Subtraction of Fractions

Subtraction of Fractions

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Introduction to Fractions

Fractions are part of a whole. Before moving to the subtraction of fractions, let us revise about fractions here


Steps to Subtracting Fractions

We have learnt to subtract whole numbers (example: \(4 -2  = 2 \)).

Similarly, we can subtract fractions.

Subtracting Fractions include: \(\)

  • Subtracting fractions with whole numbers
    \[\begin{align} \left(3\frac {1}{2} - 2 \right) \end{align}\]
  • Subtracting fractions with like denominators
    \[\begin{align} \left(\frac {3}{4} -  \frac {1}{4} \right) \end{align}\]
  • Subtracting fractions with different denominators
    \[\begin{align} \left(\frac {3}{5} -  \frac {1}{2} \right) \end{align}\]
  • Subtracting  fractions with variables
    \[\begin{align} \left(\frac {3}{5} x -  \frac {1}{4}x \right) \end{align}\]

Subtracting Fractions with the Same Denominator

Let us subtract the fractions \(\begin{align} \frac {2}{5} \end{align}\) and \(\begin{align} \frac {1}{5} \end{align}\) using rectangular models.

 A rectangular area model to add fractions

We will represent \(\begin{align} \frac {2}{5} \end{align}\)  in this model by shading \(2\) out of \(5\) parts.

We will further shade out \(1\) part from our shaded parts of the model to represent removing \(\begin{align} \frac {1}{5} \end{align}\)

 Subtracting fraction with area models

We are now left with \(1\) part in the shaded parts of the model.

Therefore,

\(\begin{align} \frac {2}{5} \end{align}\) - \(\begin{align} \frac {1}{5} \end{align}\) = \(\begin{align} \frac {1}{5} \end{align}\)


Subtract Fractions with Different Denominators

Let us understand how to subtract unlike fractions using the area model.

\(\begin{align} \frac {3}{5} \end{align}\) - \(\begin{align} \frac {1}{3} \end{align}\) 

This indicates that we have to remove \(\begin{align} \frac {1}{3}^{rd} \end{align}\) part from \(\begin{align} \frac {3}{5} \end{align}\)

Let us use our rectangle model and represent \(\begin{align} \frac {3}{5} \end{align}\) as shown.

 model of three fifth of a fraction

\(\begin{align} \frac {1}{3} \end{align}\)  when represented vertically will have \(1\) column shaded out of \(3\) columns.

one third area model of a fraction

Now let us represent both in the same model.

 Subtraction of one third from three fifth of a fraction

We see that our model is divided into \(15\) parts. This is our denominator.

This is nothing but the LCM of the denominators of the given fraction.

As we need to remove \(\begin{align} \frac {1}{3} \end{align}\)  from \(\begin{align} \frac {3}{5} \end{align}\), we will move the selected two parts which are not part of the \(\begin{align} \frac {3}{5} \end{align}\), to remove it from\(\begin{align} \frac {3}{5} \end{align}\)

Now, our model will look like this.

 Removing one third from three fifth of a fraction

We see that there are only \(4\) parts of the remaining which are not shaded out.

Therefore,

\(\begin{align} \frac {3}{5} \end{align}\)  - \(\begin{align} \frac {1}{3} \end{align}\) = \(\begin{align} \frac {4}{15} \end{align}\)

To subtract unlike fractions,

  1. Take the LCM of the denominators
  2. Convert the given fractions to equivalent fractions with the denominator as the LCM
  3. Subtract the numerators

Subtracting Fractions With Whole Numbers

To subtract a fraction from a whole number, consider the following example.

\(\begin{align}3 - \frac {1}{2} \end{align}\)

Convert the whole number to a fractional form 

\(\begin{align} 3= \frac {3}{1}\end{align}\)

 Subtract them like unlike fraction

\[\begin{align} \frac {3}{1} - \frac {1}{2} \end{align}\]

\[\begin{align}
&= \left(\frac {3 }{1} \times \frac {2}{2}\right) - \frac {1}{2} \\\
&=  \frac {6}{2} - \frac {1}{2} \\\
&= \frac {5}{2} \\\
 &= 2\frac {1}{2}
\end{align}\]

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Subtracting Fractions with Variables

Consider this example,  \(\begin{align} \frac {2}{5} x\end{align}\) - \(\begin{align} \frac {1}{5} x\end{align}\)

They are like terms as they have the same denominator and \( \text x\) is common. We can take the common factor out and rewrite it as: 

\(\begin{align} \frac {2}{5} x\end{align}\) - \(\begin{align} \frac {1}{5} x\end{align}\) = \((\begin{align} \frac {2}{5} \end{align}\) - \(\begin{align} \frac { 1}{5} \end{align}\))\(x\) =  \(\begin{align} \frac {1}{5} x\end{align}\)

Similarly, when we have to subtract \(\begin{align} \frac {1}{3} x\end{align}\) from \(\begin{align} \frac {1}{2} x\end{align}\)

We can take the LCM of the denominator and convert them to like terms and take the common variable out are rewrite it as follows:

LCM (\(2 , 3\) = 6 )

 \(\begin{align} \frac {1}{2} x\end{align}\) =  \(\begin{align} \frac {1}{2} x\end{align}\) \(\times\)  \(\begin{align} \frac {3}{3} \end{align}\) =  \(\begin{align} \frac {3}{6} x\end{align}\)

 \(\begin{align} \frac {1}{3} x\end{align}\) =  \(\begin{align} \frac {1}{3} x\end{align}\) \(\times\)  \(\begin{align} \frac {2}{2} \end{align}\) =  \(\begin{align} \frac {2}{6} x\end{align}\)

 \(\begin{align} \frac {1}{2} x - \frac {1}{3} x &= \frac {3}{6} x - \frac {2}{6} x \\ &= \left(\frac {3-2}{6} \right)x \\ &= \frac {1}{6} x\end{align}\)

 
important notes to remember
Important Notes
  1. Recall the steps to subtracting  fractions with same denominator:

    \[\begin{align}  \frac {\text{Nr\(_1\)}}{ \text{Dr} }  -  \frac {  \text{Nr\(_2\)} }{ \text{Dr}}   = \frac{ \text{Nr\(_1\)} -  \text{Nr\(_2\)} }{ \text{Dr} }  \end{align}\]

  2. Steps to subtracting fractions with different denominators: 

    a) Convert the given fractions to like fractions by taking the LCM of the denominator.  
    b) Find the equivalent fractions of the given fractions whose denominator is the LCM.  
    c) Subtract the numerators and retain the same denominator.

