The simplification of `3.bar(36) - 2.bar(05) + 1.bar(33)` equals:

Options

A. `2.6`
B. `2.bar(61)`
C. `2.64`
D. `2.bar(64)`

Answer

To simplify the expression \(3.\overline{36} - 2.\overline{05} + 1.\overline{33}\), we will follow these steps:

### Step 1: Understand the notation
The notation \(x.\overline{y}\) indicates that the digits in \(y\) repeat indefinitely. For example, \(3.\overline{36}\) means \(3.363636...\).

### Step 2: Split the numbers
We can separate the whole number part from the decimal part:
\[ 3.\overline{36} = 3 + 0.\overline{36} \]
\[ 2.\overline{05} = 2 + 0.\overline{05} \]
\[ 1.\overline{33} = 1 + 0.\overline{33} \]

### Step 3: Rewrite the expression
Now, we can rewrite the expression as:
\[ (3 + 0.\overline{36}) - (2 + 0.\overline{05}) + (1 + 0.\overline{33}) \]

### Step 4: Combine the whole numbers
Combine the whole number parts:
\[ 3 - 2 + 1 = 2 \]

### Step 5: Combine the decimal parts
Now, we need to simplify the decimal parts:
\[ 0.\overline{36} - 0.\overline{05} + 0.\overline{33} \]

### Step 6: Convert repeating decimals to fractions
Convert each repeating decimal to a fraction:
1. \(0.\overline{36} = \frac{36}{99}\) (since \(0.\overline{36} = \frac{36}{99}\))
2. \(0.\overline{05} = \frac{5}{99}\) (since \(0.\overline{05} = \frac{5}{99}\))
3. \(0.\overline{33} = \frac{33}{99}\) (since \(0.\overline{33} = \frac{33}{99}\))

### Step 7: Substitute fractions into the expression
Now substitute these fractions back into the expression:
\[ \frac{36}{99} - \frac{5}{99} + \frac{33}{99} \]

### Step 8: Combine the fractions
Combine the fractions:
\[ \frac{36 - 5 + 33}{99} = \frac{64}{99} \]

### Step 9: Add the whole number and the decimal part
Now, combine the whole number part (which is 2) with the fraction:
\[ 2 + \frac{64}{99} \]

### Step 10: Write the final answer
This can be expressed as:
\[ 2.\overline{64} \]

### Final Answer:
Thus, the simplification of \(3.\overline{36} - 2.\overline{05} + 1.\overline{33}\) equals \(2.\overline{64}\).