Question

If a = 1/x+1/y , what is the value of 1/a

Answer 1

We are given that a =(1/x) + (1/y) , and need to find the value of (1/a) , as a is the sum of two fractions so we can't reciprocal both sides directly without finding their sum and converting them into a single fraction . So , let's start by simplifying a first

${:\implies \quad \sf a=\dfrac{1}{x}+\dfrac{1}{y}}$

${:\implies \quad \sf a=\dfrac{x+y}{xy}}$

Now , we knows an indentity ;

• ${\boxed{\bf{{\left(\dfrac{a}{b}\right)}^{-1}=\dfrac{b}{a}}}}$

Now , raising to the power -1 on both sides :

${:\implies \quad \sf {\left(\dfrac{a}{1}\right)}^{-1}={\left(\dfrac{x+y}{xy}\right)}^{-1}}$

${:\implies \quad \bf \therefore \quad \underline{\underline{\dfrac{1}{a}=\dfrac{xy}{x+y}}}}$

Answer 2

Solution

$a=\frac{1}{x}+\frac{1}{y}$

so the value of 1/a is…

$\begin{aligned}\frac{1}{a}&=\frac{1}{\frac{1}{x}+\frac{1}{y}}\\&\Rightarrow\frac{1}{\frac{y+x}{xy}}\\&\Rightarrow1\times\frac{xy}{y+x}\\&\Rightarrow\frac{xy}{x+y}\end{aligned}$