QUESTION

# The decimal expansion of a number may terminate in which case the number is called a regular number or finite decimal. In the given options:$A.\dfrac{{77}}{{210}} = .366666666.......or 0.3\overline 6 \\ B.\dfrac{{23}}{{30}} = .766666666.....or 0.7\overline 6 \\ C.\dfrac{{125}}{{441}} = .283446712.........\infty \\ D.\dfrac{{23}}{8} = 2.875 \\$

Hint: A terminating decimal that ends. It’s a decimal with a finite number of digits. For example $\dfrac{1}{4} = 0.25$, it has only two decimal digits. So, it is a terminating decimal as it is a finite number of digits.

According to the definition of terminating decimal that there should be a finite number of digits . So, in order to arrive at the answer , we can solve this question through accepting and rejecting the options method .
So we will check options one by one
Now consider the fraction $\dfrac{{77}}{{210}}$whose decimal form will be :
$\dfrac{{77}}{{210}} = .366666666.....$
The resulting decimal of the first option ends with $n$ no. of digits . Hence option $A$ doesn't satisfy the definition of terminating decimal . hence , it is rejected.
Now consider the fraction $\dfrac{{23}}{{30}}$ whose decimal form will be :
$\dfrac{{23}}{{30}} = .766666666....$
The resulting decimal of the second option ends with $n$ no. of digits . Hence option $B$ don’t satisfy the definition of terminating decimal . hence , it is rejected.
Now consider the fraction $\dfrac{{125}}{{441}}$ whose decimal form will be:
$\dfrac{{125}}{{441}} = .283446712.......\infty$
The resulting decimal of third option ends with $n$no. of digits . Hence option $C$don’t satisfy the definition of terminating decimal . hence , it is rejected.
Now consider the fraction $\dfrac{{23}}{8}$ whose decimal form will be:
$\dfrac{{23}}{8} = 2.875$
The resulting decimal number ends with three decimal digits and therefore, it is terminating decimal.
Hence, $\dfrac{{23}}{8}$ has a terminating decimal expansion.

Note: It is advisable that for such types of questions one should try various methods like hit and trial method , eliminating the options methods etc. , in order to save time and solve the question easily. According to the definition of terminating decimal, there should be a finite number of digits . So, in order to arrive at the answer, we can solve this question through accepting and rejecting the options method.