Question

Which of the following numbers has a terminating decimal expression?
(A) $\dfrac{{37}}{{45}}$
(B) $\dfrac{{21}}{{{2^3}{5^6}}}$
(C) $\dfrac{{17}}{{49}}$
(D) $\dfrac{{89}}{{{2^2}{3^2}}}$

Answer 1

Hint: The decimal expression of a fraction is terminating if the prime factorization of the denominator takes the form ${2^x}{5^y}$, where $x$ and $y$ are non-negative integer. Firstly, convert the given fraction into its simplest form then write the prime factors of the denominator, if only $2$or $5$, or both $2$ and $5$ are the factors then only the decimal expression of the given fraction is terminating otherwise non terminating.

Complete step-by-step solution:
Consider option (A).
The given fraction is $\dfrac{{37}}{{45}}$. It is already in the simplest form.
Now, the prime factors of the denominator $45$ is
$45 = 3 \times 3 \times 5$.
Here, we get that one of the factors is other than $2$ or $5$. So, the decimal expression of the given fraction is non terminating.
Consider option (B).
The given fraction is $\dfrac{{21}}{{{2^3}{5^6}}}$. It is already in the simplest form.
Now, the denominator ${2^3}{5^6}$ can be written as ${2^3}{5^6} = 2 \times 2 \times 2 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5$.
Here, we get that the prime factors of denominator are only $2$ and $5$. So, the decimal expression of the given fraction is terminating.
Consider option (C).
The given fraction is $\dfrac{{17}}{{49}}$. It is already in the simplest form.
Now, find the prime factors of the denominator $49$is
$49 = 7 \times 7$.
Here, we get that the factors are other than $2$ or $5$. So, the decimal expression of the given fraction is non terminating.
Consider option (D).
The given fraction is $\dfrac{{89}}{{{2^2}{3^2}}}$. It is already in the simplest form.
Now, the denominator ${2^2}{3^2}$ can be written as ${2^2}{3^2} = 2 \times 2 \times 3 \times 3$.
Here, we get that the factors are other than $2$ or $5$. So, the decimal expression of the given fraction is non terminating.

Thus, option (B) is the correct answer.

Note: The simplest form of a fraction means that there is no any common factor of the numerator and the denominator other than $1$.
If the decimal expression of a number is a terminating or non-terminating recurring decimal then the given number is a rational number.