Question

The triplicate ratio of \[{a^2}:{b^2}\] is ______.
A. \[\sqrt[3]{{{a^2}}}:\sqrt[3]{{{b^2}}}\]
B. \[{a^3}:{b^3}\]
C. \[{a^6}:{b^6}\]
D. \[{a^9}:{b^9}\]

Answer 1

Hint: First we will use the concept of composition of ratios, that is, triplicate ratio, which is the cube of the original ratio like we know that \[{a^3}:{b^3}\] is the triplicate ratio of \[a:b\]. Use this value to find the triplicate ratio of \[{a^2}:{b^2}\] to compute the required answer.

Complete step by step answer:

We are given that the ratio is \[{a^2}:{b^2}\].

We know that the triplicate ratio is the compound ratio of three equal ratios, that is, the triplicate ratio of the ratio \[a:b\] is the ratio \[{a^3}:{b^3}\].

We will now find the triplicate ratio \[{a^2}:{b^2}\] using the above triplicate ratio by taking the cube of the ratio, we get
\[ \Rightarrow {\left( {{a^2}:{b^2}} \right)^3}\]

Using the rule of ratios that \[{\left( {a:b} \right)^3} = {a^3}:{b^3}\] in the above equation, we get

\[ \Rightarrow {\left( {{a^2}} \right)^3}:{\left( {{b^2}} \right)^3} = {a^6}:{b^6}\]

Thus, the triplicate ratio of \[{a^2}:{b^2}\] is \[{a^6}:{b^6}\].

Hence, option C is correct.

Note: In solving these types of questions, students should know that the concept plays a vital role so it is advisable to remember the definitions and concepts of the basic composition of ratios to save a lot of time. One should know that the triplicate ratio is the compound ratio of three equal ratios, that is, the triplicate ratio of the ratio cube of the given ratio.