In this post we will learn about adding fractions with different denominators.
For the addition, you should have basic understanding of the fraction and LCM concept.
Adding fractions with different denominator
In order to add the fractions, its important to have a common denominator.
In this method we will try to convert the fractions with the help of LCM so that they have same denominator.
Follow the below steps for the whole process;
(a) Find LCM of the denominator.
(b) Multiply the fractions so that the denominator is same as LCM value.
(c) Now all the fractions have same denominator.
Simply add the numerator and retain the denominator.
Why its important to have same denominator in fraction addition?
It is very difficult to add fractions manually with different denominator.
Let us understand this with help of graphical representation.
Consider the fraction 1/4 and 1/5.
The fraction 1/4 mean that the object is divided into 4 equal part and 1 part of it is shaded.
While fraction 1/5 mean that the object is divided into 5 equal part and 1 part of it is shaded.
Observe the above image.
You will note that adding the fractions is quite difficult as the object is divided into different parts.
In order to add fractions, it is important to have object which is divided into equal parts. (Hence, same denominator is necessary)
Now consider the fraction 1/4 and 2/4.
Fraction 1/4 mean that the object is divided into 4 part and 1 part is shaded.
Fraction 2/4 mean that object is divided into 4 part and 2 part is shaded.
Since the object is divided into equal parts the addition gets easy and straight forward.
Observe the above image.
Note that the addition is done simply by adding numerator and retaining the denominator.
Examples of fraction addition with different denominator
Example 01
Add \mathtt{\frac{1}{3} \ \&\ \frac{1}{5}}
Solution
For fraction addition, follow the below steps:
(a) Find LCM of the denominators
LCM (3, 5) = 15
(b) Multiply the fraction to make denominator equal to 15
For Fraction 1/3
Multiply numerator and denominator by 5
\mathtt{\Longrightarrow \frac{1\ \times \ 5}{3\ \times \ 5} \ =\ \frac{5}{15}}
For Fraction 1/5
Multiply numerator and denominator by 3
\mathtt{\Longrightarrow \frac{1\ \times \ 3}{5\ \times \ 3} \ =\ \frac{3}{15}}
(c) Now we have fractions we same denominator.
Simply add the numerator and retain the same denominator.
\mathtt{\Longrightarrow \ \frac{5}{15} \ +\ \frac{3}{15} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{5\ +\ 3\ }{15}}\\\ \\ \mathtt{\Longrightarrow \ \frac{8}{15}}
Hence, 8/15 is the solution.
Example 02
Add the fractions \mathtt{\frac{2}{6} \ \&\ \frac{1}{8} \ }
Solution
Follow the below steps:
(a) Find LCM of denominators
LCM (6, 8) = 24
(b) Multiply the fraction to make denominator 24
Fraction 2/6
Multiply numerator and denominator by 4
\mathtt{\Longrightarrow \frac{2\ \times \ 4}{6\ \times \ 4} \ =\ \frac{8}{24}}
Fraction 1/8
Multiply numerator and denominator by 3
\mathtt{\Longrightarrow \frac{1\ \times \ 3}{8\ \times \ 3} \ =\ \frac{3}{24}}
(c) Now we have fractions with same denominator.
Just add the numerator and leave the denominator as it is.
\mathtt{\Longrightarrow \ \frac{8}{24} \ +\ \frac{3}{24} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{8\ +\ 3\ }{24}}\\\ \\ \mathtt{\Longrightarrow \ \frac{11}{24}}
Hence, 11/24 is the solution.
Example 03
Add the fractions \mathtt{\frac{6}{13} \ \&\ \frac{7}{15}}
Solution
Follow the below steps:
(a) Find LCM of the denominators
LCM (13, 15) = 195
(b) Multiply the fractions to make denominator 195
Fraction 6/13
Multiply numerator and denominator by 15
\mathtt{\Longrightarrow \frac{6\ \times \ 15}{13\ \times \ 15} \ =\ \frac{90}{195}}
Fraction 7/15
Multiply numerator and denominator by 13
\mathtt{\Longrightarrow \frac{7\ \times \ 13}{15\ \times \ 13} \ =\ \frac{91}{195}}
(c) Now the fractions have same denominator.
Add the numerators and retain the denominator.
\mathtt{\Longrightarrow \ \frac{90}{195} \ +\ \frac{91}{195} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{90\ +\ 91\ }{195}}\\\ \\ \mathtt{\Longrightarrow \ \frac{181}{195}}
Hence, 181/195 is the solution.
Example 04
Add the fractions \mathtt{\frac{2}{9} \ \&\ \frac{13}{45} \ }
Solution
Do the following steps:
(a) Take LCM of denominator
LCM (9, 45) = 45
(b) Multiply the fractions to make denominator 45.
Fraction 2/9
Multiply numerator and denominator by 5
\mathtt{\Longrightarrow \frac{2\ \times \ 5}{9\ \times \ 5} \ =\ \frac{10}{45}}
Fraction 13/45
The denominator is already 45.
No need to do anything.
(c) Now we have fractions with same denominator.
Add the numerator and retain the denominator.
\mathtt{\Longrightarrow \ \frac{10}{45} \ +\ \frac{13}{45} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{10\ +\ 13\ }{45}}\\\ \\ \mathtt{\Longrightarrow \ \frac{23}{45}}
Hence, 23/45 is the solution.
Example 05
Add the fractions; \mathtt{\frac{1}{5} \ ,\ \frac{1}{10} \ \ \&\ \ \frac{1}{15}}
Solution
Do the following steps:
(a) Find LCM of denominators.
LCM (5, 10, 15) = 30
(b) Multiply fractions to make denominator 30
Fraction 1/5
Multiply numerator and denominator by 6
\mathtt{\Longrightarrow \frac{1\ \times \ 6}{5\ \times \ 6} \ =\ \frac{6}{30}}
Fraction 1/10
Multiply numerator and denominator by 3
\mathtt{\Longrightarrow \frac{1\ \times \ 3}{10\ \times \ 3} \ =\ \frac{3}{30}}
Fraction 1/15
Multiply numerator and denominator by 2
\mathtt{\Longrightarrow \frac{1\ \times \ 2}{15\ \times \ 2} \ =\ \frac{2}{30}}
(c) Now we have fractions with same denominator.
Add the fractions and retain the denominator.
\mathtt{\Longrightarrow \ \frac{6}{30} \ +\ \frac{3}{30} \ +\ \frac{2}{30} \ \ \ }\\\ \\ \mathtt{\Longrightarrow \ \frac{6\ +\ 3\ +\ 2\ }{30}}\\\ \\ \mathtt{\Longrightarrow \ \frac{11}{30}}
Hence, 11/30 is the solution of the addition.