Expression of `2.bar (44)` as a rational in form of `(p)/(q)`is

Options

A. `(231)/(98)`
B. `(230)/(99)`
C. `(22)/(9)`
D. `(233)/(99)`

Answer

To express the repeating decimal \(2.\overline{44}\) as a rational number in the form \(\frac{p}{q}\), we can follow these steps:

### Step 1: Let \(x\) be the repeating decimal
Let \(x = 2.\overline{44}\).

### Step 2: Multiply by a power of 10 to shift the decimal point
Since the repeating part "44" has 2 digits, we multiply \(x\) by \(100\) to shift the decimal point two places to the right:
\[ 100x = 244.\overline{44} \]

### Step 3: Set up an equation to eliminate the repeating part
Now, we can set up the equation:
\[ 100x = 244.\overline{44} \]
\[ x = 2.\overline{44} \]

### Step 4: Subtract the second equation from the first
Subtract the second equation from the first to eliminate the repeating decimal:
\[ 100x - x = 244.\overline{44} - 2.\overline{44} \]
This simplifies to:
\[ 99x = 244 - 2 \]
\[ 99x = 242 \]

### Step 5: Solve for \(x\)
Now, divide both sides by \(99\):
\[ x = \frac{242}{99} \]

### Step 6: Simplify the fraction
Next, we simplify \(\frac{242}{99}\). We can find the greatest common divisor (GCD) of \(242\) and \(99\). The GCD is \(11\):
\[ \frac{242 \div 11}{99 \div 11} = \frac{22}{9} \]

### Final Result
Thus, the expression of \(2.\overline{44}\) as a rational number in the form \(\frac{p}{q}\) is:
\[ \frac{22}{9} \]

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