Question

Simplify and reduce to a simplest fraction : \[\left( {\dfrac{4}{5}} \right) \div \left( {\dfrac{7}{{15}}} \right) \times \left( {\dfrac{8}{9}} \right)\]

Answer 1

Hint:We have to first solve the given expression to get a fractional value and then reduce the fraction into its simplest terms by cancelling the terms of the numerator and the denominator . We will solve this question using the concept of fractional dividing of terms . We should also have the knowledge of the concept of BODMAS . Using the concept of BODMAS , we will solve the given expression to form a fractional term 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 : \[\left( {\dfrac{4}{5}} \right) \div \left( {\dfrac{7}{{15}}} \right) \times \left( {\dfrac{8}{9}} \right)\]
Now we will solve the given expression using BODMAS . Using the concept of BODMAS we will first do the division and then the multiplication of the terms of the given expression .
So , on applying BODMAS we get the expression as :
\[\left( {\dfrac{4}{5}} \right) \div \left( {\dfrac{7}{{15}}} \right) \times \left( {\dfrac{8}{9}} \right) = \left[ {\dfrac{{\left( {\dfrac{4}{5}} \right)}}{{\left( {\dfrac{7}{{15}}} \right)}}} \right] \times \left( {\dfrac{8}{9}} \right)\]

Now , we also know that the fractional terms of division can be written as :
\[\left( {\dfrac{4}{5}} \right) \div \left( {\dfrac{7}{{15}}} \right) \times \left( {\dfrac{8}{9}} \right) = \left[ {\dfrac{{4 \times 15}}{{5 \times 7}}} \right] \times \left( {\dfrac{8}{9}} \right)\]
On further simplifying , we get
\[\left( {\dfrac{4}{5}} \right) \div \left( {\dfrac{7}{{15}}} \right) \times \left( {\dfrac{8}{9}} \right) = \left[ {\dfrac{{4 \times 15 \times 8}}{{5 \times 7 \times 9}}} \right]\]
\[\Rightarrow \left( {\dfrac{4}{5}} \right) \div \left( {\dfrac{7}{{15}}} \right) \times \left( {\dfrac{8}{9}} \right) = \left[ {\dfrac{{480}}{{315}}} \right]\]
Thus , we get the fractional value of the given expression as \[\dfrac{{480}}{{315}}\] .

Now , we will reduce the fraction into its simplest fraction.
Dividing both the numerator and denominator by \[5\] , we get
\[\left( {\dfrac{4}{5}} \right) \div \left( {\dfrac{7}{{15}}} \right) \times \left( {\dfrac{8}{9}} \right) = \left[ {\dfrac{{96}}{{63}}} \right]\]
Now , Dividing both the numerator and denominator by \[3\] , we get
\[\left( {\dfrac{4}{5}} \right) \div \left( {\dfrac{7}{{15}}} \right) \times \left( {\dfrac{8}{9}} \right) = \left[ {\dfrac{{32}}{{21}}} \right]\]
Now , it is not possible to further cancel the terms of numerator and denominator.

Hence , the simplest form of the given expression \[\left( {\dfrac{4}{5}} \right) \div \left( {\dfrac{7}{{15}}} \right) \times \left( {\dfrac{8}{9}} \right)\] is \[\dfrac{{32}}{{21}}\].

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.