Question

Find the value of $\dfrac{p+2x}{p-2x}+\dfrac{p+2y}{p-2y}$ if $p=\dfrac{4xy}{x+y}$.

Answer 1

Hint: We first explain the expression of the given function where $p=\dfrac{4xy}{x+y}$. We find the individual form of $\dfrac{p}{2x}$ and $\dfrac{p}{2y}$. Then we use the componendo-dividendo formula to find the value of the terms in the summation of $\dfrac{p+2x}{p-2x}+\dfrac{p+2y}{p-2y}$. We find the final solution.

Complete step by step solution:
We have been given $p=\dfrac{4xy}{x+y}$. To find the summation form of $\dfrac{p+2x}{p-2x}+\dfrac{p+2y}{p-2y}$, we first need to find the individual form of $\dfrac{p}{2x}$ and $\dfrac{p}{2y}$.
Therefore, $p=\dfrac{2x\times 2y}{x+y}\Rightarrow \dfrac{p}{2x}=\dfrac{2y}{x+y}$ and $p=\dfrac{2x\times 2y}{x+y}\Rightarrow \dfrac{p}{2y}=\dfrac{2x}{x+y}$
Now we use the componendo-dividendo formula on both relations.
Therefore, from $\dfrac{p}{2x}=\dfrac{2y}{x+y}$ we get $\dfrac{p+2x}{p-2x}=\dfrac{2y+x+y}{2y-x-y}$.
Simplifying, we get $\dfrac{p+2x}{p-2x}=\dfrac{x+3y}{y-x}$.
From $\dfrac{p}{2y}=\dfrac{2x}{x+y}$ we get $\dfrac{p+2y}{p-2y}=\dfrac{2x+x+y}{2x-x-y}$.
Simplifying, we get $\dfrac{p+2y}{p-2y}=\dfrac{3x+y}{x-y}$.
Now we complete the summation of the $\dfrac{p+2x}{p-2x}+\dfrac{p+2y}{p-2y}$.
So, $\dfrac{p+2x}{p-2x}+\dfrac{p+2y}{p-2y}=\dfrac{x+3y}{y-x}+\dfrac{3x+y}{x-y}$.
We can convert the denominators into the same form.
We get $\dfrac{p+2x}{p-2x}+\dfrac{p+2y}{p-2y}=\dfrac{x+3y}{y-x}-\dfrac{3x+y}{y-x}$.
We conduct the subtraction in the numerators and get
$\dfrac{p+2x}{p-2x}+\dfrac{p+2y}{p-2y}=\dfrac{x+3y}{y-x}-\dfrac{3x+y}{y-x}=\dfrac{x+3y-3x-y}{y-x}$.
The simplified form becomes $\dfrac{p+2x}{p-2x}+\dfrac{p+2y}{p-2y}=\dfrac{2y-2x}{y-x}$.
We take 2 common from the numerator to get $\dfrac{p+2x}{p-2x}+\dfrac{p+2y}{p-2y}=\dfrac{2\left( y-x \right)}{y-x}$.
So, $\dfrac{p+2x}{p-2x}+\dfrac{p+2y}{p-2y}=2$,.
Therefore, the value of $\dfrac{p+2x}{p-2x}+\dfrac{p+2y}{p-2y}$ is 2 if $p=\dfrac{4xy}{x+y}$.

Note: Componendo and dividendo is a theorem on proportions that allows for a quick way to perform calculations and reduce the number of expansions needed. The ratio or the proportionality gives the increased value of the numerator and denominator by 1.