# How to Simplify Complex Fractions?

• Last Updated : 23 Aug, 2022

Fractions are defined as numerical figure that represents a portion of a whole. A fraction is a portion or section of any quantity taken from a whole, which might be any number, a specified value, or an item.

Every fraction has a numerator and a denominator separated by a horizontal bar known as the fractional bar.

• The number of parts into which the whole has been divided is indicated by the denominator. It is placed below the fractional bar at the fraction lower part.
• The numerator indicates how many fractional parts are depicted or selected. It is placed above the fractional bar at the fraction upper part.

Examples: 2/3, 5/4, 9/8 etc

## Complex Fraction

A complex fraction is one in which the denominator and numerator, or both, include fractions. A complex rational expression is a complex fraction with a variable.

For Example:

• 4/(1/3) is a complex fraction in which 4 is the numerator and 1/3 is the denominator.
• (2/7)/8 is also a complex fraction, with the numerator and denominator being 2/7 and 8, respectively.
• (5/4)/(2/10) is another complicated fraction having a numerator of 5/4 and a denominator of 2/10.

## How to Simplify Complex Fractions?

Solution: To simplify complex fractions, we have two methods:

### Method 1

Step 1: Create a single fraction from both the denominator and the numerator.

Step 2: Apply the division rule by multiplying the top of the fraction by the reciprocal of the bottom.

Step 3: Simplify the fraction to its simplest terms.

Example: Simplify Complex fraction (5/2)/(2/4). (Method 1)

Solution:

Given: (5/2)/(2/4)

Step 1: Create a single fraction from both the denominator and the numerator.

Step 2: Apply the division rule by multiplying the top of the fraction by the reciprocal of the bottom

= (5/2)/(2/4)                                                    { Reciprocal of denominator 2/4 = 4/2 }

Therefore,

= 5/2 × 4/2

=  20/4

Step 3: Simplify the fraction to its simplest terms.

= 20/4

= 5

### Method 2

Step 1: Begin by calculating the Least Common Multiple of each denominator in the complex fractions.

Step 2: Multiply the complex fraction’s numerator and denominator by this L.C.M.

Step 3: Reduce the result to the simplest terms feasible.

Example: Simplify Complex fraction (5/2)/(2/4). (Method 2)

Solution:

Given: (5/2)/(2/4)

Step 1: Begin by calculating the Least Common Multiple of each denominator in the complex fractions.

So we have (5/2)/(2/4)

LCM of denominator 2 and 4 is  4

Step 2: Multiply the complex fraction’s numerator and denominator by this L.C.M i.e 4

= {(5/2) × 4 } / {(2/4) × 4}

= (5×2)/(2)

= 10/2

Step 3:  Reduce the result to the simplest terms feasible.

= 10/2

= 5

## Solved Examples based on Complex Fractions

Example 1: Simplify {(1 + 1/x) / (1 -1/x) } ?

Solution:

Given: {(1 + 1/x) / (1 -1/x) }

Step 1: Create a single fraction from both the denominator and the numerator.

=   {(1 + 1/x) / (1 -1/x) }

= [{(x+1)/x } / {(x-1)/x}]

Step 2: Apply the division rule by multiplying the top of the fraction by the reciprocal of the denominator

= [{(x+1)/x } / {(x-1)/x}]                        { Reciprocal of denominator {(x-1)/x} is {x /(x-1)}   }

Therefore ,

=  [{(x+1)/x } × {x /(x-1)} ]

=  {(x+1)x / {x(x-1) }

Step 3:  Reduce the result to the simplest terms feasible.

= {(x+1)x / {x(x-1) }

= (x+1)/(x-1)

Example 2: Simplify Complex fraction (40/3)/(10/12).

Solution:

Given: (40/3)/(10/12)

Step 1: Create a single fraction from both the denominator and the numerator.

Step 2: Apply the division rule by multiplying the top of the fraction by the reciprocal of the bottom

= (40/3)/(10/12)                                          { Reciprocal of denominator 10/12 = 12/10 }

Therefore ,

= 40/3 × 12/10

=  480 /30

Step 3: Simplify the fraction to its simplest terms.

= 480/30

= 16

Example 3: Simplify {(4+2x)/x}/ (2/x).

Solution:

Given fraction: {(4+2x)/x}/ (2/x)

Apply the division rule by multiplying the top of the fraction by the reciprocal of the bottom

=  {(4+2x)/x}/ (2/x)

=  (4+2x)/x} × (x/2)

=  (4+2x)/2

=  {2(2+x)}/2

= 2+x

Example 4: Simplify Complex fraction (5)/(15/6).

Solution:

Given: (5)/(15/6)

Step 1: Create a single fraction from both the denominator and the numerator.

Step 2: Apply the division rule by multiplying the top of the fraction by the reciprocal of the bottom

= (5/1)/(15/6)                                       { Reciprocal of denominator 15/6 = 6/15 }

Therefore ,

= 5/1 × 6/15

=  30/15

Step 3: Simplify the fraction to its simplest terms.

= 30/15

= 2

## Frequently Asked Questions on Fractions

Question 1: What are fractions in Maths?

Fractions are the numerical values that are a part of the whole. A whole can be an object or a group of objects. If a number or a thing is divided into equal parts, then each part will be a fraction of the whole. A fraction is denoted as a/b, where a is the numerator and b is the denominator.

Question 2: How to solve fractions?

To add or subtract fractions, we have to check if the denominators are the same or different. For the same denominators, we can directly add or subtract the numerators, keeping the denominator common. But if the denominators are different, then we need to simplify them by finding the LCM.

Question 3: What are the 3 types of fractions in Maths?

The 3 types of fractions in Maths are Proper fractions, Improper fractions, and mixed fractions.

Question 4: Give real-life examples of fractions.

If a watermelon is divided into four equal parts, then each part is a fraction of ¼.
Similarly, if a pizza is divided into three equal parts, then each part shows 1/3rd of the pizza.

Questions 5: What is a unit fraction?