6th Grade - Addition and Subtraction of Fractions

Introduction

• "Fraction" represents a relation of a part(s) to the whole, where the whole is divided into equal parts. 
• Fraction =$\frac{\mathrm{numerator}}{\mathrm{denominator}}=\frac{part\left(s\right) of a whole}{whole}$
• Any whole number can be written as a fraction, such as 5 can be written as  $\frac{5}{1}$. This is helpful while adding or subtracting fractions.

• Proper Fraction -  The numerator is smaller than the denominator, eg. $\frac{3}{5}$.
• Improper Fraction - The numerator is larger than the denominator, e.g. $\frac{5}{3}$.
• Mixed Fraction- Consists of a whole number and a proper fraction, e.g. $2\frac{3}{5}$.
• Equivalent Fractions - Fractions that are equal when reduced to their lowest terms. e.g. $\frac{3}{5} =\frac{9}{15}$. Fractions can also be changed into equivalent fractions by multiplying the numerator and the denominator by the same desired number.

Equivalent Fraction

• Fractions that hold the same value are called equivalent fractions. For example: $\frac{3}{5} = \frac{3}{5} \mathrm{or} \frac{3}{5} = \frac{6}{10}$
• Unequal fractions can be changed into equivalent fractions by:

1. Multiplying the numerator and the denominator by the same number.

OR

2. Reducing the fraction by dividing the numerator and denominator evenly by the same number.

• To check if the fractions are equivalent:

1. Cross-multiply the fractions.

Example:  Is $\frac{3}{4}=\frac{9}{12}$?

         Solution: $12 \times 3 = 9 \times 4$

                        $36 = 36$

OR

2. Reduce fractions to their lowest terms.

Example:  Is $\frac{4}{5} = \frac{9}{12}$?

          Solution: $\frac{4}{5} \mathrm{cannot} \mathrm{be} \mathrm{reduced}$

                $\frac{9}{12} (can be reduded by dividing by 3) \frac{9 ÷3}{12 ÷ 3} = \frac{3}{4}$

                 $\frac{4}{5} \ne \frac{9}{12}$

 

Addition and Subtraction of Fractions with Like Denominators

•  To add/subtract fractions with the same (like) denominators, add/subtract the numerator and keep the common denominator unchanged.

e.g. $$\begin{gathered} \frac{2}{3}+ \frac{2}{3} \\ =\frac{4}{3} \end{gathered}$$

Addition and Subtraction of Fractions with Unlike Denominators

• To add or subtract fractions with unlike (different) denominators:

1. Find the Least Common Denominator (LCD).  

To find LCD:

• • • List the multiples of the denominators until the least common multiple(LCM) is found. The LCM is the LCD.

OR

• • • Multiply the denominators of both fractions; the product will be the LCD.

2. Create equivalent fractions by multiplying the denominator and the numerator of a fraction with the desired number that can yield the Least common denominators.  

3. Add/Subtract the numerators keeping the same common denominator.

For e.g.  $\frac{1}{2} + \frac{2}{3}$

Multiples of 2: 2, 4, 6 (Stop at this number as it is the least (smallest) common multiple.)

Multiples of 3: 3, 6  (Stop at this number because it is the least (smallest) common multiple.)

OR multiply the denominators (3 x 2) = 6.

Since the least common denominator is 6,  the denominators should be multiplied by a number that can yield 6. The numerator should also be multiplied with the same number as the denominator to create an equivalent fraction.

$$\begin{gathered} =\left(\frac{1\mathbf{\times }\mathbf{3}}{2\mathbf{\times }\mathbf{3}}\right) + \left(\frac{2\mathbf{\times }\mathbf{2}}{3\mathbf{\times }\mathbf{2}}\right) \\ =\frac{3}{6} + \frac{4}{6} \\ = \frac{7}{6} \end{gathered}$$

Mixed Fraction/Number

• A mixed Fraction consists of a whole number and a proper fraction, e.g. $2\frac{3}{5}$.
• To change a mixed number to an improper fraction:

1. Keep the denominator the same.

2. For the numerator, multiply the whole number by the denominator and add the numerator.

• Example: Change $2\frac{3}{5} \mathrm{to} \mathrm{an} \mathrm{improper} \mathrm{fraction}.$

$= 2\frac{+3}{\times 5} =\frac{13}{5}$

• To change an improper fraction to a mixed number 

1. 1. Divide the numerator by the denominator.
   2. The quotient will be the whole number of the mixed number.
   3. Keep the denominator the same.
   4. The numerator will be the remainder of the division.

