The problem asks for the decimal result of the following expression:

$R = \frac{(0.254444 \dots) \cdot (0.181818 \dots)}{0.22222 \dots}$

Step 1: Convert repeating decimals into fractions

1. $0.254444 \dots$ is a repeating decimal. The decimal $0.254444 \dots$ can be written as: $0.25\overline{4} = 0.25 + \frac{4}{90} = \frac{23}{90}$

2. $0.181818 \dots$ is a repeating decimal. We recognize that: $0.\overline{18} = \frac{18}{99} = \frac{2}{11}$

3. $0.22222 \dots$ is also a repeating decimal. We convert it to a fraction as: $0.\overline{2} = \frac{2}{9}$

Step 2: Substitute fractions into the expression for $R$

Now substitute these values into the expression:

$R = \frac{\left( \frac{23}{90} \right) \cdot \left( \frac{2}{11} \right)}{\frac{2}{9}}$

Step 3: Simplify the expression

First, multiply the numerators and denominators in the numerator of the fraction:

$\left( \frac{23}{90} \right) \cdot \left( \frac{2}{11} \right) = \frac{23 \cdot 2}{90 \cdot 11} = \frac{46}{990}$

Now, divide by $\frac{2}{9}$:

$R = \frac{\frac{46}{990}}{\frac{2}{9}} = \frac{46}{990} \cdot \frac{9}{2} = \frac{46 \cdot 9}{990 \cdot 2} = \frac{414}{1980}$

Step 4: Simplify the fraction

Simplifying $\frac{414}{1980}$ by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 18:

$R = \frac{414 \div 18}{1980 \div 18} = \frac{23}{110}$

Step 5: Convert to decimal

Finally, $\frac{23}{110}$ as a decimal is approximately:

$R \approx 0.2091$

Thus, the decimal result of the operation is approximately 0.2091.

Would you like further details or explanations?

Related Questions:

1. How do you convert repeating decimals to fractions in general?
2. What is the greatest common divisor (GCD), and how do you find it?
3. What steps are involved in multiplying and dividing fractions?
4. How can you convert a fraction to a decimal manually?
5. Can you explain how to identify the repeating part of a decimal?

Tip:

When dealing with repeating decimals, recognize patterns quickly to convert them into fractions for easier calculations.