Which of the following rational numbers have terminating decimal?

Question:

Which of the following rational numbers have terminating decimal?

(i) $\frac{16}{225}$

(ii) $\frac{5}{18}$

(iii) $\frac{2}{21}$

(iv) $\frac{7}{250}$

(a) (i) and (ii)

(b) (ii) and (iii)

(c) (i) and (iii)

(d) (i) and (iv)

Solution:

(i) We have,

$\frac{16}{225}=\frac{16}{3^{2} \times 5^{2}}$

Theorem states:

Let $x=\frac{p}{q}$ be a rational number, such that the prime factorization of $q$ is not of the form $2^{m} \times 5^{n}$, where $m$ and $n$ are nonnegative integers.

Then, x has a decimal expression which does not have terminating decimal.

(ii) We have,

$\frac{5}{18}=\frac{5}{2 \times 3^{2}}$

Theorem states:

Let $x=\frac{p}{q}$ be a rational number, such that the prime factorization of $q$ is not of the form $2^{m} \times 5^{n}$, where $m$ and $n$ are nonnegative integers.

Then, x has a decimal expression which does not have terminating decimal.

(iii) We have,

$\frac{2}{21}=\frac{2}{7 \times 3}$

Theorem states:

Let $x=\frac{p}{q}$ be a rational number, such that the prime factorization of $q$ is not of the form $2^{m} \times 5^{n}$, where $m$ and $n$ are nonnegative integers.

Then, x has a decimal expression which does not have terminating decimal.

(iv) We have,

$\frac{7}{250}=\frac{7}{2^{1} \times 5^{3}}$

Theorem states:

Let $x=\frac{p}{q}$ be a rational number, such that the prime factorization of $q$ is of the form $2^{m} \times 5^{n}$, where $m$ and $n$ are nonnegative integers.

Then, x has a decimal expression which terminates after k places of decimals, where k is the larger of m and n.

Then, x has a decimal expression which will have terminating decimal after 3 places of decimal.

Hence the (iv) option will have terminating decimal expansion.

There is no correct option.