Improper Fractions

Improper fractions, with the name, signifying that the fractions are not done in a proper manner for any number, object or element. In Maths, a fraction is a part of a whole. Fractions have two parts, numerator and denominator. If  ⅓ is a fraction, then 1 is the numerator, and 3 is the denominator. An improper fraction has a numerator greater than the denominator. For example, 3/2 is an improper fraction, but 2/3 is a proper fraction, whose denominator is greater than the numerator.  

In this article, you will find all the topics related to improper fractions. Also, we perform various arithmetic operations on these fractions, such as addition, division, multiplication, etc.

What are Improper Fractions?

An improper fraction is defined as a fraction whose numerator is greater than the denominator. Suppose, x/y is an improper fraction, such that x > y. It is therefore, the improper fraction is always greater than one.

Examples of Improper Fractions are:

• 17/5
• 9/4
• 13/4
• 16/3
• 5/2

How to Simplify Improper Fractions?

We have understood what an improper fraction is. Now, let us discuss here how to simplify such fractions here.

• Step 1: Determine if the given fraction is improper or not.
• Step 2: Now interpret the denominator and check for how many parts it is dividing the numerator.
• Step 3: Check the common factors of numerator and denominator
• Step 4: Cancel the like terms both from numerator and denominator.

The resulted fraction is the simplified fraction. Let us understand these steps with the help of examples.

Example: Simplify the fraction 24/10.

Solution: Since we know 24>10, thus 24/10 is an improper fraction.

Let us factorise the numerator and denominator.

24 = 2 x 2 x 2 x 3

10 = 2 x 5

Since we can see 2 is the common factor for 24 and 10, so after cancelling 2 from the numerator and denominator, we get;

(2 x 2 x 3)/5 = 12/5

Improper Fractions to Mixed Fractions

To convert an improper fraction to a mixed fraction, divide the denominator by the numerator. The quotient becomes the whole number, the remainder will be the numerator, and the divisor becomes the denominator.

Let us consider an example. Convert 17/4 to a mixed fraction.

When we divide 17 by 4, the quotient is 4 and the remainder is 1. So the mixed fraction is 4  1/4.

Although an improper fraction can be converted to a mixed fraction, there are situations where expressing a fraction as an improper fraction minimizes a lot of confusion, especially when the fractions are expressed in calculations.

For example, consider 3 + 6  2/3.

Is it 3 + 6  +  2/3 or 3 + 6  × 2/3 ?

This confusion can be removed by writing it as  3 + 20/3.

Mixed to Improper Fraction

Let us say 31/4 is a mixed fraction to be converted into an improper fraction.
Multiply 4 by 3 and add with the numerator 1, we get;
4 x 3 + 1 = 13
Hence, the required numerator of the improper fraction is 13 while the denominator remains the same, i.e., 4
Therefore, the required fraction is 13/4

Another Method:
Write the mixed number as a sum of the whole number and proper fraction, such that;
3 + ¼
Now rationalise the denominators and write the two parts with a common denominator.
12/4 + ¼
Now add the fractions
13/4

Addition of Improper Fractions

The addition of improper fractions can have two scenarios. If the denominators of all the fractions are equal, we can add all the numerators and keep the same denominator.

For example, (17/4) + (9/4) + (5/4)

Since the denominator of all three fractions is equal, we add all the numerators by keeping the common denominator as 4.

Hence, we get the following:

(17/4) + (9/4) + (5/4) = (17 + 9 + 5) / 4

(17/4) + (9/4) + (5/4) = 31/4.

The other case is that denominators of fractions are not equal. Here, the process is slightly different and involves the calculation of the least common multiple of the denominators.

Consider the example, (15/3) + (3/4) + (5/2)

Here, the denominators of the fractions are different. (i.e., 3, 4, 2), and take the LCM of 2, 3, and 4.

Hence, to add the given fractions, follow the below steps:

• The least common multiple of 3, 4, and 2 is 12.
• Now the fraction we get by adding all these fractions will have 12 as the denominator.
• Divide the LCM by each of the denominators and multiply the quotient by the numerators.
• The numerator of the new fraction will be the sum of all the numbers obtained in the previous step.
• Now, in the first fraction, the denominator is 3. LCM 12 divided by 3 is 4. The numerator is 15. 15 x 4 is 60. Similarly, for the other two fractions, the numbers are 9, and 30.

Therefore, (15/3) + (3/4) + (5/2) = (60/12) + (9/12) + (30/12)

(15/3) + (3/4) + (5/2) = (60 + 9 + 30) / 12

(15/3) + (3/4) + (5/2) = 99/12.

Now, the sum of fractions can be simplified as:

(15/3) + (3/4) + (5/2) = 33/4

convert into mixed fractions, if required.

Hence, the mixed fraction is:

(15/3) + (3/4) + (5/2) = 8  1/4.

Subtraction of Improper Fractions

In a similar manner, as we added the improper fractions, we can also subtract them. 

• The first step is to check if denominators are the same or not
• The second is to rationalise the denominators 
• The last step is to subtract the given fractions and simply, if required

Let us see an example:

Subtract 5/2 – 7/3.

LCM (2,3) = 6

Therefore,

(5/2 x 3/3) – (7/3 x 2/2) 

= 15/6 – 14/6

= (15-14)/6

= 1/6.

Solved Examples of Improper Fractions

Example 1:

Multiply 3 by 9/7.

Solution:

Given: 3 x 9/7 

3 x 9/7  = 27/7

Example 2:

Convert the improper fraction 5/4 into a mixed fraction.

Solution:

Given, 5/4

To convert an improper fraction into a mixed fraction, first, divide the number 5 by 4.

If we divide 5 by 4, we get the quotient as 1 and the remainder as 1.

Hence, keep the quotient as the whole number part and the remainder as the fraction’s numerator part and the denominator as 4.

Hence, 5/4 written in the mixed fraction is 1  1/4.

i.e., 5/4 = 1  1/4.

Example 3:

Convert the mixed number 4  2/3 into an improper fraction.

Solution:

Given: Mixed fraction = 4  2/3

To convert the mixed fraction into an improper fraction, follow the below steps:

Step 1: Multiply 3 by 4, and we get 12.

Step 2: Now, add 12 and 2, we get 14

I,e 3(4) + 2 =14

Now, put the number 14 in the fraction’s numerator and keep the number 3 as the denominator.

So, 14/3 is the required fraction.

Example 4:

Write 3 1/4 as an improper fraction.

Solution:

Given that 3 1/4 is a mixed number.

Now, multiply denominator 4 by the whole number 3.

4 × 3 = 12

Now, add 12 to the numerator.

12 + 1 = 13

Hence, 13 is the new numerator of the improper fraction, whereas the denominator remains the same.

So,

3 1/4 = 13/4

&

Win