Find the value of `9.46bar(7)` in a vulgar fraction.

Options

A. `9(421)/(900)`
B. `9(422)/(900)`
C. `9(435)/(900)`
D. `9(437)/(900)`

Answer

To convert the repeating decimal \( 9.467\overline{467} \) into a vulgar fraction, we can follow these steps:

### Step 1: Assign a variable to the repeating decimal
Let \( x = 9.467467467... \)

### Step 2: Multiply by a power of 10 to shift the decimal point
Since the repeating part "467" has 3 digits, we multiply \( x \) by \( 1000 \):
\[ 1000x = 9467.467467467... \]

### Step 3: Multiply by a power of 10 to shift the decimal point again
Now, we also multiply \( x \) by \( 10 \) to shift the decimal point:
\[ 10x = 94.674674674... \]

### Step 4: Set up the equation
Now, we have two equations:
1. \( 1000x = 9467.467467467... \)
2. \( 10x = 94.674674674... \)

### Step 5: Subtract the second equation from the first
Now, we subtract the second equation from the first:
\[ 1000x - 10x = 9467.467467467... - 94.674674674... \]
This simplifies to:
\[ 990x = 9372.792792792... \]

### Step 6: Solve for \( x \)
Now, we can isolate \( x \):
\[ x = \frac{9372.792792792...}{990} \]

### Step 7: Convert the decimal to a fraction
To convert \( 9372.792792792... \) to a fraction, we can express it as:
\[ 9372.792792792... = 9372 + \frac{792}{999} \]
So, we can write:
\[ x = \frac{9372 \times 999 + 792}{990 \times 999} \]

### Step 8: Simplify the fraction
Now, we need to simplify the fraction. First, calculate \( 9372 \times 999 + 792 \) and \( 990 \times 999 \):
- Calculate \( 9372 \times 999 = 9364328 \)
- Add \( 792 \) to get \( 9364328 + 792 = 9365120 \)
- Calculate \( 990 \times 999 = 989010 \)

Thus, we have:
\[ x = \frac{9365120}{989010} \]

### Step 9: Reduce the fraction
Now, we need to reduce \( \frac{9365120}{989010} \) to its simplest form. We can find the GCD of the numerator and denominator and divide both by it.

After performing the calculations, we find:
\[ x = \frac{468256}{49455} \]

### Final Answer
Thus, the value of \( 9.467\overline{467} \) in vulgar fraction form is:
\[ \frac{468256}{49455} \]