Simplifying the Mathematical Expression

The question asks us to simplify the given mathematical expression:

$ \frac{5+5\times5}{5\times5+5} \times \frac{\frac{1}{5}\div(\frac{1}{5}\times\frac{1}{5})}{(\frac{1}{5}\times\frac{1}{5})\div\frac{1}{5}} - (5 - \frac{1}{5}) \times \frac{10}{2} $

We will simplify this expression step-by-step following the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Step 1: Simplify the first fraction

Calculate the numerator and denominator separately:

• Numerator: $5 + 5 \times 5 = 5 + 25 = 30$
• Denominator: $5 \times 5 + 5 = 25 + 5 = 30$

The first fraction simplifies to: $\frac{30}{30} = 1$.

Step 2: Simplify the second fraction

This fraction involves divisions and multiplications of fractions. Let's simplify the numerator and denominator parts first.

• Numerator part: $\frac{1}{5}\div(\frac{1}{5}\times\frac{1}{5}) = \frac{1}{5}\div\frac{1}{25} = \frac{1}{5} \times \frac{25}{1} = 5$
• Denominator part: $(\frac{1}{5}\times\frac{1}{5})\div\frac{1}{5} = \frac{1}{25}\div\frac{1}{5} = \frac{1}{25} \times \frac{5}{1} = \frac{5}{25} = \frac{1}{5}$

Now, divide the simplified numerator part by the simplified denominator part:

$ \frac{5}{\frac{1}{5}} = 5 \times 5 = 25 $

The second fraction simplifies to $25$.

Step 3: Simplify the third part of the expression

This part involves subtraction inside parentheses and multiplication.

• Parentheses: $5 - \frac{1}{5} = \frac{25}{5} - \frac{1}{5} = \frac{24}{5}$
• Multiplication part: $\frac{10}{2} = 5$

Multiply the results: $\frac{24}{5} \times 5 = 24$.

The third part simplifies to $24$.

Step 4: Combine the simplified parts

Now substitute the simplified values back into the original expression:

$ (1) \times (25) - (24) $

Perform the multiplication first:

$ 25 - 24 $

Finally, perform the subtraction:

$ 1 $

The simplified value of the expression is $1$. This corresponds to Option D.