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The rational numbers are numbers that can be written in the form [beautiful math coming... please be patient] $\displaystyle\,\frac{a}{b}\,$,
where $\,a\,$ and $\,b\,$ are integers, and $\,b\,$ is nonzero.

Thus, the rational numbers are ratios of integers.

For example, [beautiful math coming... please be patient] $\,\frac25\,$ and $\,\frac{-7}{4}\,$ are rational numbers.

Every real number is either rational, or it isn't.
If it isn't rational, then it is said to be irrational.

By doing a long division, every rational number can be written
as a finite decimal or an infinite repeating decimal.

A finite decimal is one that stops, like [beautiful math coming... please be patient] $\,0.157\,$.

An infinite repeating decimal is one that has a specified sequence of digits that repeat,
like [beautiful math coming... please be patient] $\,0.263737373737\ldots = 0.26\overline{37}\,$.
Notice that in an infinite repeating decimal, the over-bar indicates the digits that repeat.

To answer this question:

• start by putting the fraction in simplest form;
• then, factor the denominator into primes.
• If there are only prime factors of [beautiful math coming... please be patient] $\,2\,$ and $\,5\,$ in the denominator,
  then the fraction has a finite decimal name.

The following example illustrates the idea:

[beautiful math coming... please be patient] $\displaystyle\frac{9}{60} = \frac{3}{20} = \frac{3}{2\cdot2\cdot 5}\cdot\frac{5}{5} = \frac{15}{100} = 0.15 $

If there are only factors of [beautiful math coming... please be patient] $\,2\,$ and $\,5\,$ in the denominator,
then additional factors can be introduced, as needed,
so that there are equal numbers of twos and fives.
Then, the denominator is a power of $\,10\,$,
which is easy to write in decimal form.

When the fraction is in simplest form,
then any prime factors other than [beautiful math coming... please be patient] $\,2\,$ or $\,5\,$ in the denominator
will give an infinite repeating decimal. For example:

[beautiful math coming... please be patient] $\displaystyle\frac{1}{6} = \frac{1}{2\cdot 3} = 0.166666\ldots = 0.1\overline{6} $     (bar over just the $6$)

[beautiful math coming... please be patient] $\displaystyle\frac{2}{7} = 0.\overline{285714} $     (bar over the digits $285714$)

$\displaystyle\frac{3}{11} = 0.\overline{27} $     (bar over the digits $27$)

Consider the given fraction.
In decimal form, determine if the given fraction is a finite decimal, or an infinite repeating decimal.