Complex Fractions

Definition: A complex fraction is a fraction whose numerator contains a fraction or whose denominator contains a fraction or both numerator and denominator contains a fraction.

Remark: Current Web constraints limit the readability of mathematical material. Complex fractions are particularly hard to denote in text format. Usually the main fraction bar in a complex fraction is the longest fraction bar or is the fraction bar denoted in bold type. On this page the main fraction will be denoted in bold type.

Examples:

(3/5)/2                   Numerator: 3/5           Denominator: 2

x/(2/3)                   Numerator: x             Denominator: 2/3

(1/5)/(2/7)               Numerator: 1/5           Denominator: 2/7

[(x²  - 1)/6]/[(x - 1)/2]  Numerator: (x²  - 1)/6    Denominator: (x - 1)/2

[1 + 1/2 ]/[1 - 3/4 ]     Numerator: 1 + 1/2       Denominator: 1 - 3/4 

There are two ways to simplify a complex fraction.

• Method 1: Divide numerator by denominator.
• Method 2: Use the fundamental theorem of fractions.

Divide numerator, 3/5, by denominator, 2.
(3/5)/2 = (3/5)(1/2) = 3/10

Divide numerator, x, by denominatoor, 2/3.
x/(2/3) = (x/1)(3/2) = (3x)/2

Divide numerator, 1/5, by denominator, 2/7.
(1/5)/(2/7) = (1/5)(7/2) = 7/10

Divide numerator, (x²  - 1)/6, by denominator, (x - 1)/2.
[(x²  - 1)/6]/[(x - 1)/2] = [(x²  - 1)/6][2/(x - 1)]
= [(x + 1)(x - 1)/6][2/(x - 1)] = (x + 1)/3

If numerator and denominator are not single fractions we write
numerator and denominator as a single fraction before we divide
numerator, 1 + ½ , by denominator 1 - ¾ .
[1 + 1/2]/[1 - 3/4] = [ 2/2 + 1/2}/[4/4 - 3/4] = [3/2]/[1/4] = [3/2][4/1] = 6/1 = 6

(3/5)/2       When there is only one simple fraction involved in
either numerator or denominator we multiply the numerator and denominator
by the denominator of that fraction.  In this case we multiply numerator
and denominator by 5.

(3/5)/2 = [(3/5)(5)]/[(2)(5)] = 3/10

x/(2/3)       When there is only one simple fraction involved in
either numerator or denominator we multiply the numerator and denominator
by the denominator of that fraction.  In this case we multiply numerator
and denominator by 3.

x/(2/3) = [x(3)]/[(2/3)(3)] = (3x)/2


(1/5)/(2/7)   When both numerator and denominator involve simple
fractions we multiply both numerator and denominator by the LCD, Least 
Common Denominator, of the two fractions.  In this case we multiply numerator
and denominator by 35.

(1/5)/(2/7) = [(1/5)(35)]/[(2/7)(35)] = 7/10

[(x²  - 1)/6]/[(x - 1)/2]  When both numerator and denominator involve simple
fractions we multiply both numerator and denominator by the LCD, Least 
Common Denominator, of the two fractions.  In this case we multiply numerator
and denominator by 6.
 
[(x²  - 1)/6]/[(x - 1)/2] = [((x²  - 1)/6)(6)]/[((x - 1)/2)(6)]
=[x²  - 1]/[3(x - 1)] = [(x + 1)(x - 1)]/[3(x - 1)] = (x + 1)/3  

[1 + ½ ]/[1 - ¾ ]  When both numerator and denominator involve simple
fractions we multiply both numerator and denominator by the LCD, Least 
Common Denominator, of the two fractions.  In this case we multiply numerator
and denominator by 4. 

[1 + ½ ]/[1 - ¾ ] = [(1 + ½ )(4)]/[(1 - ¾ )(4)] = [4 + 2]/[4 - 3] = 6/1 = 6

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