Addition of Fractions

Fractions represent parts of a whole and consist of two components: a numerator and a denominator, separated by a horizontal line. A fraction is written as a/b, where a is the numerator and b is the denominator.

Addition fractions involves finding the sum of two or more fractions. The method of adding fractions can vary depending on the types of fractions involved, such as like fractions, unlike fractions, or mixed fractions.

\(\textbf{Example 1: } \frac{2}{5} + \frac{1}{5} = \frac{3}{5} \)
 

\(\textbf{Example 2: } \frac{1}{4} + \frac{2}{3} = \frac{3}{12} + \frac{8}{12} = \frac{11}{12} \)

\(\textbf{Example 3: } \frac{4}{5} + \frac{1}{5} = \frac{5}{5} = 1 \)
 

\(\textbf{Example 4: } \frac{2}{3} + \frac{3}{2} = \frac{4}{6} + \frac{9}{6} = \frac{13}{6} \)

Fractions can have the same denominators and different numerators. To add fractions with the same denominators, we simply need to add the numerators over the common denominator.

Example: Add the fractions: 3/9 + 4/9 
We can see that the fractions have the same denominator and different numerators. These are called like fractions.
 

We add the fractions as: 3/9 + 4/9 → result 7/9.

\(\textbf{Example 1: } \frac{2}{7} + \frac{3}{7} = \frac{2+3}{7} = \frac{5}{7} \\[6pt] \\ \ \\ \textbf{Example 2: } \frac{4}{9} + \frac{1}{9} = \frac{4+1}{9} = \frac{5}{9} \\[6pt] \\ \ \\ \textbf{Example 3: } \frac{7}{5} + \frac{2}{5} = \frac{7+2}{5} = \frac{9}{5} \\[6pt] \\ \ \\ \textbf{Example 4: } \frac{5}{12} + \frac{3}{12} = \frac{5+3}{12} = \frac{8}{12} = \frac{2}{3} \\[6pt] \\ \ \\ \textbf{Example 5: } \frac{5}{8} + \frac{6}{8} = \frac{5+6}{8} = \frac{11}{8} \\[6pt] \)

To add fractions with different or unlike denominators, there are a few steps that must be followed:

Step 1: Identify the denominators of the given fractions before adding them. 
 

Step 2: Determine the lowest common denominator of the denominators of the fractions.
 

Step 3: Multiply the numerator and denominator by the same number to make the denominators of all fractions equal.
 

Step 4: Add the numerators of all the fractions while maintaining the denominator.
 

\(\textbf{Example 1: } \frac{1}{4} + \frac{1}{6} \\[3pt] \text{Finding the LCM of 4 and 6: } \\[2pt] \text{LCM}(4,6) = 12 \\[3pt] \frac{1}{4} = \frac{3}{12}, \quad \frac{1}{6} = \frac{2}{12} \\[3pt] \frac{3}{12} + \frac{2}{12} = \frac{5}{12} \)

\(\textbf{Example 2: } \frac{2}{3} + \frac{5}{8} \\[3pt] \text{The LCM of 3 and 8 is 24} \\[2pt] \text{Finding the equivalent fractions: } \\[2pt] \frac{2}{3} \times \frac{8}{8} = \frac{16}{24}, \quad \frac{5}{8} \times \frac{3}{3} = \frac{15}{24} \\[2pt] \frac{16}{24} + \frac{15}{24} = \frac{31}{24} \\[2pt] \frac{31}{24} = 1 \frac{7}{24} \\[6pt]\)

\(\textbf{Example 3: } \frac{3}{5} + \frac{4}{7} \\[3pt] \text{LCM of 5 and 7 is 35} \\[2pt] \frac{3}{5} = \frac{21}{35}, \quad \frac{4}{7} = \frac{20}{35} \\[2pt] \frac{21}{35} + \frac{20}{35} = \frac{41}{35} \\[2pt] \frac{41}{35} = 1 \frac{6}{35} \\[6pt] \)

\(\textbf{Example 4: } \frac{2}{9} + \frac{5}{6} \\[3pt] \text{LCM of 9 and 6 is 18} \\[2pt] \frac{2}{9} = \frac{4}{18}, \quad \frac{5}{6} = \frac{15}{18} \\[2pt] \frac{4}{18} + \frac{15}{18} = \frac{19}{18} \\[6pt]\)

