Question

How do you write a complex fraction that when simplified results in $ \dfrac{1}{x} $?

Answer 1

Hint: In order to determine a complex fraction which results in $ \dfrac{1}{x} $ , remember that every complex fraction $ \dfrac{{\dfrac{a}{b}}}{{\dfrac{c}{d}}} $ can be written in the product form as $ \left( {\dfrac{a}{b}} \right)\left( {\dfrac{d}{c}} \right) $ . To obtain the denominator as $ x $ , set \[b = d.x\] and to make the result of numerator 1 , set $ a = c $ . $ a,b,c,d $ can be any values satisfying the both the conditions i.e. $ a = c $ and \[b = d.x\].

Complete step by step solution:
We are given a fraction in variables $ x $ as $ \dfrac{1}{x} $ .
As per the question we have to determine a complex fraction which when simplified result in $ \dfrac{1}{x} $ .
Before proceeding to the actual solution, let’s understand what is a complex fraction.
Remember, dividing two fractions is actually the same thing as multiplying the 1st fraction with the reciprocal of 2nd fraction.
Suppose we have a fraction as $ \dfrac{{\dfrac{a}{b}}}{{\dfrac{c}{d}}} $ , the equivalent of this complex fraction will be same as:
 $ \Rightarrow \left( {\dfrac{a}{b}} \right)\left( {\dfrac{d}{c}} \right) $ This form is easier to understand and also to solve.
Now in this question, the numerator of the resulting fraction should be equal to 1.So as per the requirement we have to set $ a = c $ so that they can be cancelled out easily to make the numerator equals 1.
Next if we see the denominator part of the resulting fraction, we have $ x $ . So to obtain the denominator of the result equal to $ x $ , we have to set \[b = d.x\]. So that when we simplify this, it results in $ x $ .
For example let's assume $ a = c = x + 9 $ and $ b = {x^2} $ $ d = x $ , then the complex fraction will look like
 $ \Rightarrow \dfrac{{\dfrac{{x + 9}}{{{x^2}}}}}{{\dfrac{{x + 9}}{x}}} $
We can rewrite the above complex fraction in simple product as
 $ \Rightarrow \dfrac{{x + 9}}{{{x^2}}} \times \dfrac{x}{{x + 9}} $
Simplifying further, by cancelling out $ \left( {x + 9} \right) $ and divide $ x\, $ with $ {x^2} $ which will give $ x $ in denominator ,we get
 $ \Rightarrow \dfrac{1}{x} $
Therefore, the required complex fraction is $ \dfrac{{\dfrac{{x + 9}}{{{x^2}}}}}{{\dfrac{{x + 9}}{x}}} $ .

Note: 1. Students can take expressions for $ a,b,c,d $ of their choice but values must satisfy the condition that $ a = c $ and \[b = d.x\] such that the simplification results in $ \dfrac{1}{x} $ .
2.The solution is just a sample solution , you can derive as many complex functions by following the steps .
3. Avoid jumping off steps while performing simplification.