`[(1+(1)/(10+(1)/(10)))xx(1+(1)/(10+(1)/(10)))-(1-(1)/(10+(1)/(10)))xx (1-(1)/(10+(1)/(10)))+(1+(1)/(10+(1)/(10)))+(1-(1)/(10+(1)/(10)))]` simplifies to

Options

A. `(100)/(101)`
B. `(90)/(101)`
C. `(20)/(101)`
D. `(101)/(100)`

Answer

To simplify the expression \[ \left[ \left( 1 + \frac{1}{10 + \frac{1}{10}} \right) \times \left( 1 + \frac{1}{10 + \frac{1}{10}} \right) - \left( 1 - \frac{1}{10 + \frac{1}{10}} \right) \times \left( 1 - \frac{1}{10 + \frac{1}{10}} \right) + \left( 1 + \frac{1}{10 + \frac{1}{10}} \right) + \left( 1 - \frac{1}{10 + \frac{1}{10}} \right) \right] \]

### Step 1: Simplify the inner fractions
First, we simplify the term \( 10 + \frac{1}{10} \):
\[ 10 + \frac{1}{10} = \frac{100 + 1}{10} = \frac{101}{10} \]

### Step 2: Substitute back into the expression
Now substitute this back into the expression:
\[ 1 + \frac{1}{10 + \frac{1}{10}} = 1 + \frac{1}{\frac{101}{10}} = 1 + \frac{10}{101} = \frac{101 + 10}{101} = \frac{111}{101} \]
And similarly,
\[ 1 - \frac{1}{10 + \frac{1}{10}} = 1 - \frac{10}{101} = \frac{101 - 10}{101} = \frac{91}{101} \]

### Step 3: Substitute \( a \) and \( b \)
Let \( a = \frac{111}{101} \) and \( b = \frac{91}{101} \). Now the expression becomes:
\[ \left( a \times a - b \times b + a + b \right) \]

### Step 4: Calculate \( a^2 \) and \( b^2 \)
Calculate \( a^2 \) and \( b^2 \):
\[ a^2 = \left( \frac{111}{101} \right)^2 = \frac{12321}{10201} \]
\[ b^2 = \left( \frac{91}{101} \right)^2 = \frac{8281}{10201} \]

### Step 5: Substitute \( a^2 \) and \( b^2 \) into the expression
Now substitute these values back into the expression:
\[ \frac{12321}{10201} - \frac{8281}{10201} + \frac{111}{101} + \frac{91}{101} \]
Combine the first two fractions:
\[ \frac{12321 - 8281}{10201} = \frac{4040}{10201} \]

### Step 6: Combine \( a + b \)
Now combine \( a + b \):
\[ a + b = \frac{111 + 91}{101} = \frac{202}{101} = 2 \]

### Step 7: Final expression
Now the expression becomes:
\[ \frac{4040}{10201} + 2 \]
Convert \( 2 \) to a fraction with the same denominator:
\[ 2 = \frac{20202}{10201} \]
So,
\[ \frac{4040 + 20202}{10201} = \frac{24242}{10201} \]

### Step 8: Simplify the final fraction
The final simplified expression is:
\[ \frac{24242}{10201} \]