`(2)/(2 + (2)/(3 + (2)/(3 + (2)/(3))) xx 0.39)` is simplified to

Options

A. `(1)/(3)`
B. 2
C. 6
D. None of these

Answer

To simplify the expression \(\frac{2}{2 + \frac{2}{3 + \frac{2}{3 + \frac{2}{3}}}} \times 0.39\), we will follow these steps:

### Step 1: Simplify the innermost fraction
First, we simplify the innermost part of the expression:
\[ 3 + \frac{2}{3} \]
To do this, we need a common denominator:
\[ 3 = \frac{9}{3} \implies 3 + \frac{2}{3} = \frac{9}{3} + \frac{2}{3} = \frac{11}{3} \]

**Hint for Step 1:** Remember to convert whole numbers to fractions with a common denominator when adding.

### Step 2: Replace the innermost fraction in the next layer
Now we replace the innermost fraction in the next layer:
\[ 3 + \frac{2}{\frac{11}{3}} = 3 + \frac{2 \times 3}{11} = 3 + \frac{6}{11} \]
Again, convert \(3\) to a fraction:
\[ 3 = \frac{33}{11} \implies 3 + \frac{6}{11} = \frac{33}{11} + \frac{6}{11} = \frac{39}{11} \]

**Hint for Step 2:** Always express whole numbers as fractions to maintain consistency in operations.

### Step 3: Replace the second layer in the outermost fraction
Now we replace the second layer in the outermost fraction:
\[ \frac{2}{2 + \frac{2}{\frac{39}{11}}} = \frac{2}{2 + \frac{2 \times 11}{39}} = \frac{2}{2 + \frac{22}{39}} \]
Convert \(2\) to a fraction:
\[ 2 = \frac{78}{39} \implies 2 + \frac{22}{39} = \frac{78}{39} + \frac{22}{39} = \frac{100}{39} \]

**Hint for Step 3:** When adding fractions, ensure they have the same denominator for easy addition.

### Step 4: Simplify the outermost fraction
Now we simplify the outermost fraction:
\[ \frac{2}{\frac{100}{39}} = 2 \times \frac{39}{100} = \frac{78}{100} \]

**Hint for Step 4:** Dividing by a fraction is the same as multiplying by its reciprocal.

### Step 5: Multiply by 0.39
Now we multiply this result by \(0.39\):
\[ \frac{78}{100} \times 0.39 = \frac{78 \times 39}{100} \]
Calculating \(78 \times 39\):
\[ 78 \times 39 = 3042 \]
Thus, we have:
\[ \frac{3042}{100} = 30.42 \]

**Hint for Step 5:** When multiplying fractions, multiply the numerators and denominators separately.

### Final Step: Simplify the result
Now, we can express \(30.42\) as a fraction:
\[ 30.42 = \frac{3042}{100} = \frac{1521}{50} \]

**Hint for Final Step:** Always check if the fraction can be simplified further.

### Conclusion
The simplified form of the expression \(\frac{2}{2 + \frac{2}{3 + \frac{2}{3 + \frac{2}{3}}}} \times 0.39\) is \(\frac{1521}{50}\).