Mixed Fractions are a combination of a whole number and a part.

Mixed fractions or mixed numbers are fractions with both whole numbers and a proper fraction. They express the same thing that an improper fraction expresses. However, instead of using an improper fraction we may use a mixed form of expression.

To create a mixed number or a mixed fraction, we need a whole number and a proper fraction in a relation where the denominator takes the role of divisor, the whole number takes the role of a quotient and the numerator takes the role of the remainder. The divisor when multiplied with the quotient gets added to the remainder and gives us the desired mixed fraction.

Mixed Fractions

A Mixed fraction is also called as a Mixed number, because it is a combination of a whole number and a proper fraction, we can either call it as a mixed fraction or a mixed number. If we take an example 1$\frac{3}{4}$, we can divide it in various parts and name it accordingly. Here 1 is called the whole number and 3 is the numerator and 4 is the denominator.

There are basically 3 types of fractions, proper fractions, improper fractions and mixed fractions. To express the same amount we can either use a mixed fraction or an improper fraction. Like in the above example we can show the same amount as $\frac{7}{4}$ also, which is an improper fraction.

There are two different methods in adding mixed fractions. They are,

  • Add mixed fractions with common denominator
  • Add mixed fractions with different denominators

Adding Mixed Fractions with Common Denominator


The sum of mixed fractions with a common or same denominator can be obtained through the following steps.

Step 1: Express the mixed fraction as an improper fraction.
Step 2: Add the numerator.
Step 3: Retain the common denominator.
Step 4: Sum of the mixed fractions = $\frac{Sum\ of\ Numerators\ from\ step\ 3}{Common\ Denominator}$

Examples on Adding Mixed Fraction with a Common Denominator:


Add the mixed fractions 2$\frac{3}{5}$+3$\frac{1}{5}$

We write the mixed fractions as improper fractions

2$\frac{3}{5}$ = $\frac{((2 \times 5)+3)}{5}$

=$\frac{13}{5}$

3$\frac{1}{5}$=$\frac{(3 \times 5)+1}{5}$

=$\frac{16}{5}$

2$\frac{3}{5}$+3$\frac{1}{5}$=$\frac{13}{5}$+$\frac{16}{5}$

Here both the fractions have a common denominator, so we can combine the numerator.

2$\frac{3}{5}$+3$\frac{1}{5}$=$\frac{13}{5}$+$\frac{16}{5}$

=$\frac{13+16}{5}$

=$\frac{29}{5}$

= 5$\frac{4}{5}$

Adding Mixed Fractions with Different Denominators

The step we need to undertake at the beginning while doing the operation of adding a mixed fraction is to convert the mixed fraction into an improper fraction.
To convert the mixed fraction to an improper fraction we need to multiply the whole number with the denominator and then add the numerator.
This gives us a new numerator with the original denominator still in the fraction.

Step 1: Find the lowest common denominator (LCD)
Step 2: Sometimes we have to go for the largest denominator as an LCD and such that it is divisible by the other denominators.
Step 3: Multiply the largest denominator by each of the denominators and check their divisibility with other denominators, until we have an LCD.
Step 4: After finding the LCD, we need to change the fractions into equivalent fractions having a common denominator.
Step 5: Divide the denominator of each fraction into LCD and then multiply the result with the fraction’s numerator and place it over the LCD.

Examples on Adding Mixed Fractions with Different Denominators

Add 1$\frac{1}{3}$ + 2 $\frac{1}{2}$ + 1$\frac{1}{6}$
Converted into $\frac{4}{3}$ + $\frac{5}{2}$ + $\frac{7}{6}$
Step 1: Find the LCD of 3, 2 and 6 = 6
Step 2: Change the fractions into equivalent fractions having a common denominator 6
Step 3: Divide 3 into 6 and multiply 2 into the numerator; divide 2 into 6 and multiply 3 into numerator; divide 6 into 6 and multiply 1 into numerator
Step 4: Add the numerators (8 + 15 + 7) = 30
Step 5: Reduce the fraction $\frac{30}{6}$ = 5

There are two different methods of subtracting mixed fractions. They are,

  • Subtracting mixed fractions with a common denominator
  • Subtracting mixed fractions with a different denominator

Subtracting Mixed Fractions with a Common Denominator


The difference of mixed fractions with common or same denominator can be obtained through the following steps
Step 1: Express the mixed fraction as improper fractions.
Step 2: Find the difference of the numerators.
Step 4: Retain the common denominator.
Step 5: Difference of the mixed fractions $\frac{Difference\ of\ numerators\ from\ step3}{Common\ Denominator}$

Examples on Subtracting Mixed Fractions with a Common Denominator


Subtract the mixed fractions 5$\frac{2}{3}$ - 3$\frac{1}{3}$

We write the mixed fractions as improper fractions
5$\frac{2}{3}$ = $\frac{(5 \times 3)+2}{3}$

$\frac{17}{3}$

3$\frac{1}{3}$ = $\frac{(3 \times 3)+1}{3}$

$\frac{10}{3}$
5$\frac{2}{3}$ - 3$\frac{1}{3}$ = $\frac{17}{3}$ - $\frac{10}{3}$

Here the both fractions have a common denominator, so we can subtract the numerators.

5$\frac{2}{3}$ - 3$\frac{1}{3}$ = $\frac{17}{3}$ - $\frac{10}{3}$

$\frac{17-10}{3}$

=$\frac{7}{3}$

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

Subtracting mixed fractions with different denominators


Convert the mixed fraction into a improper fraction to begin the operation of subtraction.
Multiply the whole number with the denominator and add the numerator to convert this into a new improper fraction.

