Convert each of the following into a vulgar fraction :
`0.57bar(4)`

Answer

To convert the repeating decimal \( 0.57\overline{4} \) into a vulgar fraction, we can follow these steps:

### Step 1: Define the variable
Let \( x = 0.5744444\ldots \) (where the digit 4 repeats indefinitely).

### Step 2: Eliminate the non-repeating part
Since there are two digits before the repeating part (57), we multiply \( x \) by 100 to shift the decimal point two places to the right:
\[ 100x = 57.44444\ldots \]

### Step 3: Eliminate the repeating part
Next, we need to eliminate the repeating part. To do this, we multiply \( x \) by 10 (since there is one digit in the repeating part):
\[ 10x = 5.74444\ldots \]

### Step 4: Set up the equations
Now we have two equations:
1. \( 100x = 57.44444\ldots \) (Equation 1)
2. \( 10x = 5.74444\ldots \) (Equation 2)

### Step 5: Subtract the two equations
Now we subtract Equation 2 from Equation 1:
\[ 100x - 10x = 57.44444\ldots - 5.74444\ldots \]
This simplifies to:
\[ 90x = 51.7 \]

### Step 6: Solve for \( x \)
Now, we can solve for \( x \):
\[ x = \frac{51.7}{90} \]

### Step 7: Convert \( 51.7 \) to a fraction
To convert \( 51.7 \) to a fraction, we can express it as:
\[ 51.7 = \frac{517}{10} \]
Thus, we can rewrite \( x \):
\[ x = \frac{517/10}{90} = \frac{517}{900} \]

### Final Answer
Therefore, the repeating decimal \( 0.57\overline{4} \) can be expressed as the vulgar fraction:
\[ \frac{517}{900} \]

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