Question

Reduce to lowest terms: $\dfrac{{4xy}}{{2xy + {y^2}}}$
A) $\dfrac{{4x}}{{2x + y}}$
B) $\dfrac{{4x}}{{2 + y}}$
C) $\dfrac{{4x}}{{2y + x}}$
D) None of these

Answer 1

Hint: We will first write the expression which is already given to us. Now, we will just rewrite it by taking something common out of numerator. Now, we can cancel out that common part of the denominator by the part of the numerator and thus we have the answer.

Complete step by step answer:
We already have the following expression which is given to us:
$\dfrac{{4xy}}{{2xy + {y^2}}}$
Now, we can rewrite it as:
$ \Rightarrow \dfrac{{4xy}}{{2xy + {y^2}}} = \dfrac{{4xy}}{{y\left( {2x + y} \right)}}$
Since, we can observe that y is in both numerator and denominator in multiplication form. So, we can cancel those out like as follows:-
$ \Rightarrow \dfrac{{4xy}}{{2xy + {y^2}}} = \dfrac{{4x}}{{2x + y}}$

Hence, the correct answer is $\dfrac{{4xy}}{{2xy + {y^2}}} = \dfrac{{4x}}{{2x + y}}$. Therefore, the correct option is (A).

Note:
The students must note that we can only cancel out as we did in this question when both the terms are in multiplication. For example, if it would have been like $\dfrac{{4xy}}{{2xy + 4}}$, we cannot cancel out any of $x$ and $y$, but however, we can still cancel out $2$ in this.
The students must also note that, if we would not have given the statement that we need to reduce it in a simple form, we just need to find its equivalent or anything, then we could not have proceeded in the same way. Because we would then require the condition of $y$ is never equal to zero. Because is any quantity is equal to $0$, it will result in $\dfrac{0}{0}$ form, which is undefined. We cannot cancel them to get $1$.
The simplest form of a fraction is that state of a fraction where the numerator and the denominator of fraction do not have anything in common which can be canceled out and they both are in the smallest form as much as possible. If we talk in a bit of complicated mathematics language, we basically need to divide numerator and demonical by their HCF individually to receive a fraction’s simplest form.