Which of the following rational numbers is expressible as a terminating decimal?

Question:

Which of the following rational numbers is expressible as a terminating decimal?

(a) $\frac{124}{165}$

(b) $\frac{131}{30}$

(c) $\frac{2027}{625}$

(d) $\frac{1625}{462}$

Solution:

(c) $\frac{2027}{625}$

$\frac{124}{165}=\frac{124}{5 \times 33}$; we know 5 and 33 are not the factors of 124 . It is in its simplest form and it cannot be expressed as the product of $\left(2^{m} \times 5^{n}\right)$ for some non-negative integers

$m, n$

So, it cannot be expressed as a terminating decimal.

$\frac{131}{30}=\frac{131}{5 \times 6}$; we know 5 and 6 are not the factors of 131 . Its is in its simplest form and it cannot be expressed as the product of $\left(2^{m} \times 5^{n}\right)$ for some non-negative integers

$m, n$

So, it cannot be expressed as a terminating decimal.

$\frac{2027}{625}=\frac{2027 \times 2^{4}}{5^{4} \times 2^{4}}=\frac{32432}{10000}=3.2432 ;$ as it is of the form $\left(2^{m} \times 5^{n}\right)$, where $m, n$ are non-negative integers.

So, it is a terminating decimal.

$\frac{1625}{462}=\frac{1625}{2 \times 7 \times 33} ;$ we know 2,7 and 33 are not the factors of 1625 . It is in its simplest form and cannot be expressed as the product of $\left(2^{m} \times 5^{n}\right)$ for some non-negative

integers $m, n$.

So, it cannot be expressed as a terminating decimal.