To solve the problem, we'll first convert each repeating decimal into a fraction and then multiply those fractions.

Step 1: Convert Each Repeating Decimal to a Fraction

Converting $0.1363636\ldots$

Let's denote $x = 0.1363636\ldots$. Notice that the repeating part "36" is two digits long.

To eliminate the repeating part, multiply $x$ by $100$ (because the repeating sequence has 2 digits): $100x = 13.6363636\ldots$

Subtract $x = 0.1363636\ldots$ from $100x$: $100x - x = 13.6363636\ldots - 0.1363636\ldots$ $99x = 13.5$

Solve for $x$: $x = \frac{13.5}{99}$

To simplify $\frac{13.5}{99}$, note that $13.5 = \frac{27}{2}$: $x = \frac{\frac{27}{2}}{99} = \frac{27}{2 \times 99} = \frac{27}{198}$

Now simplify $\frac{27}{198}$. The greatest common divisor (GCD) of 27 and 198 is 9: $x = \frac{27 \div 9}{198 \div 9} = \frac{3}{22}$

So, $0.1363636\ldots = \frac{3}{22}$.

Converting $0.2222\ldots$

Let's denote $y = 0.2222\ldots$.

Since the repeating part "2" is one digit long, multiply $y$ by $10$: $10y = 2.2222\ldots$

Subtract $y = 0.2222\ldots$ from $10y$: $10y - y = 2.2222\ldots - 0.2222\ldots$ $9y = 2$

Solve for $y$: $y = \frac{2}{9}$

So, $0.2222\ldots = \frac{2}{9}$.

Step 2: Multiply the Fractions

Now, multiply $\frac{3}{22}$ and $\frac{2}{9}$: $\frac{3}{22} \times \frac{2}{9} = \frac{3 \times 2}{22 \times 9} = \frac{6}{198}$

Simplify $\frac{6}{198}$ by dividing the numerator and the denominator by their GCD, which is 6: $\frac{6 \div 6}{198 \div 6} = \frac{1}{33}$

Answer

The value of $0.1363636\ldots \times 0.2222\ldots$ as a fraction in its simplest form is: $\frac{1}{33}$

Would you like more details or have any questions?

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Here are some related questions:

1. How can we convert any repeating decimal to a fraction?
2. What is the process to find the greatest common divisor (GCD) of two numbers?
3. Why does multiplying fractions involve multiplying both the numerators and the denominators?
4. What are some common repeating decimals and their fraction equivalents?
5. How can you check if a fraction is in its simplest form?

Tip: When dealing with repeating decimals, identify the length of the repeating sequence to determine how many places to multiply by (10, 100, etc.).