`3.00bar(72)`=?

Options

A. `29772/9900`
B. `29772/9999`
C. `29772/9990`
D. None of these

Answer

To solve the problem \( 3.00\overline{72} \), we will convert the repeating decimal into a fraction step by step.

### Step 1: Define the repeating decimal
Let \( x = 3.00\overline{72} \). This means that \( x = 3.0072727272...\)

### Step 2: Separate the integer part and the decimal part
We can express \( x \) as:
\[ x = 3 + 0.00\overline{72} \]

### Step 3: Convert the repeating decimal to a fraction
Let \( y = 0.00\overline{72} \). Then:
\[ y = 0.0072727272... \]

To eliminate the repeating part, we can multiply \( y \) by 1000 (since the repeating part starts after two decimal places):
\[ 1000y = 7.272727... \]

Now, we also multiply \( y \) by 100:
\[ 100y = 0.727272... \]

### Step 4: Set up an equation
Now we can set up the equation:
\[ 1000y - 100y = 7.272727... - 0.727272... \]
This simplifies to:
\[ 900y = 7 \]
So,
\[ y = \frac{7}{900} \]

### Step 5: Substitute back into the equation for \( x \)
Now substitute \( y \) back into the equation for \( x \):
\[ x = 3 + \frac{7}{900} \]

### Step 6: Convert to a single fraction
To combine these, we convert 3 into a fraction with a denominator of 900:
\[ x = \frac{3 \times 900}{900} + \frac{7}{900} = \frac{2700 + 7}{900} = \frac{2707}{900} \]

### Step 7: Final simplification
Now we have:
\[ x = \frac{2707}{900} \]
This fraction is already in its simplest form.

### Conclusion
Thus, \( 3.00\overline{72} = \frac{2707}{900} \).