Correct expression of `0.02bar(36)` = ?
A. `(13)/(550)`
B. `(236)/(1000)`
C. `2(36)/(1000)`
D. `(13)/(555)`

Options

A. D
B. A
C. C
D. B

Answer

To convert the repeating decimal \(0.02\overline{36}\) into a fraction, we can follow these steps:

### Step 1: Define the repeating decimal
Let \(x = 0.0236363636...\)

### Step 2: Eliminate the repeating part
To eliminate the repeating part, we can multiply \(x\) by a power of 10 that shifts the decimal point to the right of the repeating part. Since "36" has 2 digits, we multiply by 100:
\[ 100x = 2.36363636... \]

### Step 3: Set up another equation
Now, we also want to shift the decimal point to the right of the non-repeating part. Since "02" has 2 digits, we multiply \(x\) by 1000:
\[ 1000x = 23.6363636... \]

### Step 4: Subtract the two equations
Now we have:
\[ 1000x - 100x = 23.6363636... - 2.36363636... \]
This simplifies to:
\[ 900x = 21.27 \]

### Step 5: Solve for \(x\)
Now, we can solve for \(x\):
\[ x = \frac{21.27}{900} \]

### Step 6: Convert \(21.27\) to a fraction
To convert \(21.27\) to a fraction, we can write it as:
\[ 21.27 = \frac{2127}{100} \]
Thus,
\[ x = \frac{2127}{100 \times 900} = \frac{2127}{90000} \]

### Step 7: Simplify the fraction
Now we need to simplify \(\frac{2127}{90000}\). We can find the GCD of 2127 and 90000 and divide both the numerator and denominator by it.

After simplification, we find:
\[ x = \frac{13}{550} \]

### Conclusion
Thus, the correct expression of \(0.02\overline{36}\) is:
\[ \boxed{\frac{13}{550}} \]