Subtracting Mixed Fractions

The subtraction of mixed fractions is a fundamental arithmetic operation that finds the difference between two mixed fractions. A mixed fraction consists of a whole number and a proper fraction.

For example, \(3\frac{1}{2}\) and \(2\frac{3}{5}\) can be subtracted to know the difference between these two numbers. 

In fractions, there can be like and unlike denominators, and the method of subtraction is different for each. Let us now learn how to subtract mixed fractions with like denominators.

Fractions that have a common denominator are known as like fractions. The mixed fractions also have the same denominators, such as \(2\frac{1}{4}\) and \(3\frac{2}{4}\). When we subtract mixed fractions with like denominators, we have to keep some points in mind. 

• Mixed fractions can be represented as \(a\frac{b}{c}\) and written as \(a + \frac{b}{c}\).
   
• The standard procedure while subtracting mixed numbers is to subtract the whole number first before subtracting the fractional part.
   
• A mixed number can be converted into an improper fraction by multiplying the denominator by the whole number. The product should then be added to the numerator and the denominator must be retained. 
  
  
  For example, a mixed fraction \(1 + \frac{6}{13}\) can be converted to an improper fraction as follows: 
  
  Multiply the whole number by the denominator:
  \(1 × 13 = 13\)
  
  Add the product to the numerator:
  \(13 + 6 = 19\)
  
  Keep the denominator the same and write the improper fraction as: 
  \(19 \over 13\). 
   
• We can convert an improper fraction into a mixed fraction by dividing the numerator by its denominator. Now, the resulting quotient is the whole number, the remainder is the numerator, and the denominator is retained. It can be expressed as:
  
  \(Q {R \over D}\)
  
  
  For instance, an improper fraction \(25 \over 3\) can be converted to a mixed fraction as follows:
  
  Divide the numerator by the denominator: 
  
  \(25 ÷ 3 \)
  
  Quotient = 8
  
  Remainder = 1 
  
  Hence, the mixed fraction will be: 
  
  \(\frac{25}{3} = 8 \frac{1}{3}\)

Now, let us subtract \(2 \frac{1}{5}\) from \(4 \frac{2}{5}\). 

Step 1: Subtract the whole numbers. 
\(4 - 2 = 2\)

Step 2: Subtract the fractional parts. 
\({2 \over 5} - {1 \over 5} = {1 \over 5}\)

Step 3: Combine the whole number and the fractional part from the answers obtained in step 1 and step 2.

Therefore, the final answer is \(2 {1 \over 5}\)

Fractions that have different denominators are known as unlike fractions. If the mixed fractions have unlike denominators, we can follow two methods to find the difference.

Method 1: Use the least common denominator (LCD) of the unlike denominators.

For example, subtract \(2 {3 \over 4}\) from \(4 {5 \over 6}\)

Step 1: Subtract the whole numbers of the mixed fractions. 
            \(4 - 2 = 2\)

Step 2: If the fractions have a different denominator, then we must find the LCD.

The LCD of 4 and 6 is 12. 

Step 3: Convert the two fractions into fractions with a common denominator of 12. 
 

• To convert 5/6, multiply both numerator and denominator by 2.
  \(\frac{5}{6} = {5 × 2 \over {6 × 2}} = {10 \over 12}\)
   
• Now, to convert 3/4, multiply both the numerator and the denominator by 3:
  \({3 \over 4} = \frac{3 × 3}{4 × 3} = {9\over 12}\)

Step 4: Now that both fractions have a common denominator, we can just subtract the numerator and retain the denominator. 
\({10 \over 12} - {9 \over 12} = {1 \over12}\)

Step 5: Write the difference by combining the results. 2 is the difference of whole numbers, and \(1 \over12 \)is the difference of the fractional part. Therefore, the final result is \(2 {1 \over12}\)

So, \(4 {5 \over 6} - 2 {3 \over 4}  = 2 {1 \over 12}\)

Method 2: Convert the mixed fractions into improper fractions.

For instance, subtract \(3 {1 \over 4} \space \text{from} \space 6 {3 \over5}\)

Step 1: Convert the mixed fractions to improper fractions: 
           \(\begin{align*} 6 {3 \over 5}  &= \frac{(6 × 5) + 3}{5} = \frac{30 + 3}{5} = \frac{33}{5}  \\  6 {3 \over 5}  &= \frac{33}{5} \end{align*}\)
   
    Next,

\(\begin{align*}3 {1 \over 4}  &= \frac{(3 × 4) + 1}{4} \\ &= \frac{12 + 1}{4} = {13 \over 4} \end{align*}\)

Step 2: Make the denominators the same by finding the LCD. 
          The LCD of 5 and 4 is 20. 

Step 3: Write the fractions with the common denominator. 
        \(\frac{33}{5} = \frac{(33 × 4)} { (5 × 4)} = \frac{132}{20} \)

        \(\frac{13}{4} = \frac{(13 × 5)} { (4 × 5)} = \frac{65}{20}\)

Step 4: Now, subtract the fractions:
         \({132 \over 20} - {65 \over 20} = {67\over 20}\)

Step 5: Convert the fraction into a mixed number.
 

• To convert, let’s divide the numerator by the denominator.
   
