A complex fraction is a fraction in which either the numerator or the denominator or both contain a fraction. When we find this kind of complexity in a fraction, we have to simplify it to a proper or improper fraction before we perform any kind of operation.

## What is a Complex Fraction?

According to definition, a Complex Fraction should have a fraction either in the numerator or the denominator of the fraction. It can also have a fraction in both the numerator and the denominator.

Points to Remember While Solving a Complex Fraction:

If the fraction is obtained in the numerator, then write '1' as the denominator in the denominator fraction and multiply the numerator by the reciprocal of the denominator and simplify it further.
$\rightarrow$ If the fraction obtained in the denominator, then write '1' as the denominator in the numerator fraction and multiply the numerator by the reciprocal of the denominator and simplify it further.
$\rightarrow$ If the fraction obtained in both the numerator and the denominator, then multiply the numerator by the reciprocal of the denominator and then simplify it further.

The following steps need to be followed while adding complex fractions.

Step 1: Convert the complex fraction into a normal fraction by multiplying the numerator by the reciprocal of the denominator.
Step 2: Now, find the lowest or least common multiple of the denominators (LCD) of the fractions.
Step 3: Express the fractions as equivalent fractions with the denominator as the LCD.
Step 4: Add the numerators of the given fractions.
Step 5: Simplify the fraction if necessary.

### Example on Adding Complex Fractions:

Find the sum of $\frac{4/8}{10}$+ $\frac{8/9}{5}$ ?

Solution:

Step 1: Multiply the numerator by the reciprocal of the denominator fraction.

$\frac{4/8}{10/1} \rightarrow \frac{4}{8} \times \frac{1}{10} \rightarrow \frac{4}{80}$

$\frac{8/9}{5/1} \rightarrow \frac{8}{9} \times \frac{1}{5} \rightarrow \frac{8}{45}$

Step 2: LCD of the fractions = least common multiple (LCM) of 80 and 45 is = 720

Step 3: Express the fractions as equivalent fractions with the common denominator 720

$\frac{4}{80} = \frac{4 \times 9}{80 \times 9} = \frac{36}{720}$
$\frac{8}{45} = \frac{8 \times 16}{45 \times 16} = \frac{128}{720}$

$\rightarrow$ $\frac{36}{720} + \frac{128}{720}$

$\rightarrow \frac{(36 + 128)}{720}$

$\rightarrow \frac{164}{720}$

Step 5: Divide the numerator and denominator by 4

$\rightarrow \frac{(164 ÷ 4)}{(720 ÷ 4)}$

= $\frac{41}{180}$

## Subtracting Complex Fractions

The following steps need to be followed while Subtracting Complex Fractions:

Step 1: Convert complex fractions into normal fractions by multiplying the numerator by the reciprocal of the denominator.
Step 2: Now, find the lowest or least common multiple of the denominators (LCD) of the fractions.
Step 3: Express the fractions as equivalent fractions with the denominator as the LCD.
Step 4: Subtract the numerators of the given fractions.
Step 5: Simplify the fraction if necessary.

### Example on Subtracting Complex Fractions:

Find the Difference of $\frac{4/9}{8} + \frac{4/7}{9}$ ?

Solution:

Step 1: Multiply the numerator by the reciprocal of the denominator fraction.

$\frac{4/9}{8/1} \rightarrow \frac{4}{9} \times \frac{1}{8} \rightarrow \frac{4}{72}$

$\frac{4/7}{9/1} \rightarrow \frac{4}{7} \times \frac{1}{9} \rightarrow \frac{4}{63}$

Step 2: LCD of the fractions = least common multiple (LCM) of 72 and 63 = 504

Step 3: Express the fractions as equivalent fractions with the common denominator 504

$\frac{4}{72} = \frac{(4 \times 7)}{(72 \times 7)} = \frac{28}{504}$
$\frac{4}{63} = \frac{(4 \times 8)}{(63 \times 8)} = \frac{32}{504}$

Step 4: Subtract the numerators

$\rightarrow \frac{28}{504} - \frac{32}{504}$

$\rightarrow \frac{(28 - 32)}{504}$

$\rightarrow \frac{-4}{504}$

Step 5: Divide the numerator and denominator by 4

$\rightarrow \frac{(4 ÷ 4)}{(504 ÷ 4)}$

= $\frac{1}{126}$

## Multiplying Complex Fractions

The following steps need to be followed while multiplying complex fractions.

Step 1: Convert complex fraction into normal fractions by multiplying the numerator by the reciprocal of the denominator.
Step 2: Now, multiply the numerators (top number) and the denominator of the new fractions.
Step 3: Simplify the fraction if necessary.

### Example on Multiplying Complex Fractions:

Simplify $\frac{3/4}{9/5}$ X $\frac{5/3}{7/4}$

Solution:

Step 1: Multiply the numerator by the reciprocal of the denominator fraction.

$\frac{3/4}{9/5} \rightarrow \frac{3}{4} \times \frac{5}{9} \rightarrow \frac{15}{36}$

$\frac{5/3}{7/4} \rightarrow \frac{5}{3} \times \frac{7}{4} \rightarrow \frac{20}{21}$

Step 2: Multiply the numerator (top number) and denominator of the new fraction.

$\frac{15}{36} \times \frac{20}{21} = \frac{(15 \times 20)}{(36 \times 21)} = \frac{300}{756}$

Step 3: Divide the numerator and denominator by 12 $\rightarrow \frac{(300 ÷ 12)}{(756 ÷ 12)} = \frac{25}{63}$

## Dividing Complex Fractions

The following steps need to be followed while dividing complex fractions.