\(\textbf{Example 5: } \frac{5}{12} + \frac{3}{4} \\[3pt] \text{LCM of 12 and 4 is 12} \\[2pt] \text{Convert to equivalent fractions: } \\[2pt] \frac{5}{12} = \frac{5}{12}, \quad \frac{3}{4} = \frac{9}{12} \\[2pt] \text{Add the numerators: } \\[2pt] \frac{5}{12} + \frac{9}{12} = \frac{5+9}{12} = \frac{14}{12} \\[2pt] \text{Simplify: } \frac{14}{12} = \frac{7}{6} \\[6pt]\)

Mixed fractions (or mixed numbers) are numbers that combine a whole number and a fraction. To add a mixed number, there are a few extra steps compared to regular addition of fractions.

Step 1: Convert the mixed fraction into improper fractions. We need to multiply the whole number by the denominator and then add the numerator. The result will become our numerator, and the denominator will be the same.

 

Step 2: Determine the least common denominator or the lowest integer that can be divided equally by each denominator.

 

Step 3: Add the fractions’ numerators, while maintaining the least common denominator as the denominator.
 

\(\textbf{Example 1: } 2 \dfrac{3}{4} + 1 \dfrac{1}{2} \\ \ \\ \text{Convert the mixed fractions to improper fractions} \\ \ \\ 2 \dfrac{3}{4} = \frac{(2 \times 4) + 3}{4} = \frac{11}{4} \\ \ \\ 1 \dfrac{1}{2} = \frac{(1 \times 2) + 1}{2} = \frac{3}{2} \\ \ \\ \text{To add } \frac{11}{4} \text{ and } \frac{3}{2}, \text{ we find the LCM of 4 and 2.} \\ \ \\ \text{LCM}(4,2) = 4 \\ \ \\ \text{Find the equivalent fraction of } \frac{11}{4} \text{ and } \frac{3}{2}. \\ \ \\ \frac{3}{2} = \frac{3 \times 2}{2 \times 2} = \frac{6}{4} \\ \ \\ \text{Adding the fractions:} \\ \ \\ \frac{11}{4} + \frac{6}{4} = \frac{17}{4} \)

\(\textbf{Example 2: } 3 \dfrac{2}{5} + 1 \dfrac{3}{10} \\ \ \\ \text{Convert the mixed fractions to improper fractions} \\ \ \\ 3 \dfrac{2}{5} = \frac{(3 \times 5) + 2}{5} = \frac{17}{5} \\ \ \\ 1 \dfrac{3}{10} = \frac{(1 \times 10) + 3}{10} = \frac{13}{10} \\ \ \\ \text{Find the LCM of 5 and 10} \\ \ \\ \text{LCM}(5, 10) = 10 \\ \ \\ \text{Convert } \frac{17}{5} \text{ to denominator 10:} \\ \ \\ \frac{17}{5} = \frac{17 \times 2}{5 \times 2} = \frac{34}{10} \\ \ \\ \text{Adding the fractions:} \\ \ \\ \frac{34}{10} + \frac{13}{10} = \frac{47}{10} \\ \ \\ \text{Converting to mixed number:} \\ \ \\ \frac{47}{10} = 4 \dfrac{7}{10} \)

\(\textbf{Example 3: } 1 \dfrac{1}{3} + 2 \dfrac{3}{6} \\ \ \\ \text{Convert the mixed fractions to improper fractions} \\ \ \\ 1 \dfrac{1}{3} = \frac{(1 \times 3) + 1}{3} = \frac{4}{3} \\ \ \\ 2 \dfrac{3}{6} = \frac{(2 \times 6) + 3}{6} = \frac{15}{6} \\ \ \\ \text{To add } \frac{4}{3} \text{ and } \frac{15}{6}, \text{ we find the LCM of 3 and 6.} \\ \ \\ \text{LCM}(3,6) = 6 \\ \ \\ \text{Convert } \frac{4}{3} \text{ to denominator } 6: \\ \ \\ \frac{4}{3} = \frac{8}{6} \\ \ \\ \text{Adding: } \frac{8}{6} + \frac{15}{6} \\ \ \\ \frac{8}{6} + \frac{15}{6} = \frac{8 + 15}{6} = \frac{23}{6} \\ \ \\ \text{Converting } \frac{23}{6} \text{ to mixed number: } \frac{23}{6} = 3 \dfrac{5}{6} \)

