1.22`bar(4)` can be expressed as a fraction in simplest form as ________.

Options

A. `(450)/(551)`
B. `(1224)/(1000)`
C. `(551)/(450)`
D. `(1000)/(1224)`

Answer

To express \( 1.22\overline{4} \) as a fraction in simplest form, we can follow these steps:

### Step 1: Define the repeating decimal
Let \( x = 1.224444\ldots \) (where \( 4 \) is the repeating digit).

### Step 2: Multiply by a power of 10
Since the repeating part has one digit (the digit \( 4 \)), we can multiply \( x \) by \( 10 \) to shift the decimal point one place to the right:
\[ 10x = 12.24444\ldots \]

### Step 3: Set up the equations
Now we have two equations:
1. \( x = 1.22444\ldots \) (Equation 1)
2. \( 10x = 12.24444\ldots \) (Equation 2)

### Step 4: Subtract the first equation from the second
Now, we will subtract Equation 1 from Equation 2:
\[ 10x - x = 12.24444\ldots - 1.22444\ldots \]
This simplifies to:
\[ 9x = 12.24444 - 1.22444 \]
Calculating the right side:
\[ 9x = 12.24444 - 1.22444 = 11.02 \]

### Step 5: Solve for \( x \)
Now, divide both sides by \( 9 \):
\[ x = \frac{11.02}{9} \]

### Step 6: Convert \( 11.02 \) to a fraction
To convert \( 11.02 \) to a fraction, we can express it as:
\[ 11.02 = \frac{1102}{100} \]
Thus:
\[ x = \frac{1102}{100} \times \frac{1}{9} = \frac{1102}{900} \]

### Step 7: Simplify the fraction
Now, we need to simplify \( \frac{1102}{900} \). We can find the greatest common divisor (GCD) of \( 1102 \) and \( 900 \).

The GCD of \( 1102 \) and \( 900 \) is \( 2 \), so we divide both the numerator and the denominator by \( 2 \):
\[ x = \frac{1102 \div 2}{900 \div 2} = \frac{551}{450} \]

### Final Answer
Thus, \( 1.22\overline{4} \) can be expressed as a fraction in simplest form as:
\[ \frac{551}{450} \]

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