Question

Find after how many decimal places, will the decimal representation of rational number \[\dfrac{{{\rm{59}}}}{{{{\rm{2}}^{\rm{3}}}{\rm{ \times }}{{\rm{5}}^{\rm{2}}}}}\] terminate?

Answer 1

Hint: Here, we have to simply solve the given fraction to find out the decimal places. Fraction is considered to be the ratio of two numbers i.e. fraction has numerator and denominator. Firstly we have to solve the denominator of the given fraction. Then after solving the denominator we will solve the fraction to its simple decimal form. So, after solving the fraction we will get the number of decimal places of the given rational number.

Complete step-by-step answer:
Rational numbers are the numbers which are present in the form of fraction.
\[{\rm{Fraction = }}\dfrac{{{\rm{numerator}}}}{{{\rm{denominator}}}}\]
Given rational number is \[\dfrac{{{\rm{59}}}}{{{{\rm{2}}^{\rm{3}}}{\rm{ \times }}{{\rm{5}}^{\rm{2}}}}}\]
By comparing the given rational number with the fraction formula we get to know that 59 is the numerator and \[{{\rm{2}}^{\rm{3}}}{\rm{ \times }}{{\rm{5}}^{\rm{2}}}\]is the denominator.
Firstly we have to solve the denominator of the given number to its simplest form by simple multiplication of the numbers.
Therefore, the given rational number becomes \[\dfrac{{{\rm{59}}}}{{{{\rm{2}}^{\rm{3}}}{\rm{ \times }}{{\rm{5}}^{\rm{2}}}}} = \dfrac{{{\rm{59}}}}{{{\rm{2 \times 2 \times }}2{\rm{ \times 5 \times }}5}} = \dfrac{{{\rm{59}}}}{{8 \times 25}} = \dfrac{{59}}{{200}}\]
Now, we have to solve this fraction to get it in the simplest form of decimals, we get
\[\dfrac{{{\rm{59}}}}{{{{\rm{2}}^{\rm{3}}}{\rm{ \times }}{{\rm{5}}^{\rm{2}}}}} = \dfrac{{59}}{{200}} = \dfrac{{29.5}}{{100}} = 0.295\]
Therefore, from the decimal representation of the given rational number we can see that there are only 3 decimal places.
Hence, after three decimal places the decimal representation of the given rational number terminates.

Note: We have to know the meaning of
Proper fraction is a fraction having the numerator less, or lowers in degree, than the denominator.
i.e., Numerator < Denominator
The value of proper fraction after simplification is always less than 1.
Improper Fraction is a fraction where the numerator is greater than or equals to the denominator, then it is known as an improper fraction.
i.e., Numerator ≥ Denominator
After the simplification of an improper fraction results in the value which is equal or greater than 1, but not less than 1.
Terminating decimals are those decimal numbers that contain a finite number of digits after the decimal point.
Non-terminating repeating decimals are those decimal numbers that contain an infinite number of digits repeating itself again and again after the decimal point.