Subtraction of Fractions

A fraction represents a part of a whole or a division of a quantity. Fractions consist of two numbers: the numerator, and the denominator. The numerator, also called the top number, represents how many parts we have. The denominator is another part of the fraction, the denominator represents the total number of equal parts that make up a whole.

 

 

For example, \(\frac{3}{4}\). This means 3 out of 4 equal parts of something. There are many types of fractions. The most common types are mentioned below:
 

 

• Proper Fractions: In proper fractions, the numerator is smaller than the denominator. The fractions are less than one. For example, \(\frac{1}{2}, \frac{1}{4}, \frac{3}{4}. \)
   

 

• Mixed Fractions: Mixed fractions are a combination of a whole number and a proper fraction. A mixed fraction can be converted into an improper fraction. For example, \(3{1\over3}\), \(1{2\over3}\), \(3{3\over4}\).
   

 

• Like Fractions: Like fractions are the fractions with the same denominator. For example, \(3\over 4\), \(1\over 4\), \(7\over4\).
   

 

• Unlike Fractions: Unlike fractions are the fractions with different denominators. For example, \(\frac{1}{2}, \frac{1}{4}, \frac{3}{4}.\)

Some key features of subtraction of fractions are as follows:
 

• Fractions can be subtracted from only when they have the same denominator. The difference between fractions is found by subtracting the numerators.
   
• When fractions have different denominators, we find the LCM or the least common denominator, then convert them to have the same denominator with the help of the LCM.
   
• Improper fractions can be subtracted as they are. To find the difference between mixed numbers, change the mixed fraction to an improper fraction and then do the subtraction.
   
• To find the difference between mixed fractions where the second fraction is less than the first fraction, subtract the integer parts and fractional parts separately.
   

Subtracting fractions involves finding the difference between two fractions.  Subtraction of fractions involves: 

• Subtracting Fractions with Like Denominators
   
• Subtracting Fractions with Unlike Denominators
   
• Subtracting Mixed Fractions
   
• Subtracting Fractions with Whole Numbers

 

Subtracting Fractions with Like Denominators
 

The fractions with the same denominator are called like denominators. As the denominator is equal, the fraction represents parts of the same whole. Subtracting fractions with like denominators is as simple as subtracting the numerators while keeping the denominator unchanged. Follow these steps to subtract fractions with like denominators: 

 

•  When subtracting the fractions with like denominators, keep the denominator the same and subtract the numerators. 
   
• Simplify the fraction if required. 

 

For example, subtract \(5\over 9\) from \(8\over 9\). 

As the denominators are the same, we keep the denominator unchanged. 
 

\(\frac{8}{9} - \frac{5}{9} = \frac{8 - 5}{9} \)

 

\( = {3\over 9}\)

 

Simplifying the fraction: 

\(\frac{3}{9} = \frac{1}{3} \)

So, \(\frac{8}{9} - \frac{5}{9} = \frac{1}{3} \)
 

 

Subtracting Fractions with Unlike Denominators

Subtracting fractions with unlike denominators means subtracting fractions with different denominators. Follow the steps below to subtract unlike fractions. 

• Find the LCM of the denominators and convert them into equivalent values. 
   
• As the denominators are the same, now we are subtracting two like fractions
   
• So subtract the numerator and keep the denominator the same
   
• Simplify the fraction

Example, subtract \(3 \over 8\) from \(5 \over 12\). 

 

Finding the LCM of 12 and 8. 

The multiples of 12 are 12, 24, 36, 48, ….

The multiples of 8 are 8, 16, 24, 32, …
 

So, the LCM is 24
 

To convert each fraction to have a denominator of 24.
 

\(\frac{5}{12} = \frac{5 \times 2}{12 \times 2} = \frac{10}{24} \)

 

\(\frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24} \)

 

Subtracting the numerator: 
 

\(\frac{10}{24} - \frac{9}{24} = \frac{10 - 9}{24} \)

 

\(= \frac{1}{24} \)

 

Subtracting Mixed Fractions

Mixed fractions involve a fractional part with a whole number. To subtract mixed fractions, follow these steps: 

 

• Convert the mixed fraction into an improper fraction. 
   
• Check the denominators:
   
  • If the denominators are the same, subtract the numerators and keep the denominator the same.
     
