Question

Witjout actual division find whether the rational number 41/37500 is a terminating or non terminating

Answer 1

Heya!!
Here is your answer :-
The question asks us to check whether the given rational number 41/37500 has a terminating or a non-terminating and recurring decimal expansion.
We know if a number is terminating then it's denominator can be represented in the form of :-
$5 ^{x} . {2}^{y}$
So , we will try to prime factorise the denominator of this number i.e. 37500
$37500 = {5}^{5} . {2}^{2} .3$
So we can see that this number has a factor other that 5 or 2.
So, the rational number 41/37500 is a non-terminating recurring rational number.

Hope it helps you.

Answer 2

For terminating number the the prime factorisation of the denominator should only contain factor 2 or 5 or both 2 and 5.

So,
Factorising 37500 = 2*2*3*5*5*5*5*5
= (2)^2 * 3 * (5)^5

So,
37500 has factors other than 2 and 5. (which is 3)

Hence, the rational number 41/37500 is non terminating.