Find the vlaue of `27xx1.bar( 2) xx 5.526bar(2) xx 0.bar( 6)`.

Options

A. `121.bar( 57)`
B. `121.bar( 75)`
C. `121.7bar(5)`
D. None of these

Answer

To solve the expression \( 27 \times 1.\overline{2} + 5.526\overline{2} \times 0.\overline{6} \), we will follow these steps:

### Step 1: Convert \( 1.\overline{2} \) to a Fraction
Let \( x = 1.\overline{2} \).
Then, \( 10x = 12.\overline{2} \).

Now, subtract the first equation from the second:
\[ 10x - x = 12.\overline{2} - 1.\overline{2} \]
\[ 9x = 11 \implies x = \frac{11}{9} \]

### Step 2: Convert \( 5.526\overline{2} \) to a Fraction
Let \( y = 5.526\overline{2} \).
Then, \( 1000y = 5526.\overline{2} \) and \( 10y = 55.262\overline{2} \).

Now, subtract:
\[ 1000y - 10y = 5526.\overline{2} - 55.262\overline{2} \]
\[ 990y = 5526 - 55.262 = 5470.737 \implies y = \frac{5470.737}{990} \]
To simplify, we can express \( 5470.737 \) as \( 5470 + 0.737 \) and convert \( 0.737 \) to a fraction.

### Step 3: Convert \( 0.\overline{6} \) to a Fraction
Let \( z = 0.\overline{6} \).
Then, \( 10z = 6.\overline{6} \).

Subtract:
\[ 10z - z = 6.\overline{6} - 0.\overline{6} \]
\[ 9z = 6 \implies z = \frac{2}{3} \]

### Step 4: Substitute Back into the Expression
Now we substitute back into the expression:
\[ 27 \times \frac{11}{9} + \left( \frac{5470.737}{990} \times \frac{2}{3} \right) \]

### Step 5: Calculate Each Term
1. Calculate \( 27 \times \frac{11}{9} \):
\[ = \frac{27 \times 11}{9} = \frac{297}{9} = 33 \]

2. Calculate \( \frac{5470.737}{990} \times \frac{2}{3} \):
\[ = \frac{5470.737 \times 2}{990 \times 3} = \frac{10941.474}{2970} \]

### Step 6: Combine the Results
Now, add the results from Step 5:
\[ 33 + \frac{10941.474}{2970} \]

### Step 7: Final Calculation
Convert \( 33 \) to have a common denominator with \( \frac{10941.474}{2970} \):
\[ 33 = \frac{33 \times 2970}{2970} = \frac{98010}{2970} \]
Now add:
\[ \frac{98010 + 10941.474}{2970} = \frac{108951.474}{2970} \]

### Step 8: Simplify the Result
Finally, divide to find the decimal:
\[ \frac{108951.474}{2970} \approx 36.69 \]

### Final Answer
The value of the expression is approximately \( 36.69 \).

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