Question

Show that in the denominator Addition /Subtraction property of proportions : If$\dfrac{a}{b}=\dfrac{c}{d}$ then$\dfrac{a+b}{b}=\dfrac{c+d}{d}$ or $\dfrac{a-b}{b}=\dfrac{c-d}{d}$ .

Answer 1

Hint: In this question, let us start solving from what we are asked to prove. We have to prove it using the piece of information that we are given which is $\dfrac{a}{b}=\dfrac{c}{d}$. Let us not cross-multiply and prove it in both the cases. But we will just share the denominator to both parts of the numerator and then use the information that we are given to prove both.

Complete step by step answer:
Let us first look at the expression $\dfrac{a+b}{b}=\dfrac{c+d}{d}$.
Let us share the denominator and separate them.
Upon doing so, we get the following :
$\begin{align}
  & \Rightarrow \dfrac{a+b}{b}=\dfrac{c+d}{d} \\
 & \Rightarrow \dfrac{a}{b}+\dfrac{b}{b}=\dfrac{c}{d}+\dfrac{d}{d} \\
\end{align}$
We know that $\dfrac{b}{b}$ or $\dfrac{d}{d}$ is equal to $1$ .
So the expression changes as the following :
$\begin{align}
  & \Rightarrow \dfrac{a}{b}+1=\dfrac{c}{d}+1 \\
\end{align}$
Let us bring the $1$ on the left hand side to the right hand side.
Upon doing so, they cancel each other and we get the following :
$\begin{align}
 & \Rightarrow \dfrac{a}{b}=\dfrac{c}{d}+1-1 \\
 & \Rightarrow \dfrac{a}{b}=\dfrac{c}{d} \\
\end{align}$
We already know that $\dfrac{a}{b}=\dfrac{c}{d}$ as it is already mentioned in the question.
Now let us look at the other expression $\dfrac{a-b}{b}=\dfrac{c-d}{d}$.
Let us repeat what we did to the first expression for the proof.
As we did, we will first share the denominator and separate them.
Upon doing so, we get the following :
$\begin{align}
  & \Rightarrow \dfrac{a-b}{b}=\dfrac{c-d}{d} \\
 & \Rightarrow \dfrac{a}{b}-\dfrac{b}{b}=\dfrac{c}{d}-\dfrac{d}{d} \\
\end{align}$
We know that $-\dfrac{b}{b}$ or $-\dfrac{d}{d}$ is equal to $-1$ .
So the expression changes as the following :
$\begin{align}
 & \Rightarrow \dfrac{a}{b}-1=\dfrac{c}{d}-1 \\
\end{align}$
Let us bring the $-1$ on the left hand side to the right hand side.
Upon doing so, they cancel each other and we get the following :
\[\begin{align}
 & \Rightarrow \dfrac{a}{b}=\dfrac{c}{d}+1-1 \\
 & \Rightarrow \dfrac{a}{b}=\dfrac{c}{d} \\
\end{align}\]
We already know that $\dfrac{a}{b}=\dfrac{c}{d}$as it is already mentioned in the question
Hence proved.

Note: It is very important to learn how to approach for the proof sums. A lot of practice is required for that. We should be careful while solving as it may lead to calculation errors. There is also another concept called the componendo and dividendo which shares the same principle as the above proof. It states that if $\dfrac{a}{b}=\dfrac{c}{d}$then upon using componendo and dividendo we get $\dfrac{a+b}{a-b}=\dfrac{c+d}{c-d}$ . This is a safe manipulation technique used especially in algebra and trigonometry.