Question

Express the following ratios in the simplest form .
\[\left( a \right)\] \[36:16\]
\[\left( b \right)\] \[324:144\]
\[\left( c \right)\] \[125:1125\]

Answer 1

Hint:We have to express the given ratios into their simplest possible form . We solve this question using the concept of cancellation of the fractional terms . We should also have the knowledge of how to write the given ratios into fractional form . First we will convert each and every ratio into its fractional form and then we will start cancelling the same terms of the numerator and the denominator . We will cancel the terms of the numerator and denominator till the point when we would not be able to further cancel the terms.

Complete step by step answer:
Given : \[\left( a \right)\] \[36:16\]
As , we know that the ratio of two number can be represented as fractional terms as :
\[a:b = \dfrac{a}{b}\]
Using this , we get the ratio as :
\[36:16 = \dfrac{{36}}{{16}}\]
Now , we will start cancelling the terms of the numerator and denominator .
Dividing both the numerator and denominator by \[2\] , the ratio becomes
\[36:16 = \dfrac{{18}}{8}\]
Again , Dividing both the numerator and denominator by \[2\] , the ratio becomes
\[36:16 = \dfrac{9}{4}\]
Now , it is not possible to further cancel the terms of numerator and denominator .
Now , we also know that the fractional value of two numbers can be represented in ratio as :
\[\dfrac{a}{b} = a:b\]
So , we get the ratio as :
\[36:16 = 9:4\]
Hence , the simplest form of the given ratio \[36:16\] is \[9:4\] .

\[\left( b \right)\] \[324:144\]
As , we know that the ratio of two number can be represented as fractional terms as :
\[a:b = \dfrac{a}{b}\]
Using this , we get the ratio as :
\[324:144 = \dfrac{{324}}{{144}}\]
Now , we will start cancelling the terms of the numerator and denominator .
Dividing both the numerator and denominator by \[2\] , the ratio becomes
\[324:144 = \dfrac{{162}}{{72}}\]
Again , Dividing both the numerator and denominator by \[2\] , the ratio becomes
\[324:144 = \dfrac{{81}}{{36}}\]
Now , Dividing both the numerator and denominator by \[3\] , the ratio becomes
\[324:144 = \dfrac{{27}}{{12}}\]
Again , Dividing both the numerator and denominator by \[3\] , the ratio becomes
\[324:144 = \dfrac{9}{4}\]
Now , it is not possible to further cancel the terms of numerator and denominator .
Now , we also know that the fractional value of two numbers can be represented in ratio as :
\[\dfrac{a}{b} = a:b\]
So , we get the ratio as :
\[324:144 = 9:4\]
Hence , the simplest form of the given ratio \[324:144\] is \[9:4\] .

\[\left( c \right)\] \[125:1125\]
As , we know that the ratio of two number can be represented as fractional terms as :
\[a:b = \dfrac{a}{b}\]
Using this , we get the ratio as :
\[125:1125 = \dfrac{{125}}{{1125}}\]
Now , we will start cancelling the terms of the numerator and denominator .
Dividing both the numerator and denominator by \[5\] , the ratio becomes
\[125:1125 = \dfrac{{25}}{{225}}\]
Again , Dividing both the numerator and denominator by \[5\] , the ratio becomes
\[125:1125 = \dfrac{5}{{45}}\]
Again , Dividing both the numerator and denominator by \[5\] , the ratio becomes
\[125:1125 = \dfrac{1}{9}\]
Now , it is not possible to further cancel the terms of numerator and denominator .
Now , we also know that the fractional value of two numbers can be represented in ratio as :
\[\dfrac{a}{b} = a:b\]
So , we get the ratio as :
\[125:1125 = 1:9\]
Hence , the simplest form of the given ratio \[125:1125\] is \[1:9\].

Note:We will cancel the terms of the numerator and denominator by the common factors of the both . Apart from the way shown above , we can also split the numbers of the numerator and the denominator of each part into its prime factors and on cancelling the common terms of the prime factor of the numerator and denominator we will get the simplest form.