Formula For Adding Fractions
Before learning the formula for adding fractions, let us recall what are fractions. In mathematics, fractions can be defined as the parts of a whole. A fraction can be a portion or section of any quantity out of a whole, where, the whole can be any number, a specific value, or a thing. Examples are 1/2, 3/7,4/5, 6/6, etc. There are different types of fractions depending upon their form. These are
 Proper Fraction
 Improper Fraction
 Unit Fraction
 Mixed Fraction
 Equivalent Fractions
 Like Fractions, and
 Unlike Fractions.
What Is Formula for Adding Fractions?
The formulas for adding fractions vary in each of the following situations: Adding fractions with whole numbers, adding fractions with like denominators, adding fractions with different denominators, and adding fractions with variables. These formulas are:

Adding fractions with whole numbers
\(\left(a+\dfrac{b}{c}\right)\) = \(\dfrac{({a×c})+b}{c}\) 
Adding fractions with like denominators
\(\left(\dfrac{a}{b} + \dfrac{d}{b}\right)\)=\(\dfrac{(a+d)}{b}\) 
Adding fractions with different denominators
\(\left(\dfrac{a}{b} + \dfrac{c}{d}\right)\)=\(\left(\dfrac{a×d+b×c}{b×d}\right)\) 
Adding fractions with variables
\(\dfrac {a}{b} x + \dfrac{ c}{b} x = \left( \dfrac{a}{b} + \dfrac{c}{b} \right) x = \left(\dfrac{a+c}{b}\right) x\)
where
 a, b, c, and, d are constants
 x is variable.
Let us see the applications of formula for adding fractions in the solved examples below.
Solved Examples Using Formula for Adding Fractions

Example 1: Using the formula for adding fractions, add 3/2 and 5.
Solution:
To find: addition of 3/2 and 5.
We have
\(\left(a+\dfrac{b}{c}\right)\) = \(\dfrac{({a×c})+b}{c}\)
Adding Fractions = \(\dfrac{({2×5})+3}{2}\)
= (10+3)/2
= 13/2
Answer: The addition of 3/2 and 5 is 13/2

Example 2: If the difference between two fractions is 2/3 and one fraction is 6/5, find the original fraction.
Solution:
To find: The original fraction.
Using formula for adding fractions,
Adding Fractions = \(\dfrac{a×d+b×c}{b×d}\)
Adding Fractions = \(\dfrac{2×5+6×3}{3×5}\)
= (10+18)/15
= 28/15
Answer: The original fraction is 28/15