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Solved Examples

Example 1

 

 

FInd the difference  of \(\begin{align} \frac {5}{7} \end{align}\) & \(\begin{align} \frac {3}{7} \end{align}\)

Solution

The given fractions are like fractions.

Subtract the numerators and retain the same denominator.

\[\begin{align} &=\frac {5}{7} + \frac {3}{7} \\\\
&=\frac {5 - 3}{7} \\\\
&= \frac {2}{7} \end{align}\]

\(\therefore \text { The difference is } \frac {2}{7}\)
Example 2

 

 

Subtract \(\begin{align} \frac {2}{5} \end{align}\) from  \(\begin{align} \frac {2}{3} \end{align}\) 

Solution

The given fractions are unlike fractions.

We have to find the LCM of the denominators and convert \(\begin{align} \frac {2}{5} \end{align}\) and \(\begin{align} \frac {2}{3} \end{align}\) to equivalent fractions of same denominator and then subtract.

LCM of  \(3 ,5 = 15 \)

\[\begin{align}
&= \left(\frac {2}{3} \times \frac{5}{5} \right) -  \left(\frac {2}{5} \times \frac {3}{3} \right) \\
&= \frac {10}{15} - \frac {6}{15} \\
&=  \frac {4}{15} \end{align}\]

\(\therefore \text { The difference is } \frac {4}{15}\)
Example 3

 

 

Subtract  \(\begin{align} \frac {1}{3} \end{align}\) from  \(\begin{align} 3 \end{align}\) 

Solution

\[\begin{align}  3 - \frac {1}{3} &= \frac {3}{1} -\frac {1}{3} \\
&= \left(\frac {3}{1} \times \frac {3}{3}\right) - \frac {1}{3}  \\
&= \frac {9}{3} - \frac {1}{3} \\
&= \frac {8}{3} \\
&= 2\frac{2}{3} \end{align}\]

\(\therefore \text { The difference is } 2\frac {2}{3}\)

Example 4

 

 

Solve \(\begin{align} \frac {2}{5} x\end{align}\) + \(\begin{align} \frac { 1}{3} y\end{align}\) - \(\begin{align} \frac {1}{5} x\end{align}\) + \(\begin{align} \frac { 1}{2} y\end{align}\)

Solution

We can group the \( x\) terms together and subtract:

\(\begin{align}\frac {2}{5} x - \frac {1}{5} x =  \frac {1}{5}x \end{align}\) 

Grouping the \(\text y\) terms together, we have: 

\(\begin{align} \frac {1}{3} y\end{align}\) + \(\begin{align} \frac {1}{2} y\end{align}\) 

LCM of the denominator (\(3 ,2\)) is \(6\)

Converting it to like terms, we have 

\(\begin{align} \frac { 1}{3} y\end{align}\) \(\times\) \(\begin{align} \frac { 2}{2} \end{align}\) =  \(\begin{align} \frac { 2}{6} y\end{align}\)  

\(\begin{align} \frac { 1}{2} y\end{align}\) \(\times\) \(\begin{align} \frac {3}{3} \end{align}\) =  \(\begin{align} \frac { 3}{6} y\end{align}\)  

\[\begin{align} \frac { 1}{3} y + \frac { 1}{2} y &= \frac { 2}{6} y + \frac { 3}{6} y\\
&= \left(\frac {2+3}{6}\right)y \\
&=   \frac{5}{6}y \end{align}\]

\(\begin{align} \frac {1}{5}x + \frac {5}{6}y \end{align}\)


Practice Questions

 
 
 
 
 
 
tips and tricks
Tips and Tricks
  1. For unlike fractions, never subtract the numerators and denominators directly.

    \(\begin{align} \frac {1}{5} \end{align}\) +  \(\begin{align} \frac { 2}{3} \end{align}\) ≠ \(\begin{align} \frac {3}{8} \end{align}\)

  2. When subtracting unlike fractions, it is not necessary to find the LCM of the denominators. Any common multiple will do.
    So, simply multiplying the two denominators gives us a common multiple. This may lead to larger looking numbers, but it can be reduced to its lowest form.

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Frequently Asked Questions (FAQs)

1. How to subtract fractions

For like fractions, subtract the numerators and retain the same denominator. 

For subtracting unlike fractions, take the LCM of the denominators and convert the fractions to equivalent fractions and subtract them as like fractions.

Reduce to its lowest term if needed.

2. Subtracting fractions from whole numbers

 Convert the Whole number part to fractional part  and subtract like unlike fractions.

It will be in a mixed form. Finally, convert it to improper form:

\[\begin{align}  2 - \frac {1}{5} &= \frac {2}{1} -\frac {1}{5} \\
&= \left(\frac {2}{1} \times \frac {5}{5}\right) - \frac {1}{5}  \\
&= \frac {10}{5} - \frac {1}{5} \\
&= \frac {9}{5} \\
&= 1\frac{4}{5} \end{align}\]

  
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