• The quotient and remainder after dividing \(67 \over 20\) are 3 and 7 respectively.
   
• Write the quotient as the whole number and the remainder as the numerator. 
   
• The denominator of the fractional part will be the same.  
   
• Hence, the mixed fraction is \(3 {7 \over 20}\)

Here, the value of \(6 {3 \over 5}  - 3 {1 \over 4}\) is equal to \(3 {7 \over 20}\).

We can use regrouping while subtracting mixed numbers when the fractional part of the subtrahend is greater than that of the minuend.

For example, subtract \(6 {3 \over 4}\) from \(9 {1 \over 5}\). 

Step 1: Compare the fractional parts and make the unlike denominators to a common denominator.
 

• Here, the fractional parts are: 
  \(1 \over 5\) and \(3 \over 4\).

Step 2: Now, find the least common denominator of 5 and 4. 
 

• 20 is the LCD of 4 and 5. 

• Let’s convert \(1 \over 5\) and \(3 \over 4\) from unlike to like denominators.

• Multiply both the numerator and the denominator of \(1 \over 5\) by 4
  \(1 \over 5\)  \(= \frac{1 × 4}{5 × 4} = {4\over 20}\)

• Now multiply the numerator and the denominator of 3/4 by 5
  \(3 \over 4\)  \(= \frac{ 3 × 5}{4 × 5}  = {15 \over 20}\)

Step 3: Since \({15 \over 20} > {4 \over 20}\), we cannot subtract directly. Therefore, borrow 1 from the whole number part of \(9{1 \over 5}\). 

1 whole can be written as \(20 \over 20\) since the LCD of 4 and 5 is 20. 

Now, add the borrowed \(20 \over 20\) to \(4 \over 20\), and we get \(24 \over 20\).

Step 4: The whole number 9 becomes 8, and the fraction \(4 \over 20\) becomes \(24 \over 20\). 

Hence, \(9{1 \over 5}\) becomes \(8{24 \over 20}\) and \(6 {3 \over 4}\) becomes \(6{15 \over 20}\).

Step 5: Subtract the fractions. 
\(8{24 \over 20}\) - \(6{15 \over 20}\) 

• Subtract the whole numbers
  \(8 - 6 = 2 \)   

• Subtract the fractional part
  \({24 \over 20} - {15 \over 20}  = \frac{24 - 15}{20}  = {9\over 20}\)  

Thus, the result is \(2 {9 \over 20}\).

By following the right approach, subtracting mixed fractions can be easy and simple. Here are a few tips and tricks:
 

• Always convert mixed fractions to improper fractions for easy calculation.
   
• Find the LCM to convert unlike denominator to like denominator. 
   
• Use visual aids like fraction bars to understand subtraction steps and avoiding mistakes.
   
• Always write the final answer in mixed fractions or proper fraction.
   
• Always convert the borrow into fraction, before adding it to the fractional part.

Understanding the process of subtracting mixed fractions will help us to solve various real-life problems easily. We can use this concept in various situations, ranging from cooking to engineering. Here are some real-life applications of subtracting mixed fractions: 

• When cooking, we need to subtract mixed fractions to ensure the correct quantity of items is used.  For example, to bake a cake, we need \(3{1 \over 2}\) cups of flour, but we have already added \(2{1 \over 2}\)cups. To find the remaining cups, we should subtract 2 1/2 from 3 1/2.
  
  \(3 {1 \over 2} - 2 {1\over 2} = 1 \)
  
  So, 1 cup of flour is still needed. 

• In engineering and construction, engineers use the subtraction of mixed fractions to measure the materials accurately. For instance, an engineer builds a room that is \(8{1 \over 4}\) square feet long. If he wants to remove \(2{3 \over 7}\) square feet, he needs to subtract: 
  
  \(8 {1 \over 4} - 2 {3\over 7} = 5 {23 \over 28}\)
  
  So, the remaining space is \(5 {23 \over 28}\) feet. 

• While shopping, we must compare the overall price of items with our total budget. For example, our total budget is \($6 {1 \over 2}\), and we buy two items that cost \($1 {1 \over 2}\) and \($2 {1 \over 2}\). To calculate how much money we have left: 
  
  \(6{ 1 \over 2} - 1 {1 \over 2} = 5\)
  
  $5 left after shopping. 

• When calculating remaining time, after a fraction of time has already passed, we use subtraction of mixed fractions. For example, if a movie is \(2 {1 \over 2}\) hours, and  {1 \over 2}\(1 {1 \over 2}\) hours has already passed. To calculate the time left for the movie to be finished:  
  
  \(2 {1 \over 2} - 1 {1 \over 2} = 1 {1 \over 2}\)
  
  The remaining movie will be finished in \(1 {1 \over 2}\) hours.

• During a marathon, to calculate the path you have left to cover, subtracting a mixed fraction is applied. For example, if you have covered \(2 {2 \over 5}\) km and the marathon is of \(6 {4 \over 5}\) km, the distance left to cover can be calculated by:
  
  \(6 {4 \over 5} - 2 {2 \over 5} = 4 {2 \over 5}\)
  
  The remaining distance is \(4 {2 \over 5}\) km.