  • If the denominators are different, first convert the fraction to an equivalent fraction with a common denominator. Then subtract the numerators and keep the denominator the same. 
     
• Convert the fraction to a mixed number.

 

For example, \(6{{2\over 5}}\) from \(10{{3\over 4}}\)

 

Converting the mixed number to an improper fraction: 

 

\({10{3\over4}} = \frac{10 \times 4 + 3}{4} = \frac{43}{4} \)

 

\( {6{2\over5} }= \frac{6 \times 5 + 2}{5} = \frac{32}{5}\)

 

As the denominators are unlike, we convert them to equivalent fractions. 

LCM of 4 and 5 is 20
 

\(\frac{43}{4} = \frac{43 \times 5}{4 \times 5} = \frac{215}{20} \)

 

\(\frac{32}{5} = \frac{32 \times 4}{5 \times 4} = \frac{128}{20} \)

 

Subtracting the numerator:

 

\(\frac{215}{20} - \frac{128}{20} = \frac{87}{20} \)

 

Converting to a mixed number: 
 

\({{87\over 20 }}= {4{7\over 20}}\)

 

Subtracting Fractions with Whole Numbers

 

Subtracting fractions from whole numbers involves converting the whole number into a fraction first. Here are the steps to subtracting fractions with whole numbers. 

• Convert the whole number into a fraction 
   
• Apply the rules for subtracting fractions by checking the denominators. 
   
• If the denominators are different, find the LCM and make them equivalent fractions. If the denominator is the same, subtract the numerator and keep the denominator the same. 
   
• Simplify the result or convert it to a mixed fraction. 

 

For example, subtract \(3 - {2\over 3}\).

 

Converting 3 into a fraction: \(3 = {3\over 1}\)

 

Since the denominators are different, find the LCM of 1 and 3

 

LCM of 1 and 3 is 3
 

\(\frac{3}{1} = \frac{3 \times 3}{1 \times 3} = \frac{9}{3} \)

 

\(\frac{2}{3} = \frac{2}{3} \)

 

\(\frac{9}{3} - \frac{2}{3} = \frac{7}{3} \)

 

Converting 7/3 into a mixed number: \({7\over3} = {2{1\over 3}}\)
 

Subtracting fractions helps students build strong number skills, understand the relationship between parts and wholes, and apply these skills in real-life situations. Here are a few tips and tricks to master subtraction of fractions. 

 

• When subtracting fractions, always check the denominators first to know whether the fractions are like or unlike. 
   
• When subtracting mixed fractions, convert them to improper fractions before subtracting. 
   
• Parents can use real-world objects like fruits, cake, pizza, or a piece of paper to help children understand fractions. 
   
• Teachers can use visual aids such as pie charts, bar graphs, and models to help students understand the concepts of fractions and their operations. 
   
• Always simplify the answer to its simplest form if the numerator and denominator have common factors.  
   
• Students can use number lines to understand how much one fraction is being subtracted from another.

The subtraction of fractions has numerous applications across various fields. Let us explore how the subtraction of fractions is used in different areas:

 

• Cooking and Baking: We use subtraction of fractions in cooking and baking, where recipes often require subtracting fractional amounts of ingredients. For example, if a recipe calls for ¾ cup of sugar, but you have already added ½ cup, you will need to subtract ½ from ¾ to determine how much more to add.
   

 

• Finance:  People use fraction subtraction when dealing with money, discounts, and budgeting. For example, If someone has \(5{3\over 4}\) dollars and spends \(2{1\over2}\) dollars, they subtract the fractions to find what remains.
   

 

• Sports: Subtraction of fractions is also useful in sports and fitness. Athletes track time and distances, often leading to subtraction of fractions to analyze performance improvements. For example, a runner who previously completed a lap in \(13\over5\) minutes and now finishes in \(12\over5\) minutes must subtract to find their improvement.

 

 

• Shopping Discounts: Subtract the fractions to find remaining price. For example, if an items costs \(\$10\) and you use a coupon for \(\frac{3}{10}\) of the price, the remaining cost is 10 -10 \( \times \frac {3} {10}\) = 7 dollars.

 

 

• Construction: In construction, workers and carpenters commonly use the fraction subtraction in their work. When cut the materials like wood, metal, or piece.