Let `L = 0.71bar5`. When L is written as a fraction in the lowest terms, the denominator exceeds the numerator by:

Options

A. A. 644
B. B. 64
C. C. 900
D. D. 744

Answer

To convert the repeating decimal \( L = 0.71\overline{5} \) into a fraction and find the difference between the denominator and the numerator, we can follow these steps:

### Step 1: Define the repeating decimal
Let \( L = 0.715555\ldots \)

### Step 2: Multiply to eliminate the repeating part
To eliminate the repeating decimal, we can multiply \( L \) by 1000 (to move the decimal point three places to the right):
\[ 1000L = 715.555\ldots \]

### Step 3: Set up the equation
Now, we can set up the equation:
\[ 1000L = 715 + L \]

### Step 4: Solve for \( L \)
Subtract \( L \) from both sides:
\[ 1000L - L = 715 \]
\[ 999L = 715 \]
Now, divide both sides by 999:
\[ L = \frac{715}{999} \]

### Step 5: Simplify the fraction
Next, we need to simplify \( \frac{715}{999} \). We can find the greatest common divisor (GCD) of 715 and 999.

- The prime factorization of 715 is \( 5 \times 11 \times 13 \).
- The prime factorization of 999 is \( 3^3 \times 37 \).

Since there are no common factors, the fraction is already in its simplest form:
\[ L = \frac{715}{999} \]

### Step 6: Find the difference between the denominator and numerator
Now, we need to find how much the denominator exceeds the numerator:
\[ 999 - 715 = 284 \]

### Final Answer
The denominator exceeds the numerator by \( 284 \).