Like denominators

Two or more fractions are said to have like denominators if they have same denominators.

Such fractions are called like fractions.

For example:

\(\dfrac {3}{5},\;\dfrac {2}{5}\)

Here, like denominators = 5

• While performing addition of fractions, two situations may arise:

1. Fractions having like denominators.
2. Fractions having unlike denominators.

Here, we are performing addition of fractions having like denominators.

For example: \(\dfrac {1}{2}+\dfrac {4}{2}\)

Both the fractions have same denominators, i.e. 2.

Step-1: Add the numerators.

1 + 4 = 5

Step-2: Sum of the numerators becomes the new numerator.

Numerator = 5

Step-3: Denominator remains the same.

Denominator = 2

Step-4: Write the resulting fraction.

Fraction = \(\dfrac {5}{2}\)

Step-5: Simplify the result.

Fraction = \(\dfrac {5}{2}\)

5 and 2 do not have any common factor other than 1.

\(\therefore\) \(\dfrac {5}{2}\) is in its simplest form.

Thus, it is our final answer.

• In order to perform addition of fractions having different denominators, we first need to convert them into fractions having like denominators.
• Follow the steps given below to convert the unlike fractions into like fractions.

Example:- \(\dfrac{2}{3} + \dfrac{1}{6}\)

Step-1: Find the least common multiple (L.C.M) of both the denominators.

 L.C.M of \(3\) and \(6\)

Multiples of \(3 = 3,\; 6,\; ...\)

Multiples of \(6 = 6, \; 12, \;...\)

 L.C.M of \(3\) and \(6 = 6\)

Step-2: The L.C.M becomes the lowest common denominator (L.C.D).

 L.C.D = \(6\)

Step-3: Convert each fraction into an equivalent fraction with L.C.D as the denominator.

Equivalent fraction of \(\dfrac{2}{3} = \dfrac{2 × 2}{3 ×2} = \dfrac{4}{6}\) 

\(\dfrac{1}{6}\) already has \(6\) as its denominator.

\(\therefore\) It does not need to be converted.

Step-4: Add the fractions having same denominators.

\(\dfrac{4}{6} + \dfrac{1}{6}\)

\(4 + 1 = 5 \leftarrow\; \text{Numerator}\)

\(6 \leftarrow \text{Denominator}\)

Fraction \( = \dfrac{5}{6}\)

Step-5: Simplify the result.

• \(5\) and \(6\) do not have any common factor other than \(1\).
• Therefore, \(\dfrac{5}{6}\) is in its simplest form. 
• Thus, \(\dfrac{5}{6}\) is our final answer.

Note:- If the given fraction is not in its simplest form, then first convert it into its simplest form. This will make the further calculations easier.

• Addition of like fractions means the addition of fractions having the same number of total parts or having the same denominators.
• Let us take an example:

Tim and Travis are celebrating Christmas at Mr. Larry's restaurant. They order a pineapple cake. They ask the waiter to cut the cake into 8 equal slices, which means each slice represents \(\dfrac {1}{8}\) part, as shown in figure (1.1).

Waiter serves the cake to them.

Tim gets 5 slices, i.e. \(\dfrac {5}{8}\) part as shown in figure (1.2).

Travis gets 3 slices, i.e. \(\dfrac {3}{8}\) part as shown in figure (1.3).

As Tim gets two slices more than Travis, he gives one slice to Travis.

Now, with the help of the diagram shown, we will find out the fraction of the cake that Travis has.

Thus, Travis has a total of \(\dfrac {4}{8}\) part of the cake.

• Addition of fractions having unlike denominators means the addition of fractions that do not have the same number of total parts or the same denominators.
• Let us consider an example:
• Jessica draws two squares of the same size.
• She divides the first square into \(3\) rows, as shown in figure (A) and the second square into \(2\) columns, as shown in figure (B).

• She shades the first row of figure (A) = \(\dfrac{1}{3}\) part, as shown in figure (C) and one column of figure (B) = \(\dfrac{1}{2}\) part, as shown in figure (D).

