# Find the ratio compounded of

$(1)$ The ratio 2a, 3b and the duplicate of $9{b^2}:ab$

$(2)$ The sub duplicate ratio of 64: 9, and the ratio 27: 56

$(3)$ The duplicate ratio of $\dfrac{{2a}}{b}:\dfrac{{\sqrt C {a^2}}}{{{b^2}}}$, and the ratio 3ax: 2by

Answer

Verified

382.2k+ views

Hint – In this question we have to deal with terms like duplicate, sub duplicate and we need to take out ratio compounded of ratio and duplicate. Sub duplicate ratio and ratio and duplicate ratio and ratio, so use the basic definition of duplicate, sub-duplicate directly along with the basic formula. Implementation of these two will get you to the answer.

Complete step-by-step answer:

Let us assume two ratios $\left( {x:y} \right){\text{ & }}\left( {p:q} \right)$

Duplicate of the ratio $\left( {p:q} \right)$ is $\left( {{p^2}:{q^2}} \right)$

So, the compound of $\left( {x:y} \right)$ and duplicate of $\left( {p:q} \right)$is $ \Rightarrow \left( {x \times {p^2}} \right):\left( {y \times {q^2}} \right)$

Now as we know that the sub duplicate of the ratio $\left( {p:q} \right)$ is $\left( {\sqrt p :\sqrt q } \right)$

So, the compound of $\left( {x:y} \right)$ and sub duplicate of $\left( {p:q} \right)$ is $ \Rightarrow \left( {x \times \sqrt p } \right):\left( {y \times \sqrt q } \right)$

So, use these properties in the given question we have,

$\left( 1 \right)$ Duplicate of the ratio $\left( {9{b^2}:ab} \right)$ is $\left( {81{b^4}:{a^2}{b^2}} \right)$

So, the compound of $\left( {2a:3b} \right)$ and duplicate of $\left( {9{b^2}:ab} \right)$is $ \Rightarrow \left( {2a \times 81{b^4}} \right):\left( {3b \times {a^2}{b^2}} \right)$

Now simplify the above ratio we have,

$ \Rightarrow \left( {2a \times 81{b^4}} \right):\left( {3b \times {a^2}{b^2}} \right)$

Divide by $3a{b^3}$ we have,

$ = 54b:a$

$\left( 2 \right)$ The sub duplicate of the ratio $\left( {64:9} \right)$ is \[\left( {\sqrt {64} :\sqrt 9 } \right) = \left( {8:3} \right)\]

So, the compound of $\left( {27:56} \right)$ and sub duplicate of $\left( {8:3} \right)$is $ \Rightarrow \left( {27 \times 8} \right):\left( {56 \times 3} \right)$

Now divide by 24 we have

$ \Rightarrow \left( {27 \times 8} \right):\left( {56 \times 3} \right) = 9:7$

$\left( 3 \right)$ Duplicate of the ratio $\left( {\dfrac{{2a}}{b}:\dfrac{{\sqrt c {a^2}}}{{{b^2}}}} \right)$ is $\left( {\dfrac{{4{a^2}}}{{{b^2}}}:\dfrac{{c{a^4}}}{{{b^4}}}} \right)$

So, the compound of $\left( {3ax:2by} \right)$ and duplicate of $\left( {\dfrac{{2a}}{b}:\dfrac{{\sqrt c {a^2}}}{{{b^2}}}} \right)$is $ \Rightarrow \left( {3ax \times \dfrac{{4{a^2}}}{{{b^2}}}} \right):\left( {2by \times \dfrac{{c{a^4}}}{{{b^4}}}} \right)$

Now simplify the above ratio we have,

$ \Rightarrow \left( {3ax \times \dfrac{{4{a^2}}}{{{b^2}}}} \right):\left( {2by \times \dfrac{{c{a^4}}}{{{b^4}}}} \right) = \left( {12x \times \dfrac{{{a^3}}}{{{b^2}}}} \right):\left( {2y \times \dfrac{{c{a^4}}}{{{b^3}}}} \right)$

Divide by $\dfrac{{2{a^3}}}{{{b^2}}}$ we have,

$ = 6x:\dfrac{{yac}}{b} = 6bx:acy$

So, these are the required answers.

Note – Whenever we face such types of problems the key point that we need to have in our mind is that these all are basic definitions along with direct based questions. So having a good understanding of this direct concept helps you solve problems of this kind.

