10. CBQ - Decimal expansion of rational numbers

Question:

4 m.

While practicing the decimal representation of rational numbers in the maths class, the teacher says that 'Certain decimal expansion can be expressed as rational numbers with denominators of the form

\(2^m 5^n\), where \(m\) and \(n\) are positive integers'

Based on this answer the following questions:

1. Express 0.25 in the form $\frac{p}{q}$ and verify the given condition.

$\frac{1}{64}$
$\frac{1}{25}$
$\frac{1}{32}$
$\frac{1}{4}$

2. Express 0.00032 in the form $\frac{p}{q}$ and verify the given condition.

$\frac{1}{64}$
$\frac{1}{25}$
$\frac{1}{5}$
$\frac{1}{3125}$

3. Express 0.00008 in the form $\frac{p}{q}$ and verify the given condition.

$\frac{1}{23}$
$\frac{1}{100}$
$\frac{1}{12500}$
$\frac{1}{71875}$

4. From the above given decimals, what do you observe with respect to the given condition?

All the rational numbers with terminating decimal expansion have denominators of the form $2^{m}5^{n}$ in their lowest form
All the rational numbers with non-terminating decimal expansion have denominators of the form $2^{m}5^{n}$
The decimal expansion of all the rational numbers have denominators of the form $2^{m}5^{n}$
All the rational numbers with terminating decimal expansion have denominators of the form $2^{m}5^{n}$

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