Multiply the following repeating decimals by 10, 100 and 1000.

: Classify as rational or irrational

Step 1: $0.53$
This is a terminating decimal.
$0.53 = \frac{53}{100}$
Terminating decimals are rational.
Rational

Step 2: $0.218$
This is a terminating decimal.
$0.218 = \frac{218}{1000} = \frac{109}{500}$
Rational

Step 3: $0.396$
This is a terminating decimal.
$0.396 = \frac{396}{1000} = \frac{99}{250}$
Rational

Step 4: $0.\overline{52}$
This is a repeating decimal.
Let $x = 0.\overline{52}$.
$100x = 52.\overline{52}$
$100x - x = 52.\overline{52} - 0.\overline{52}$
$99x = 52$
$x = \frac{52}{99}$
Repeating decimals are rational.
Rational

: Convert repeating decimals to fractions

Step 1: $0.\overline{16}$
Let $x = 0.\overline{16}$.
$100x = 16.\overline{16}$
$100x - x = 16$
$99x = 16$
$x = \frac{16}{99}$
\dfrac{16{99}}

Step 2: $0.\overline{6}$
Let $x = 0.\overline{6}$.
$10x = 6.\overline{6}$
$10x - x = 6$
$9x = 6$
$x = \frac{6}{9} = \frac{2}{3}$.
\dfrac{2{3}}

Step 3: $0.\overline{25}$
Let $x = 0.\overline{25}$.
$100x = 25.\overline{25}$
$100x - x = 25$
$99x = 25$
$x = \frac{25}{99}$.
\dfrac{25{99}}

Step 4: $0.\overline{46}$
Let $x = 0.\overline{46}$.
$100x = 46.\overline{46}$
$100x - x = 46$
$99x = 46$
$x = \frac{46}{99}$.
\dfrac{46{99}}