Question

Evaluate the value of the following: \[{{\left( \dfrac{30\times 25}{60\times 5} \right)}^{2}}\]
A. \[\dfrac{15}{4}\]
B. $\dfrac{25}{3}$
C. $\dfrac{12}{4}$
D. $\dfrac{25}{4}$

Answer 1

Hint: To find the solution of \[{{\left( \dfrac{30\times 25}{60\times 5} \right)}^{2}}\] , simplify numerator and denominator before applying the square. Then further simplify them by eliminating their common terms. Then apply the squares separately on the numerator and denominator. Next, if possible, simplify them.

Complete step-by-step answer:
We need to find the solution of \[{{\left( \dfrac{30\times 25}{60\times 5} \right)}^{2}}\] .
First let’s simplify the terms inside the bracket.
Now, multiply the numerator.
\[\Rightarrow {{\left( \dfrac{750}{60\times 5} \right)}^{2}}\]
Next, do the same operation on the denominator.
\[\Rightarrow {{\left( \dfrac{750}{300} \right)}^{2}}\]
Now, numerator and denominator can be further simplified.
First, cancel the zeroes. So we get
${{\left( \dfrac{75}{30} \right)}^{2}}$
Now cancel the numerator and denominator by $15$ . So the above equation becomes,
\[{{\left( \dfrac{5}{2} \right)}^{2}}\]
We know that ${{\left( \dfrac{a}{b} \right)}^{2}}=\dfrac{{{a}^{2}}}{{{b}^{2}}}$ .
Therefore, ${{\left( \dfrac{5}{2} \right)}^{2}}=\dfrac{{{5}^{2}}}{{{2}^{2}}}$ .
Now apply the squares. The above term will become
$\Rightarrow \dfrac{25}{4}$

So, the correct answer is “Option D”.

Note: In these types of questions, do not do the operations outside the bracket first. Simplify the inner terms first. Try to simplify the fraction terms as much as possible to get a small value. This can also be solved by another method:
First cancel the common terms using multiples of $5$ from numerator and denominator. We will get
\[{{\left( \dfrac{30\times 5}{60} \right)}^{2}}\]\[\]
Now let us cancel the common zeroes from numerator and denominator. The above equation becomes
\[{{\left( \dfrac{3\times 5}{6} \right)}^{2}}\]
Now, by multiplying the numerator, we get
\[{{\left( \dfrac{15}{6} \right)}^{2}}\]
We can again simplify this by cancelling the numerator and denominator by $3$ .
So the above equation becomes \[{{\left( \dfrac{5}{2} \right)}^{2}}\] .
We know that ${{\left( \dfrac{a}{b} \right)}^{2}}=\dfrac{{{a}^{2}}}{{{b}^{2}}}$ .
Therefore, ${{\left( \dfrac{5}{2} \right)}^{2}}=\dfrac{{{5}^{2}}}{{{2}^{2}}}$ .
Now apply the squares. The above term will become
$\Rightarrow \dfrac{25}{4}$