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
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
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
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
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
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
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
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
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
In our final example, we will see
that when adding more than two fractions together, we will still need a common
Calculate one-sixth plus
five-eighths plus one-half. Give your answer as a mixed
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.