Step 1: Find the LCD of the fractions
Step 2: After finding the LCD, we need to change the fractions into equivalent fractions having a common denominator
Step 3: Divide the denominator of each fraction into a LCD and then multiply the result with the fraction’s numerator and place it over the LCD.
Step 4: Subtract the numerator and get the final answer
Example of subtracting mixed fractions with different denominators:

Subtract 1 $\frac{1}{4}$ – 1$\frac{1}{3}$ – 1$\frac{1}{6}$
Convert into $\frac{5}{4}$$\frac{4}{3}$$\frac{7}{6}$
Step 1: Find the LCD of the fractions
Step 2: LCD is found to be 12
Step 3: Divide the denominator of each fraction into LCD and then multiply the result with the fraction’s numerator and place it over the LCD $\frac{(15 – 16 – 14)}{12}$
Step 3: Subtract the numerators (-15)
Reduce the answer if required $\frac{-15}{12}$ = $\frac{-5}{4}$ = - 1 $\frac{1}{4}$

In order to multiply two fractions, we convert the mixed fractions into improper fractions. Then multiply the numerators together and multiply the denominators. We need to write the answer in the reduced or simplest form.

The product of two mixed fractions can be obtained through the following steps.
Step 1: Rewrite the mixed fractions as improper fractions.
Step 2: Multiply the numerators
Step 3: Multiply the denominators.
Step 4: The product of the fractions = $\frac{Product\ of\ Numerators\ from\ step2}{Product\ of\ denominators\ from\ step3}$
Step 5: Write the product in its reduced form and then write as a mixed fraction.

Examples on Multiplying Mixed Fractions


Given below are some examples on multiplying mixed fractions.

Example 1:

Find the product of 3$\frac{2}{9}$ and 2
$\frac{3}{7}$

Solution:

Step 1: Rewrite the mixed fractions as improper fractions

3$\frac{2}{9}$ = $\frac{29}{9}$

2$\frac{3}{7}$ = $\frac{17}{7}$

Step 2: Multiply the numerators, 29 x 17=493

Step 3: Multiply the denominators, 9 x 7=63

Step 4: The product of the factors =

= $\frac{493}{63}$

=7 $\frac{52}{63}$

We can divide a mixed fraction by another mixed fraction through the following steps.

Step 1: Convert both mixed fractions into improper fractions
Step 2: Find the reciprocal of the divisor
Step 3: Multiply the dividend (after converting as an improper fraction from step1) and the divisor from step 2
Step 4: Write the answer in the reduced form.

Examples on Dividing Mixed Fractions


Given below are examples on dividing mixed fractions.

Example 1:

Divide 3$\frac{1}{2}$ and 2$\frac{3}{4}$


Solution:

Step 1: Convert both mixed fractions into improper fractions

3$\frac{1}{2}$ = $\frac{7}{2}$

2$\frac{3}{4}$ = $\frac{11}{4}$

Step 2: Find the reciprocal of the divisor, reciprocal of $\frac{11}{4}$ is $\frac{4}{11}$

Step 3: Multiply the dividend (after converting as an improper fraction from step1) and the divisor from step 2

$\frac{7}{2}$ X $\frac{4}{11}$

$\frac{7x4}{2x11}$

$\frac{28}{22}$

Step 4: Write the answer in the reduced form

=$\frac{14}{11}$, by removing the common factor, 2, from the numerator and the denominator.

A mixed fraction can be expressed as an improper fraction as

$\frac{(whole\ \times\ Denominator)\ + Numerator}{Denominator}$

Example
Consider the mixed fraction, 9$\frac{11}{13}$ . Here 9 is the whole, 11 is the numerator and 13 is the denominator
so 9$\frac{11}{13}$ = $\frac{(9 \times13 + 11)}{13}$ =$\frac{128}{13}$
Thus, all mixed fractions can be represented as improper fractions and vice versa.

When we convert a mixed fraction into a decimal, the following steps are required to followed.

Step1: Multiply the whole number of the mixed fraction by the denominator of the mixed fraction.
Step 2: Now the multiplied value is added to the numerator of the mixed fraction.
Step 3: Divide the numerator by the denominator of the new fraction.
Step 4: Now, round off the answer to the desired precision.

Examples on Converting Mixed Fraction to Decimal:


Given below are some examples that explain how to convert mixed fractions to decimals.

Example 1: 

Express 2 $\frac{3}{8}$  as a decimal

Solution:
 
Step 1: Multiply the whole number of the mixed fraction by the denominator of the mixed fraction = $\frac{(2 \times 8 + 3)}{8}$ = $\frac{16 + 3}{8}$
                     
Step 2: The multiplied value is added to the numerator of the mixed fraction.
                      = $\frac{(16 + 3)}{8}$ = $\frac{19}{8}$
Step 3: Divide the numerator by the denominator of the new fraction.
                  8 ) 19 ( 2.375
                       16
                     (-) 
                    _______
                         30
                         24
                      (-)
                   _________
                          60
                          56
                      (-)
                   _________
                           40
                           40
                      (-)
                   _________
                            0
Step 4: 2.375 

Example 2:

Express 2 $\frac{1}{3}$ as a decimal

Solution:

Step 1: Multiply the whole number of the mixed fraction by the denominator of the mixed fraction = $\frac{(2 \times 3 + 1)}{3}$ = $\frac{(6 + 1)}{3}$
 
Step 2: The multiplied value is added to the numerator of the mixed fraction
                            = $\frac{(6 + 1)}{3}$ = $\frac{7}{3}$                
Step 3: Divide the numerator by the denominator of the new fraction.
                  3 ) 7 ( 2.3333
                       6
                     (-) 
                    _______
                       10
                         9
                      (-)
                   _________
                          10
                            9
                      (-)
                   _________
                           10
                             9
                      (-)
                   _________
                             10
                               9
                          (-)
                  __________
                               1  
Step 4: 2.333