When the denominators of two or more fractions are same, they are the common denominators. For example, in the fractions ${\dfrac{2}{9}}$, ${\dfrac{5}{9}}$, and ${\dfrac{8}{9}}$, 9 is the common denominator.
Common Denominator
Why Is It Important
We can only add, subtract, or compare two or more fractions when the denominators are same. Fractions with common denominators are called like fractions. In contrast, fractions with different denominators are called unlike fractions.
For example, we can easily add the two fractions, ${\dfrac{4}{11}}$ and ${\dfrac{3}{11}}$, as their denominators are the same.
Adding Subtracting Comparing Fractions with Common Denominator
${\dfrac{14}{19}+\dfrac{3}{19}}$
Solution:
Given, ${\dfrac{14}{19}+\dfrac{3}{19}}$ The denominator is common here, that is, 19 Now, adding the numerators, we get ${\dfrac{14}{19}+\dfrac{3}{19}=\dfrac{14+3}{19}=\dfrac{17}{19}}$ Thus, the sum is ${\dfrac{17}{19}}$
Subtract ${\dfrac{8}{21}}$ from ${\dfrac{11}{21}}$
Solution:
Here, ${\dfrac{11}{21}-\dfrac{8}{21}}$ The denominator is common here, that is, 21 Now, subtracting the numerators, we get ${\dfrac{11}{21}-\dfrac{8}{21}=\dfrac{11-8}{21}=\dfrac{3}{21}}$ ${=\dfrac{1}{7} }$ Thus, the difference is ${=\dfrac{1}{7} }$
Compare and order ${\dfrac{17}{26}}$, ${\dfrac{15}{26}}$, and ${\dfrac{21}{26}}$ in ascending order.
Solution:
Here, the common denominator is 26 Arranging the numerators in ascending order, 15 < 17 < 21 Thus, the given fractions are ordered as ${\dfrac{15}{26} <\dfrac{17}{26} <\dfrac{21}{26}}$
How to Find the Common Denominator for Unlike Fractions
We can add or subtract the unlike fractions more easily by converting them to like fractions.
LCM Method
This method involves finding the least common denominator (LCD) for the two fractions. To do this, we list the multiples of each denominator and identify the lowest common multiple (LCM) among them.
Let us add ${\dfrac{4}{5}}$ and ${\dfrac{7}{10}}$
Now, finding the LCD (Least Common Denominator) and Adding
Subtract by LCM Method: ${\dfrac{9}{13}-\dfrac{7}{26}}$
Solution:
Given, ${\dfrac{9}{13}-\dfrac{7}{26}}$ The denominators are 13 and 26 The LCM of 13 and 26 is 26 Now, the fractions with the common denominator are: ${\dfrac{9}{13}}$ = ${\dfrac{9\times 2}{13\times 2}}$ = ${\dfrac{18}{26}}$ ${\dfrac{7}{26}}$ = ${\dfrac{7\times 1}{26\times 1}}$ = ${\dfrac{7}{26}}$ Here, ${\dfrac{9}{13}-\dfrac{7}{26}}$ = ${\dfrac{18}{26}-\dfrac{7}{26}}$ = ${\dfrac{18-7}{10}}$ = ${\dfrac{11}{10}}$ = ${1\dfrac{1}{10}}$ Thus, the difference is ${1\dfrac{1}{10}}$
Using LCM Method, Find the Difference: ${\dfrac{8}{15}-\dfrac{3}{20}}$
Solution:
Given, ${\dfrac{8}{15}-\dfrac{3}{20}}$ The denominators are 15 and 20 The LCM of 15 and 20 is 60 Now, the fractions with the common denominator are: ${\dfrac{8}{15}}$ = ${\dfrac{8\times 4}{15\times 4}}$ = ${\dfrac{32}{60}}$ ${\dfrac{3}{20}}$ = ${\dfrac{3\times 3}{20\times 3}}$ = ${\dfrac{9}{60}}$ Here, ${\dfrac{8}{15}-\dfrac{3}{20}}$ = ${\dfrac{32}{60}-\dfrac{9}{60}}$ = ${\dfrac{32-9}{60}}$ = ${\dfrac{23}{60}}$ Thus, the difference is ${1\dfrac{23}{60}}$
When the HCF of Denominators is 1
If the HCF of the denominators is 1, then the LCD of the given fractions is obtained by multiplying the denominators.
For example, in the fractions ${\dfrac{3}{4}}$ and ${\dfrac{2}{5}}$, the HCF of 4 and 5 is 1.
This method involves multiplying the numerator and denominator of each fraction by the denominator of the other fraction.
For example, in the fractions ${\dfrac{2}{3}}$ and ${\dfrac{7}{9}}$, we multiply 2 and 3 of the fraction ${\dfrac{2}{3}}$ by 9, and 7 and 9 of the fraction ${\dfrac{7}{9}}$ by 3
Let us subtract ${\dfrac{2}{3}}$ from ${\dfrac{7}{9}}$
