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) $\frac{1625}{462}$

Answer 1

(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 ^{$(2^m\times 5^n)$} 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 ( $2^m\times 5^n$) 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 $(2^m\times 5^n)$, 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 $(2^m\times 5^n)$ for some non-negative integers $m,n$.
So, it cannot be expressed as a terminating decimal.