Question

Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating decimal expansion:
$\frac{13}{3125}$ and $\frac{13}{21}$

• A. $\frac{13}{3125}$: Terminating decimal
  $\frac{13}{21}$: Non-terminating decimal
• B. Both have terminating decimal expansions.
• C. Both have non-terminating decimal expansion.
• D. $\frac{13}{3125}$: Non-terminating decimal
  $\frac{13}{21}$: Terminating decimal

Answer 1

The correct option is A

$\frac{13}{3125}$: Terminating decimal
$\frac{13}{21}$: Non-terminating decimal

The given numbers are $\frac{13}{3125}$ and $\frac{13}{21}$

We know a rational number is expressed in simplest form $\frac{p}{q}$ and if $q$ can be expressed as

$2^m\times 5^n$, then it is a terminating decimal

Here, 3125 can be expressed in terms of its prime as 5 x 5 x 5 x 5 x 5 = $5^5$
3125 = $2^m{.5}^n$, where m = 0 and n = 5

Since $q$ can be expressed in the form of $2^m{.5}^n$, we can say $\frac{13}{3125}$ has a terminating decimal expansion.

Similarly, 21 can be expressed in terms of its prime as $3\times 7$

Since $q$ cannot be expressed in the form of $2^m{.5}^n$, we can say $\frac{13}{21}$ has a non - terminating decimal.