Next, we multiply the numerator and denominator of five-eighths by three. This shows that five-eighths is equivalent to fifteen twenty-fourths. Finally, we multiply the numerator and denominator of one-half by 12. Therefore, we know that one-half is equivalent to twelve twenty-fourths.

Now that we have equivalent fractions where the denominators are the same, we can simply add the numerators, four, 15, and 12, to get 31. Altogether, we have thirty-one twenty-fourths, which can be written as a mixed number. The whole number part, one, is equivalent to twenty-four twenty-fourths, so seven out of the 31 parts remain. So we write our mixed number answer as one and seven twenty-fourths. This is the sum of one-sixth plus five-eighths plus one-half.

We can now summarize some of the key points of this video.

To find the sum of fractions, we must have parts of equal size. A bar model is a helpful tool to picture how we might split parts of different size into parts of equal size. We learned that fractions must have a common denominator in order to be added together. An area model can be a helpful way of working out a new common denominator. Using this model, we create an array from the two denominators. The number of equal parts is our new common denominator.

However, this may not be our least common denominator. To find the least common denominator of fractions, we can list out the multiples of the denominators. Then we select the smallest multiple they have in common. And we use the least common denominator in our equivalent fractions. Once we have equivalent fractions with the same denominator, then we can add the numerators together to find the sum. The sum may be greater than or less than one. When the sum is greater than one, we often write it as a mixed number.