Write the value of `2.1bar(34)` in the form of a simple fraction.

Explanation

To convert the repeating decimal \(2.1\overline{34}\) into a simple fraction, we can follow these steps:

### Step 1: Define the variable
Let \( x = 2.134343434...\)

### Step 2: Separate the non-repeating and repeating parts
The non-repeating part is \(2.1\) and the repeating part is \(34\). We can express \(x\) as:
\[ x = 2.1 + 0.034343434...\]

### Step 3: Convert the non-repeating part to a fraction
The non-repeating part \(2.1\) can be written as:
\[ 2.1 = \frac{21}{10} \]

### Step 4: Define the repeating decimal
Let \( y = 0.034343434...\)

### Step 5: Eliminate the repeating part
To eliminate the repeating decimal, we multiply \(y\) by \(1000\) (to shift the decimal point three places to the right):
\[ 1000y = 34.343434...\]

Now, we also multiply \(y\) by \(100\) (to shift the decimal point two places to the right):
\[ 100y = 3.434343...\]

### Step 6: Set up the equation
Now we can set up the equation:
\[ 1000y - 100y = 34.343434... - 3.434343...\]
This simplifies to:
\[ 900y = 34 - 3 = 31 \]

### Step 7: Solve for \(y\)
Now, we can solve for \(y\):
\[ y = \frac{31}{900} \]

### Step 8: Combine the fractions
Now we can combine \(x\):
\[ x = 2.1 + y = \frac{21}{10} + \frac{31}{900} \]

### Step 9: Find a common denominator
The common denominator for \(10\) and \(900\) is \(900\):
\[ \frac{21}{10} = \frac{21 \times 90}{10 \times 90} = \frac{1890}{900} \]

### Step 10: Add the fractions
Now we can add the two fractions:
\[ x = \frac{1890}{900} + \frac{31}{900} = \frac{1890 + 31}{900} = \frac{1921}{900} \]

### Final Answer
Thus, the value of \(2.1\overline{34}\) in the form of a simple fraction is:
\[ \frac{1921}{900} \]

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