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)
(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.
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