Question

The decimal expansion of number ${\displaystyle \frac{46}{2^2\times 5\times 3}}$ is
A.terminating
B.non-terminating repeating
C.non-terminating non-repeating
D.none of these

Answer 1

Hint: According to the question the given fraction is first converted into the decimal and then we have to define the type of that decimal expansion. The decimal expansion of the fraction can be categorised as terminating and non-terminating or repeating and non-repeating. Suppose the decimal expansion that has a finite number of terms or digits is termed as terminating decimal and the decimal expansion that has infinite number of terms and digits is termed as non-terminating decimal. As we know The decimal expansion that has a repeating set of terms is termed as repeating decimal and the decimal expansion that doesn’t have a repeating set of terms is termed as non-repeating decimal.

Complete step-by-step answer:
Let us evaluate the decimal expansion of given fraction:
Fraction$={\displaystyle \frac{46}{2^2\times 5\times 3}}$
So, its decimal expansion is given by:
Decimal expansion$={\displaystyle \frac{46}{2^2\times 5\times 3}}={\displaystyle \frac{46}{60}}=0.766666666...$
From the above decimal expansion, we can say that the decimal expansion has an infinite number of terms and also the digit $6$ is recurring infinitely.
Therefore, the evaluated decimal expansion is non-terminating repeating decimal.
Hence, the option B is correct.

Note: To evaluate the decimal we can simply solve the fraction into decimal and can categorize it as terminating, non-terminating, repeating and non-repeating. We can also tell whether the given fraction is terminating or non-terminating without solving it. On observing the factors of the denominator, the given fraction can be termed as terminating or on-terminating. If the factors of denominator contain only $2$ or only $5$ or both as the factors, then it is termed as terminating decimal. If there is any other factor of denominator, then it is non-terminating decimal.