A Complex Fraction can be defined as the ratio of two Rational numbers where both Numerator and Denominator are represented in Ratio. or in other words, a complex fraction is a rational expression that has a fraction in its numerator, denominator, or both. Some examples of Complex Fraction are: (a /b)/(c/d), 4/(1/2), (1/3)/(2/5), (4 + 1/5)/(1 – 3/2).
Types of Complex fractions
There are majorly three types of complex fractions. They are proper fractions, improper fractions, and mixed fractions. Let’s learn about these three types with examples and basic definitions,
- Proper fraction: Denominator > Numerator (D > N)
Example: 3/6, 5/7, 2/9
- Improper fraction: Numerator > Denominator (N > D)
Example: 7/2, 5/3, 6/5
Mixed fraction: Represented in the form of q – R/D. Where q = Quotient, R = remainder, D = Divisor.
Example: 2 – 1/2 (Read aloud as “two-and-a-half”)
In order to add 2 complex Fractions first, it is needed to convert them into simple fractions.
- Dividing negative number
(N/D)/(-N/D) = N/D × D/-N = -1. Here, N & D represents Numerator and Denominator of fractional number.
Examples: (-1/3)/(-2/3) = -1/3 × -3/2 = +1/2.
Complex Fraction to Proper Fraction
Let’s understand how to convert complex fractions into proper fractions by an example, take (4 + 1/5)/(1 – 3/2) is a complex fraction were (4 + 1/5) is Numerator and (1 – 3/2) is Denominator.
- Method 1: By division Rule
Step-1: Simplify Numerator and Denominator in a single fraction.
Solution:
(4 + 1/5)/(1 – 3/2) = (4/1 + 1/5)/(1/1 – 3/2)
(4 + 1/5)/(1 – 3/2) = (21/5)/(-1/2)
- Step-2: Keep numerator as it is and then multiply numerator by the reciprocal of the denominator.
(4 + 1/5)/(1 – 3/2) = (21/5) / (-1/2)
(4 + 1/5)/(1 – 3/2 ) = (21/5 ) × (-2/1)
(4 + 1/5)/(1 – 3/2) = (21 × -2)/(5 × 1)
(4 + 1/5)/(1 – 3/2 ) = -42/5
- Method-2: By LCM of Denominator
This is the easiest method of simplifying complex fractions. Here are the steps for this method:
Example: (2/5) % (3/10) = (2/5) / (3/10)
- Step-1: Start by finding the Least Common Multiple of all the denominator in the complex fractions,
LCM(5, 10) = 10
- Step-2: Multiply the both the numerator and denominator of the complex fraction by this L.C.M.
(2/5) / (3/10) = (2/5 × 10) /(3/10 × 10)
- Step-3: Simplify the result to the lowest terms possible.
(2/5) / (3/10) = 4/3
Addition of 2 fraction numbers
There are 2 types of fractions one is Like fractions while the other is Unlike fraction. Examples are, 1/2 & 3/2 are in a Like fraction because their Denominators are the same. 3/4 & 1/3 are Unlike fractions due to different Denominators.
- Addition of two Like fraction
It involves 2 steps:
Step-1: Simply add the numerators of both numbers because a bottom number is already the same or common.
Example: 1/4 + 3/4 = (1 + 3)/4
=4/4
Step -2: Simplify fractions as much as possible.
1/4 + 3/4 = 1/1
- Addition of two Unlike fraction
For adding two Unlike fraction first you need to make it as Like fraction by making the base or denominator the same.
Step-1: To make the base the same Multiply top and bottom of each fraction by the denominator of the other.
Example: 1/3 + 1/5 = (1 × 5)/(3 × 5) + (1 × 3)/(5 × 3)
1/3 + 1/5 = 5/15 + 3/15
Step-2: Now the base is the same, repeat the process discussed above.
1/3 + 1/5 = (5 + 3)/ 15
1/3 + 1/5 = 8/15
Sample problems
Question 1: Solve, (x + 3)/12 / (4x – 5)/15
Solution:
(x + 3) / 12 × 15/(4x – 5)
= (x + 3) × 15 /(4x – 5) × 12
= 5(x + 3) / 4(4x – 5)
Question 2: Solve, (15/2x) / (5/3x)
Solution:
15/2x × 3x/5
= (15 × 3x) / (2x × 5)
= 9/2
Question 3: (1 – x/y) / (y2/x2 – 1)
Solution:
(y – x)/y / (y2 – x2)/x2
= (y – x) × x2 / y × (y2 – x2)
= x2 / y(y + x)
Question 4: (x/9 – 1/3) / (x – 3)/6
Solution:
(x – 3)/LCM(9, 3) / (x – 3)/6
= (x – 3)/9 / (x – 3)/6
= 2/3
Question 5: (a-1 + 2) / (a-1 – 2)
Solution:
(1/a + 2) / (1/a – 2)
= (1 + 2a)/a / (1 – 2a)/a
= ((1 + 2a) × a) / ((1 – 2a) × a)
= (1 + 2a) / (1 – 2a)