Convert `0.234bar5` in `p/(q)` form of rational number

Options

A. a) `129/(550)`
B. b) `469/(1980)`
C. c) `2368/(9900)`
D. d) None of these

Answer

To convert the repeating decimal \(0.234\overline{5}\) into the form \(\frac{p}{q}\), we can follow these steps:

### Step 1: Let \(x = 0.234\overline{5}\)
Define \(x\) as the repeating decimal we want to convert.

### Step 2: Multiply by a power of 10 to shift the decimal point
Since the repeating part "5" starts after three decimal places, we multiply \(x\) by \(10^4 = 10000\) to shift the decimal point four places to the right:
\[ 10000x = 2345.5555\ldots \]

### Step 3: Multiply by a power of 10 to isolate the repeating part
Next, we multiply \(x\) by \(10^3 = 1000\) to shift the decimal point three places to the right:
\[ 1000x = 234.5555\ldots \]

### Step 4: Set up the equation
Now we have two equations:
1. \(10000x = 2345.5555\ldots\)
2. \(1000x = 234.5555\ldots\)

### Step 5: Subtract the second equation from the first
Subtract the second equation from the first to eliminate the repeating part:
\[ 10000x - 1000x = 2345.5555\ldots - 234.5555\ldots \]
This simplifies to:
\[ 9000x = 2111 \]

### Step 6: Solve for \(x\)
Now, divide both sides by 9000 to solve for \(x\):
\[ x = \frac{2111}{9000} \]

### Conclusion
Thus, the repeating decimal \(0.234\overline{5}\) can be expressed as the rational number:
\[ \frac{2111}{9000} \]