Simplifying Decimal Squares Fraction Calculation

This problem requires us to simplify a fraction containing the squares of several decimal numbers. We need to evaluate the expression:

Let's analyze the relationship between the numbers in the numerator and the denominator.

Analyzing Decimal Relationships

Observe the numbers involved:

• Numerator terms: 0.05, 0.41, 0.073
• Denominator terms: 0.005, 0.041, 0.0073

We can see that each number in the denominator is exactly one-tenth (1/10) of the corresponding number in the numerator:

• $0.005 = \frac{0.05}{10}$
• $0.041 = \frac{0.41}{10}$
• $0.0073 = \frac{0.073}{10}$

Step-by-Step Fraction Simplification

Now, let's substitute these relationships into the squares in the denominator:

1. Square the first term in the denominator: $$(0.005)^2 = \left(\frac{0.05}{10}\right)^2$$ Using the property $(a/b)^n = a^n / b^n$, we get: $$ \left(\frac{0.05}{10}\right)^2 = \frac{(0.05)^2}{10^2} = \frac{(0.05)^2}{100} $$
2. Square the second term in the denominator: $$(0.041)^2 = \left(\frac{0.41}{10}\right)^2 = \frac{(0.41)^2}{10^2} = \frac{(0.41)^2}{100}$$
3. Square the third term in the denominator: $$(0.0073)^2 = \left(\frac{0.073}{10}\right)^2 = \frac{(0.073)^2}{10^2} = \frac{(0.073)^2}{100}$$

Now substitute these back into the original fraction:

Notice that the denominator has a common factor of $\frac{1}{100}$. Let's factor it out:

Let the sum of the squares in the numerator be represented by 'S':

The expression now becomes:

To simplify this complex fraction, we multiply the numerator by the reciprocal of the denominator:

The 'S' terms cancel out, leaving:

Final Calculation Result

The simplified value of the given fraction is 100. This corresponds to the third option.