Step 1: Convert complex fractions into a normal fraction by multiplying the numerator by the reciprocal of the denominator.
Step 2: Find the reciprocal of the second fraction (the one you want to divide by) and change the division sign to multiplication sign.
Step 3: Multiply the numerator and denominator of both the new fractions.
Step 4: Simplify the fraction if necessary.

### Example on Dividing Complex Fractions

Divide $\frac{2/9}{5/4}$ with $\frac{3/5}{2/3}$

Solution :

Step 1: Multiply the numerator by the reciprocal of the denominator fraction.

$\frac{2/9}{5/4} \rightarrow \frac{2}{9} \times \frac{4}{5} \rightarrow \frac{8}{45}$

$\frac{3/5}{2/3} \rightarrow \frac{3}{5} \times \frac{3}{2} \rightarrow \frac{9}{10}$

Step 2: Find the reciprocal of the second fraction (the one you want to divide) and change the division sign to multiplication sign.

$\frac{8}{45} \times \frac{10}{9}$

Step 3: Multiply the numerator and denominators of both the new fractions:

$\frac{8}{45} \times \frac{10}{9} = \frac{(8 \times 10)}{(45 \times 9)} = \frac{80}{405}$

Step 4: Divide the numerator and denominator by 5 $\rightarrow \frac{(80 ÷ 5)}{(405 ÷ 5)} = \frac{16}{81}$

## Simplifying Complex Fractions

Since in all Complex Fractions we will have a fraction in the both numerator and the denominator or in any one of them, we should simplify the complex fraction first before we perform any operation like addition, subtraction, multiplication or division on it.

### Problems on Simplifying Complex fractions

1) Find the sum of $\frac{2}{1/4} + \frac{3}{5/7}$ ?

Solution:

Step 1: Multiply the numerator by the reciprocal of the denominator fraction.

$\frac{2/1}{1/4} \rightarrow \frac{2}{1} \times \frac{4}{1} \rightarrow \frac{8}{1}$

$\frac{3/5}{7/1} \rightarrow \frac{3}{5} \times \frac{1}{7} \rightarrow \frac{3}{35}$

Step 2: LCD of the fractions = 35

Step 3: Express the fractions as equivalent fractions with the common denominator 35

$\frac{8}{1} = \frac{(8 \times 35)}{(1 \times 35)} = \frac{280}{35}$

$\frac{3}{35} = \frac{(3 \times 1)}{(35 \times 1)} = \frac{3}{35}$

$\frac{280}{35} + \frac{3}{35} \rightarrow \frac{(280 + 3)}{35} \rightarrow \frac{283}{35}$

Step 5: $\frac{283}{35}$

2) Find the sum of $\frac{8/7}{9/7}$+ $\frac{1/5}{9/8}$ ?

Solution:

Step 1: Multiply the numerator by the reciprocal of the denominator fraction.

$\frac{2/1}{1/4} \rightarrow \frac{2}{1} \times \frac{4}{1} \rightarrow \frac{8}{1}$

$\frac{3/5}{7/1} \rightarrow \frac{3}{5} \times \frac{1}{7} \rightarrow \frac{3}{35}$

Step 2: LCD of the fractions = 35

Step 3: Express the fractions as equivalent fractions with the common denominator 35

$\frac{8}{1} = \frac{8}{35} = \frac{280}{35}$

$\frac{3}{35} = \frac{3 \times 1}{35 \times 1} = \frac{3}{35}$

$\frac{280}{35} + \frac{3}{35} \rightarrow \frac{(280 + 3)}{35} \rightarrow \frac{283}{35}$

Step 5: $\frac{283}{35}$

3) Simplify $\frac{7/3}{8/7}$ X $\frac{5/2}{5/7}$

Solution:

Step 1: Multiply the numerator by the reciprocal of the denominator fraction.

$\frac{7/3}{8/7} \rightarrow \frac{7}{3} \times \frac{7}{8} \rightarrow \frac{49}{24}$

$\frac{5/2}{5/7} \rightarrow \frac{5}{2} \times \frac{7}{5} \rightarrow \frac{7}{2}$

Step 2: Multiply the numerator (top number) and denominator of the new fraction.

$\frac{49}{24} \times \frac{7}{2} = \frac{(49 x 7)}{(24 x 2)} = \frac{343}{48}$

Step 3: $\frac{343}{48}$

4) Divide $\frac{8/3}{2/3}$ with $\frac{9/5}{4/3}$

Solution :

Step 1: Multiply the numerator by the reciprocal of the denominator fraction.

$\frac{8/3}{2/3} \rightarrow \frac{8}{3} \times \frac{3}{2} \rightarrow \frac{4}{1}$

$\frac{9/5}{4/3} \rightarrow \frac{9}{5} \times \frac{3}{4} \rightarrow \frac{27}{20}$

Step 2: Find the reciprocal of the second fraction (the one you want to divide by) and change the division sign to multiplication sign.

$\frac{4}{1} \times \frac{20}{27}$

Step 3: Multiply the numerator and denominator of both the new fractions:

$\frac{4}{1} \times \frac{20}{27}$ = $\frac{(4 \times 20)}{(1 \times 27)}$ = $\frac{80}{27}$

Step 4:$\frac{80}{27}$