Complex Fraction
How to solve a complex fraction: formula, 2 examples, and their solutions.
Formula
A complex fraction is a fraction
whose numerator or denominator is also a fraction
(or both are also fractions).
To solve a complex fraction [a/b]/[c/d],
write the product of the outer factors,
ad,
in the numerator
and write the product of the inner factors,
bc,
in the denominator:
[a/b]/[c/d] = ad/bc.
Example[(x + 2)/x] / [6/(x - 1)]
Write the product of the outer factors,
(x + 2)(x - 1),
in the numerator.
And write the product of the inner factors,
x⋅6 = 6x,
in the denominator.
So
(x + 2)(x - 1)/6x
is the answer.
Example[4x - 1/x] / (2x - 1)2
To solve the given complex fraction,
first combine the numerator part 4x - 1/x.
4x = 4x2/x
4x2/x - 1/x
= [4x2 - 1]/x
4x2 - 1
= (2x)2 - 12
(2x)2 - 12 = (2x + 1)(2x - 1)
Factor the Difference of Two Squares: a2 - b2
4x - 1/x
= (2x + 1)(2x - 1)/x
So (given) = [(2x + 1)(2x - 1)/x] / [(2x - 1)2/1].
To use the complex fraction formula,
change (2x - 1)2 to (2x - 1)2/1.
Cancel the common factor (2x - 1)
in both of
the main numerator and the main denominator.
(2x + 1)(2x - 1)/x → (2x + 1)/x
(2x - 1)2/1 → (2x - 1)/1
Solve the complex fraction.
Write the product of the outer factors,
(2x + 1)⋅1,
in the numerator.
And write the product of the inner factors,
x(2x - 1),
in the denominator.
(2x + 1)⋅1 = 2x + 1
So
(2x + 1)/[x(2x - 1)]
is the answer.