• To add or subtract mixed numbers:

1. 1. Change the mixed number to an improper fraction. 
   2. Add/Subtract following the rules of addition/subtraction of fractions.
   3. Simplify to the lowest terms if possible.

Solved Examples

Example1: Add: $\frac{3}{5} + \frac{1}{5}$

Solution: $\frac{3+1}{5} = \frac{4}{5}$

 

Example2:  Solve:  $\frac{5}{4} -\frac{ 3}{4}$

Solution: $\frac{5-1}{4} = \frac{4}{4}(\mathrm{reducing} \mathrm{to} \mathrm{the} \mathrm{lowest} \mathrm{terms}) = 1$

 

Example3: Add: $\frac{3}{7} + \frac{3}{5}$

Solution: Find the least common denominator by listing the multiples of the denominators:

Multiples of 7: 7,14, 21, 28, 35, 42

Multiples of 5: 5, 10, 15, 20, 25, 30, 35

The LCD = 35

or by multiplying the denominators: 7 x 5 = 35, LCD = 35

$= \frac{3\times 5}{7\times 5}+\frac{3\times 7}{5\times 7}$

$= \frac{15}{35} + \frac{21}{35}$

$= \frac{36}{35}$

 

Example4: Subtract: $\frac{8}{3} - \frac{1}{4}$

Solution: Find the least common denominator by listing the multiples of the denominators:

Multiples of 3: 3, 6, 9, 12, 15

Multiples of 4: 4, 8, 12, 16

LCD = 12

or by multiplying the denominators 4 x 3 = 12, LCD = 12

$= \frac{8\times 4}{3\times 4} + \frac{1\times 3}{4\times 3}$

$= \frac{32}{12} + \frac{3}{12}$

$= \frac{35}{12}$

 

Example 5: $1\frac{1}{2}+2\frac{1}{3}$

Solution:

Change mixed numbers to improper fractions.

$=1 \frac{+1}{\times 2} + 2 \frac{+1}{\times 3}$

$=\frac{3}{2} + \frac{7}{3}$

Find the LCD by listing the multiples of the denominators:

Multiples of 2: 2, 4, 6, 8, 10, 12, 14

Multiples of 3: 3, 6, 9, 12, 15

LCD = 6

or by multiplying the denominators 3 x 2 = 6, LCD = 6

Write the Equivalent Fraction

$= \frac{3\times 3}{2\times 3} + \frac{7\times 2}{3\times 2}$

$= \frac{9}{6} + \frac{14}{6}$

$= \frac{23}{6}$

Change to a mixed number.

$= 3\frac{5}{6}$

        

Example 6: Subtract: $1\frac{1}{3}-\frac{1}{2}$

Solution: Change the mixed number into an improper fraction.

$= 1\frac{+1}{\times 3} - \frac{1}{2}$

$= \frac{4}{3} - \frac{1}{2}$

Find the LCD by listing the multiples of the denominators:

Multiples of 3: 3, 6, 9, 12

Multiples of 2: 2, 4, 6, 8

LCD = 6

or by multiplying the denominators 3 x 2 = 6, LCD = 6

Write the Equivalent Fraction.

$= \frac{4\times 2}{3\times 2} - \frac{1\times 3}{2\times 3}$

$= \frac{8}{6} - \frac{3}{6}$

Subtract:

$=\frac{8}{6}-\frac{3}{6}$         

$=\frac{5}{6}$

     

Example 7: Solve $2-1\frac{1}{3}$.

Solution: 

Change mixed numbers to improper fractions.

$=\frac{2}{1} -1 \frac{+1}{\times 3}$

$= \frac{2}{1} - \frac{4}{3}$

Find the LCD by listing the multiples of the denominators:

Multiples of 1: 1, 2, 3, 4

Multiples of 3: 3, 6, 9

LCD = 3

or by multiplying the denominators 3 x 1 = 3, LCD = 3

Write the Equivalent Fraction

$= \frac{2\times 3}{1\times 3} - \frac{4\times 1}{3\times 1}$

$= \frac{6}{3} - \frac{4}{3}$

$= \frac{2}{3}$

 

Cheat Sheet

• Any whole number can be written as a fraction; for example, 5 can be written as $\frac{5}{1}$ . This is helpful while adding or subtracting fractions and whole numbers.
• The proper fraction is less than 1.
• An improper fraction or mixed fraction is greater than 1.

Blunder Areas

• While creating an equivalent fraction both the numerator and the denominator of the fraction must be multiplied/divided by the same number. 
• Always simplify the fraction whenever needed and possible.