Question

计算下列各题。

(0.dot{3}dot{6} \times 0.dot{3} =)

Explanation:

🆕 New Concept Discovered: Converting Repeating Decimals to Fractions
Turn repeating decimals into fractions to calculate easily.

Step 1: Convert \(0.\dot{3}\dot{6}\) to a fraction

A repeating decimal with two repeating digits under the dots can be written as a fraction with a denominator of \(99\):

$$ 0.\dot{3}\dot{6} = \frac{36}{99} $$

Simplify the fraction by dividing the numerator and the denominator by their greatest common divisor, which is \(9\):

$$ \frac{36 \div 9}{99 \div 9} = \frac{4}{11} $$

Step 2: Convert \(0.\dot{3}\) to a fraction

A repeating decimal with one repeating digit under the dot can be written as a fraction with a denominator of \(9\):

$$ 0.\dot{3} = \frac{3}{9} $$

Simplify the fraction by dividing the numerator and the denominator by \(3\):

$$ \frac{3 \div 3}{9 \div 3} = \frac{1}{3} $$

Step 3: Multiply the fractions

Now, multiply the two simplified fractions together:

$$ \frac{4}{11} \times \frac{1}{3} = \frac{4 \times 1}{11 \times 3} = \frac{4}{33} $$

Step 4: Convert back to a repeating decimal (optional)

To write the fraction \(\frac{4}{33}\) back as a decimal, we can make the denominator \(99\) by multiplying the numerator and denominator by \(3\):

$$ \frac{4 \times 3}{33 \times 3} = \frac{12}{99} = 0.\dot{1}\dot{2} $$

Answer:

\(0.\dot{1}\dot{2}\) (or \(\frac{4}{33}\))