Now, she wants to add the shaded parts of figures (C) and (D).

But we can see, both the squares do not have the same number of parts (they represent unlike fractions).

So, first, we have to subdivide their rows and columns in such a way that the number of parts becomes equal.

On adding the subdivided shaded parts of both the squares, we get:

Thus, figure (G) shows the addition of shaded part of figures (E) and (F) and the fraction represented by the shaded part equals \(\dfrac{5}{6}\).

• Mixed number = A whole number + A fraction 
• To understand the addition of mixed numbers easily, let us consider the following steps:

For example: \(6\dfrac{1}{5} + 4\dfrac{2}{5}\)

Step-1: First add the fraction part.

\(\dfrac{1}{5} + \dfrac{2}{5}\;=\; \dfrac{1+2}{5}\;=\; \dfrac{3}{5}\)

Step-2: Simplify the resulting fraction, \(\dfrac{3}{5}\).

\( 3\) and \(5\) do not have any common factor other than \(1\).

\(\therefore\; \dfrac{3}{5}\) is in its simplest form.

Step-3: Add the whole numbers.

\(6 + 4 = 10\)

Step-4: Put the sum of whole numbers and fractions together in the form of a mixed number  \(=10\dfrac{3}{5}\).

Thus, \(10\dfrac{3}{5}\) is our final answer.

Consider two cases:-

Case (i): If we get an improper fraction by adding the fraction part.

For example: \(4 \dfrac{7}{8} + 3\dfrac{5}{8}\)

• Add \(\dfrac{7}{8}\) and \(\dfrac{5}{8}\).

\(\dfrac{7}{8} + \dfrac{5}{8} = \dfrac{7+5}{8} = \dfrac{12}{8}\;\leftarrow \text{Improper fraction}\)

• Simplify \(\dfrac{12}{8}\),

\(\dfrac{12\div4}{8\div4} = \dfrac{3}{2}\)

[G.C.F of \(12\) and \(8\) is \(4\)]

\(\dfrac{3}{2}\) is an improper fraction as \(3>2\).

• Change the improper fraction into a mixed number.

\(\dfrac{3}{2} = 1\dfrac{1}{2}\)

• Add the whole numbers.

\(4 + 3 + 1 = 8 \)

\(\Bigg[\;1\) is the whole number in \(1\dfrac{1}{2}\;\Bigg]\)

• Put the sum of whole numbers and fractions together in the form of a mixed number \(=8\dfrac{1}{2}\).

\(\Bigg[\;\dfrac{1}{2}\) is the fraction part of \(1\dfrac{1}{2}\;\Bigg]\)

 Thus, \(8\dfrac{1}{2}\) is our answer.

Case (ii): If we get a whole number by adding the fraction part.

For example: \(2\dfrac{3}{5} + 1\dfrac{2}{5}\)

• Add \(\dfrac{3}{5}\) and \(\dfrac{2}{5} \).

\(\dfrac{3}{5} + \dfrac{2}{5} = \dfrac{3+ 2}{5} = \dfrac{5}{5} = 1 \; \leftarrow \text{Whole number}\)

• Add this whole number to the sum of the wholes.

\(2 + 1 + 1 = 4\)

Thus, \(4\) is our answer.

• Consider a fraction as a segment on a number line.
• Addition of fractions on a number line is represented by combining the segments together.
• We have to make sure that each fraction has the same denominator.
• For example: Represent \(\dfrac{2}{3}+\dfrac{4}{3}\) on a number line.

Step 1: Represent the first fraction \(\dfrac{2}{3}\) on the number line.

Step 2: Represent the second fraction \(\dfrac{4}{3}\) on the number line.

Step 3: Combine the segments of both the fractions.

Step 4:  Mark the final point as the answer of the addition.

Thus, the result of

 \(\dfrac{2}{3}+\dfrac{4}{3}=2\)  (as we have reached at 2 on the number line)