Question

Assertion: \[\dfrac{{13}}{{3125}}\]is a terminating decimal fraction.
Reason: If \[q = {2^n}{.5^m}\], where n, m are non-negative integers, then\[\tfrac{p}{q}\]is a terminating decimal fraction.
A. Both Assertion and Reason are correct, and Reason is the correct explanation for Assertion.
B. Both Assertion and Reason are correct, but Reason is not the correct explanation for Assertion.
C. Assertion is correct, but Reason is incorrect
D. Assertion is incorrect, but Reason is correct

Answer 1

Hint:
Terminating decimal numbers are the numbers that contain a finite number of digits after the decimal point. A number 2.56 can be a terminating decimal if it is represented as 2.5600000000….. where 0 is terminating.
In this question, expand the rational numbers in terminating form, then check for terminating numbers form\[{2^n} \times {5^m}\], and if the condition satisfies, then find the terminating number.

Complete step by step solution:
Factorize the denominator of the given rational number in the form \[{2^n} \times {5^m}\]to check the terminating numbers,
 \[\dfrac{{13}}{{3125}}\]
By factoring the denominator, we can write \[3125 = 5 \times 5 \times 5 \times 5 \times 5 = {5^5}\]
Multiply the factor by \[{2^0}\] which is equal to\[{2^0} = 1\] bring the factors in terminating form\[{2^0} \times {5^5}\]; hence we can say the rational number is in terminating form.
As we know, if the denominator is in the form of \[{2^n} \times {5^m}\], then the fraction is always terminating; hence we can say the number is correct, and both Assertion and Reason are correct, and Reason is the correct explanation for Assertion.

Option A is correct.

Note:
Any rational number can be written as either a terminating or repeating decimal by dividing the numerator by denominator, and if the remainder is 0, then the number is a terminating decimal. Every fraction number can be either terminating or non-termination (or repeating).