\(\textbf{Example 4: } 4 \dfrac{1}{8} + 3 \dfrac{3}{4} \\ \ \\ \text{Convert the mixed fractions to improper fractions} \\ \ \\ 4 \dfrac{1}{8} = \frac{(4 \times 8) + 1}{8} = \frac{33}{8} \\ \ \\ 3 \dfrac{3}{4} = \frac{(3 \times 4) + 3}{4} = \frac{15}{4} \\ \ \\ \text{Finding the LCM of 8 and 4:} \\ \ \\ \text{LCM}(8,4) = 8 \\ \ \\ {\text {Convert}} {15 \over 4} {\text {to an equivalent fraction}} \\ \ \\ \frac{15}{4} = \frac{30}{8} \\ \ \\ \text{Adding: } \frac{33}{8} + \frac{30}{8} \\ \ \\ \frac{33}{8} + \frac{30}{8} = \frac{63}{8} \\ \ \\ \text{Converting } \frac{63}{8} \text{ to a mixed number: } \frac{63}{8} = 7 \dfrac{7}{8} \)

When converting a whole number into a fraction, we give it a denominator of 1. This helps in adding it to other fractions. 

Step 1: Convert the whole number into an improper fraction. We multiply the whole number by the denominator of the given fraction. The result becomes the new numerator, and the denominator remains the same.
 

Step 2: Add the improper fractions and the given fraction.
 

Step 3: Simplify the resulting fraction if it is possible.

\(\textbf{Example 1: } 4 + \frac{3}{5} \\ \ \\ \text{To add a whole number with a fraction, first convert the whole number to a fraction.} \\ \ \\ 4 = \frac{4}{1} \\ \ \\ \text{Finding the equivalent fraction of } \frac{4}{1} \text{ by multiplying it by 5,} \\ \ \\ (4 \times 5) / (1 \times 5) = \frac{20}{5} \\ \ \\ \text{Adding the fractions:} \\ \ \\ \frac{20}{5} + \frac{3}{5} = \frac{23}{5} \)

\(\textbf{Example 2: } 6 + \frac{2}{3} \\ \\ \text{Converting 6 to a fraction,} \\ 6 = \frac{6}{1} \\ \\ \text{Find the equivalent fraction of } \frac{6}{1}: \\ \\ \frac{6}{1} = \frac{(6 \times 3)}{(1 \times 3)} = \frac{18}{3} \\ \\ \text{Add: } \frac{18}{3} \text{ and } \frac{2}{3} \\ \\ \frac{18}{3} + \frac{2}{3} = \frac{18 + 2}{3} = \frac{20}{3} \)

\(\textbf{Example 3: } \frac{5}{8} + 9 \\ \\ \text{Converting the whole number to an equivalent fraction:} \\ \\ 9 = \frac{(9 \times 8)}{(1 \times 8)} = \frac{72}{8} \\ \\ \text{Add: } \frac{5}{8} \text{ and } \frac{72}{8} \\ \\ \frac{5}{8} + \frac{72}{8} = \frac{5 + 72}{8} = \frac{77}{8} \\ \\ \text{Converting } \frac{77}{8} \text{ into a mixed number} \\ \\ \frac{77}{8} = 9 \dfrac{5}{8} \)

\(\textbf{Example 4: } 3 + \frac{7}{4} \\ \\ \text{Convert the whole number:} \\ 3 = \frac{(3 \times 4)}{(1 \times 4)} = \frac{12}{4} \\ \\ \text{Adding: } \frac{12}{4} \text{ and } \frac{7}{4} \\ \\ \frac{12}{4} + \frac{7}{4} = \frac{19}{4} \\ \\ \text{Converting to a mixed number: } \frac{19}{4} = 4 \dfrac{3}{4} \)

\(\textbf{Example 5: } 7 + \frac{1}{4} \\ \\ \text{Converting the whole number to an equivalent fraction:} \\ 7 = \frac{(7 \times 4)}{(1 \times 4)} = \frac{28}{4} \\ \\ \text{Add: } \frac{28}{4} \text{ and } \frac{1}{4} \\ \\ \frac{28}{4} + \frac{1}{4} = \frac{28 + 1}{4} \\ = \frac{29}{4} \)

When adding fractions with variables, we follow the same rules as adding numerical fractions. In this section, we will learn how to add fractions with variables. 
 

