State true or false.
If $\left( {4a + 9b} \right)\left( {4c - 9d} \right) = \left( {4a - 9b} \right)\left( {4c + 9d} \right)$, then $a:b = c:d$.
A.True
B.False

Hint: We will expand the terms and solve the brackets of both sides of the given equation. We will then combine the terms with same coefficients and cancel whatever gets cancelled and finally reduce it into the terms having the variables $ad$ and $bc$. By re – arranging them, we can check if the given statement is true or not.

Complete step-by-step answer:
We are given a statement that: If $\left( {4a + 9b} \right)\left( {4c - 9d} \right) = \left( {4a - 9b} \right)\left( {4c + 9d} \right)$, then $a:b = c:d$.
We are required to check if this is true or false.
Let us simplify the given equation by opening the brackets. We get
$ \Rightarrow \left( {4a + 9b} \right)\left( {4c - 9d} \right) = \left( {4a - 9b} \right)\left( {4c + 9d} \right)$$ \Rightarrow 16ac + 36bc - 36ad - 81bd = 16ac - 36bc + 36ad - 81bd$
Here we can see that the terms containing variables $ac$ and $bd$ are getting cancelled from the left – hand side and the right – hand side. So, the equation now becomes:
$ \Rightarrow 36bc - 36ad = - 36bc + 36ad$
Combining the terms with same coefficients, we get
$ \Rightarrow 36bc + 36bc = 36ad + 36ad$
$ \Rightarrow 72bc = 72ad$
Or, we can write this as: $ad = bc$
Therefore, we get
$ \Rightarrow \dfrac{a}{b} = \dfrac{c}{d}$
$ \Rightarrow a:b = c:d$
Therefore, the given statement is true.
Hence, option (A) is correct.

Note: We have just divided both sides with $bd$ i.e., $\dfrac{{ad}}{{bd}} = \dfrac{{bc}}{{bd}} \equiv \dfrac{a}{b} = \dfrac{c}{d}$. And after this, we can say that the proportionality constant of $\dfrac{a}{b}$ and $\dfrac{c}{d}$ are equal and that’s why they are equal. So, we have applied the ratio method to write it as $a:b = c:d$. In such questions where the justification of a given equation is asked, you simply solve it in simpler terms (or expand the terms) and reduce it to the required form.