Complete step-by-step answer:

Let us assume two ratios $\left( {x:y} \right){\text{ & }}\left( {p:q} \right)$

Duplicate of the ratio $\left( {p:q} \right)$ is $\left( {{p^2}:{q^2}} \right)$

So, the compound of $\left( {x:y} \right)$ and duplicate of $\left( {p:q} \right)$is $ \Rightarrow \left( {x \times {p^2}} \right):\left( {y \times {q^2}} \right)$

Now as we know that the sub duplicate of the ratio $\left( {p:q} \right)$ is $\left( {\sqrt p :\sqrt q } \right)$

So, the compound of $\left( {x:y} \right)$ and sub duplicate of $\left( {p:q} \right)$ is $ \Rightarrow \left( {x \times \sqrt p } \right):\left( {y \times \sqrt q } \right)$

So, use these properties in the given question we have,

$\left( 1 \right)$ Duplicate of the ratio $\left( {9{b^2}:ab} \right)$ is $\left( {81{b^4}:{a^2}{b^2}} \right)$

So, the compound of $\left( {2a:3b} \right)$ and duplicate of $\left( {9{b^2}:ab} \right)$is $ \Rightarrow \left( {2a \times 81{b^4}} \right):\left( {3b \times {a^2}{b^2}} \right)$

Now simplify the above ratio we have,

$ \Rightarrow \left( {2a \times 81{b^4}} \right):\left( {3b \times {a^2}{b^2}} \right)$

Divide by $3a{b^3}$ we have,

$ = 54b:a$

$\left( 2 \right)$ The sub duplicate of the ratio $\left( {64:9} \right)$ is \[\left( {\sqrt {64} :\sqrt 9 } \right) = \left( {8:3} \right)\]

So, the compound of $\left( {27:56} \right)$ and sub duplicate of $\left( {8:3} \right)$is $ \Rightarrow \left( {27 \times 8} \right):\left( {56 \times 3} \right)$

Now divide by 24 we have

$ \Rightarrow \left( {27 \times 8} \right):\left( {56 \times 3} \right) = 9:7$

$\left( 3 \right)$ Duplicate of the ratio $\left( {\dfrac{{2a}}{b}:\dfrac{{\sqrt c {a^2}}}{{{b^2}}}} \right)$ is $\left( {\dfrac{{4{a^2}}}{{{b^2}}}:\dfrac{{c{a^4}}}{{{b^4}}}} \right)$

So, the compound of $\left( {3ax:2by} \right)$ and duplicate of $\left( {\dfrac{{2a}}{b}:\dfrac{{\sqrt c {a^2}}}{{{b^2}}}} \right)$is $ \Rightarrow \left( {3ax \times \dfrac{{4{a^2}}}{{{b^2}}}} \right):\left( {2by \times \dfrac{{c{a^4}}}{{{b^4}}}} \right)$

Now simplify the above ratio we have,

$ \Rightarrow \left( {3ax \times \dfrac{{4{a^2}}}{{{b^2}}}} \right):\left( {2by \times \dfrac{{c{a^4}}}{{{b^4}}}} \right) = \left( {12x \times \dfrac{{{a^3}}}{{{b^2}}}} \right):\left( {2y \times \dfrac{{c{a^4}}}{{{b^3}}}} \right)$

Divide by $\dfrac{{2{a^3}}}{{{b^2}}}$ we have,

$ = 6x:\dfrac{{yac}}{b} = 6bx:acy$

So, these are the required answers.

Note – Whenever we face such types of problems the key point that we need to have in our mind is that these all are basic definitions along with direct based questions. So having a good understanding of this direct concept helps you solve problems of this kind.

Recently Updated Pages

Define absolute refractive index of a medium

Find out what do the algal bloom and redtides sign class 10 biology CBSE

Prove that the function fleft x right xn is continuous class 12 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Find the values of other five trigonometric ratios class 10 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Trending doubts

Which of the following Chief Justice of India has acted class 10 social science CBSE

Green glands are excretory organs of A Crustaceans class 11 biology CBSE

What if photosynthesis does not occur in plants class 11 biology CBSE

What is 1 divided by 0 class 8 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference Between Plant Cell and Animal Cell

10 slogans on organ donation class 8 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

What is the past tense of read class 10 english CBSE