• When adding fractions with variables, first check the denominators. 
   
• If the denominators and variables are the same, we add the numerical parts and multiply the variables. 
   
• If the denominators are different, we first find the LCM of the denominators. Then convert each fraction to an equivalent fraction using the LCM, add the numerators, and keep the denominator the same.
   

Always remember that fractions with different variables cannot be added.  For example, \({x\over 2 }+ {y\over3 }\) cannot be added.

\(\textbf{Example 1: } \frac{y}{5} + \frac{2y}{5} \)

As the denominators are the same, taking out the common variable

\( \frac{y}{5} + \frac{2y}{5} = \left(\frac{1}{5} + \frac{2}{5}\right)y = \frac{3y}{5} \)

\(\textbf{Example 2: } \frac{y}{2} + \frac{y}{3} \)

As the denominators are unlike, we find the LCM of the denominators:

LCM(2, 3) = 6

\( \\ \frac{y}{2} = \frac{3y}{6} \\ \frac{y}{3} = \frac{2y}{6} \\ \ \\ \text{Add: } \frac{3y}{6} + \frac{2y}{6} = \frac{5y}{6} \)

\(\textbf{Example 3: } \frac{x}{2} + \frac{y}{3} \)

\({x\over2} { \text{ and }}{y\over3} \) cannot be added as the variables are different.

They are unlike terms.

\( \text{So, } \frac{x}{2} + \frac{y}{3} \text{ cannot be simplified.} \)
 

\(\textbf{Example 4: } \frac{4x}{7} + \frac{3x}{7} \\ \ \\ \text{Factor out } x: \\ \ \\ \frac{4x}{7} + \frac{3x}{7} = \frac{(4 + 3)x}{7} = \frac{7x}{7} = x \)

\(\textbf{Example 5: } \frac{5y}{4} + \frac{y}{6} \\ \ \\ \text{Finding the LCM of 4 and 6:} \\ \ \\ \text{LCM}(4, 6) = 12 \\ \ \\ \\ \text{Converting to like fractions:} \\ \ \\ \frac{5y}{4} = \frac{5y \times 3}{4 \times 3} = \frac{15y}{12} \\ \ \\ \frac{y}{6} = \frac{y \times 2}{6 \times 2} = \frac{2y}{12} \\ \ \\ \text{Adding: } \frac{15y}{12} + \frac{2y}{12} = \frac{17y}{12} \)

Here are some tips and tricks to make fraction addition easier and more intuitive. 

• Start with fractions that have the same denominator. When the denominators are identical, you can simply add the numerators while keeping the denominator unchanged. 
   
• For fractions with different denominators, always find the least common denominator(LCD) first. This ensures that both fractions are expressed with the same denominator. 
   
• Convert mixed numbers to improper fractions before adding them. This simplifies the addition process and avoids mistakes. 
   
• Always simplify the resulting fraction after addition. Dividing the numerator and denominator by their (GCD) ensures that the fraction is in its simplest form. 
   
• Practice with real life scenarios to strengthen your understanding. Use the addition of fractions in cooking, in time management, etc. 
   

• Parents can encourage students to apply fractions to addition during activities like sharing food, measuring items, or planning. 
   

• Teachers can use visual tools such as fraction strips, grids, or charts to demonstrate how to add fractions. 
   

• Teachers can break down the process like finding the LCD, converting, adding, and simplifying.

The addition of fractions is used widely in various fields. Here are a few real-world applications that use the addition of fractions:

• Cooking: Recipes often require fractional measurements of ingredients. Adding the fractions would ensure accurate measurements for a perfect dish.
  
   
• Construction: Builders often use fractional measurements of inches and feet. Adding fractions would ensure precise cutting and fitting of materials.
  
   
• Sports: Athletes track distances in fractions. Adding the fractions of their distances would help in monitoring training progress and performance. 
  
   
• Time management: Time spend on any activities can be calculated by adding them. The time will be in fractional form, like 1/3 or 3/4.
  
   
• Resource sharing: For sharing resources like food, land or money we have to add the quantities in fractional form.