Question

Convert the following rational number into decimals and state the kind of decimal expansion.
(i) $ \dfrac{{42}}{{100}} $
(ii) $ 8\dfrac{2}{7} $
(iii) $ \dfrac{{13}}{{55}} $
(iv) $ \dfrac{{459}}{{500}} $
(v) $ \dfrac{1}{{11}} $
(vi) $ - \dfrac{3}{{13}} $
(vii) $ \dfrac{{19}}{3} $
(viii) $ - \dfrac{7}{{32}} $

Answer 1

Hint: The given questions are in fraction form. To solve these we have to divide numerator by denominator to get its decimal form. After getting an answer we can conclude if they are terminating or non-terminating or recurring decimal expansion.

Complete step-by-step solution:
(i) $ \dfrac{{42}}{{100}} $
Converting it into decimal number we get,
 $ = 0.42 $
$ \therefore $ As it has a finite number of digits after decimal point, it is a terminating decimal expansion.
(ii) $ 8\dfrac{2}{7} $
Converting it into real rational number we get,
 $ = \dfrac{{58}}{7} $
Converting it into decimal number we get,
 $ = 8.285714285714............ $
We can write it as,
 $ = 8.\overline {285714} $
$ \therefore $ As it does not have a finite number of digits after decimal point and decimal representation is periodic, it is a non-terminating and recurring decimal expansion.
(iii) $ \dfrac{{13}}{{55}} $
Converting it into decimal number we get,
 $ = 0.236363636..... $
We can write it as,
 $ = 0.2\overline {36} $
$ \therefore $ As it does not have a finite number of digits after decimal point and decimal representation is periodic, it is a non-terminating and recurring decimal expansion.
(iv) $ \dfrac{{459}}{{500}} $
Converting it into decimal number we get,
 $ = 0.918 $
$ \therefore $ As it has a finite number of digits after decimal point, it is a terminating decimal expansion.
(v) $ \dfrac{1}{{11}} $
Converting it into decimal number we get,
 $ = 0.09090909...... $
We can write it as,
 $ = 0.\overline {09} $
$ \therefore $ As it does not have a finite number of digits after decimal point and decimal representation is periodic, it is a non-terminating and recurring decimal expansion.
(vi) $ - \dfrac{3}{{13}} $
Converting it into decimal number we get,
 $ = - 0.230769230769...... $
We can write it as,
 $ = - 0.\overline {230769} $
$ \therefore $ As it does not have a finite number of digits after decimal point and decimal representation is periodic, it is a non-terminating and recurring decimal expansion.
(vii) $ \dfrac{{19}}{3} $
Converting it into decimal number we get,
 $ = 6.3333333..... $
We can write it as,
 $ = 6.\overline 3 $
$ \therefore $ As it does not have a finite number of digits after decimal point and decimal representation is periodic, it is a non-terminating and recurring decimal expansion.
(viii) $ - \dfrac{7}{{32}} $
Converting it into decimal number we get,
 $ = - 0.21875 $
$ \therefore $ As it has a finite number of digits after decimal point, it is a terminating decimal expansion.

Note: In algebra a decimal number can be defined as a number whose whole number part and fractional part is separated by a decimal point.
A terminating decimal is usually defined as a decimal number that contains a finite number of digits after the decimal point. E.g. 1.34, 6.876
A non-terminating decimal is usually defined as a decimal number that contains an infinite number of digits after the decimal point. E.g. 5.456899632248088….., 6.789789789789…….
A recurring decimal is a number whose decimal representation becomes periodic. E.g. $ 0.55555... = 0.\overline 5 $
A fraction represents a part of a whole.