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
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.
A \(\dfrac {13}{5}\)
B \(\dfrac {5}{13}\)
C \(\dfrac {56}{15}\)
D \(\dfrac {1}{5}\)
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.
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.
A \(\dfrac{2}{3}\)
B \(\dfrac{49}{3}\)
C \(57\)
D \(\dfrac{80}{9}\)
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.
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 \(\dfrac{5}{6}\).
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: \(4 \dfrac{7}{8} + 3\dfrac{5}{8}\)
\(\dfrac{7}{8} + \dfrac{5}{8} = \dfrac{7+5}{8} = \dfrac{12}{8}\;\leftarrow \text{Improper fraction}\)
\(\dfrac{12\div4}{8\div4} = \dfrac{3}{2}\)
[G.C.F of \(12\) and \(8\) is \(4\)]
\(\dfrac{3}{2}\) is an improper fraction as \(3>2\).
\(\dfrac{3}{2} = 1\dfrac{1}{2}\)
\(4 + 3 + 1 = 8 \)
\(\Bigg[\;1\) is the whole number in \(1\dfrac{1}{2}\;\Bigg]\)
\(\Bigg[\;\dfrac{1}{2}\) 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: \(2\dfrac{3}{5} + 1\dfrac{2}{5}\)
\(\dfrac{3}{5} + \dfrac{2}{5} = \dfrac{3+ 2}{5} = \dfrac{5}{5} = 1 \; \leftarrow \text{Whole number}\)
\(2 + 1 + 1 = 4\)
Thus, \(4\) is our answer.
A \(5\dfrac{7}{8}\)
B \(6\dfrac{2}{8}\)
C \(7\dfrac{1}{4}\)
D \(1\dfrac{3}{4}\)
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)