`0.12bar(3)` can be expressed in rational form as

Options

A. `(900)/(111)`
B. `(111)/(900)`
C. `(123)/(10)`
D. `(121)/(900)`

Answer

To express \(0.12\overline{3}\) in rational form, we can follow these steps:

### Step 1: Define the repeating decimal
Let \(x = 0.123333...\) (where the 3 is repeating).

### Step 2: Eliminate the non-repeating part
Since there are two digits before the repeating part (12), we multiply both sides by 100 to shift the decimal point two places to the right:
\[ 100x = 12.3333... \]

### Step 3: Isolate the repeating part
Next, we multiply both sides by 10 to shift the decimal point one more place to the right:
\[ 10x = 1.23333... \]

### Step 4: Set up the equations
Now we have two equations:
1. \(100x = 12.3333...\)
2. \(10x = 1.23333...\)

### Step 5: Subtract the second equation from the first
Subtract the second equation from the first to eliminate the repeating part:
\[ 100x - 10x = 12.3333... - 1.23333... \]
This simplifies to:
\[ 90x = 11.1 \]

### Step 6: Solve for \(x\)
Now, we can solve for \(x\):
\[ x = \frac{11.1}{90} \]

### Step 7: Convert to a fraction
To convert \(11.1\) to a fraction, we can express it as:
\[ 11.1 = \frac{111}{10} \]
Thus, we have:
\[ x = \frac{111/10}{90} = \frac{111}{900} \]

### Step 8: Simplify the fraction
Now, we can simplify \(\frac{111}{900}\). The greatest common divisor (GCD) of 111 and 900 is 3:
\[ \frac{111 \div 3}{900 \div 3} = \frac{37}{300} \]

### Final Result
Thus, the rational form of \(0.12\overline{3}\) is:
\[ \frac{37}{300} \]

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