Question

Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.

Use the infinite geometric sum formula to write 0.757575… as a fraction in reduced form. Show all steps neatly.

(3 points)

Answer 1

To convert the repeating decimal \( 0.757575\ldots \) into a fraction, we can recognize this as an infinite geometric series. We can express this decimal more conveniently.

### Step 1: Identify the repeating part
Notice that the repeating part is \( 75 \). We can express \( 0.757575\ldots \) as:
\[
x = 0.757575\ldots
\]

### Step 2: Rewrite using the geometric series formula
This decimal can be rewritten as:
\[
x = 0.75 + 0.0075 + 0.000075 + \ldots
\]

The first term \( a \) of our geometric series is \( 0.75 \) and the common ratio \( r \) is \( 0.01 \).

### Step 3: Use the formula for the sum of an infinite geometric series
The sum of an infinite geometric series can be calculated using the formula:
\[
S = \frac{a}{1 - r}
\]
where \( |r| < 1 \).

### Step 4: Substitute values into the formula
Substituting the values we identified:
\[
a = 0.75 \quad \text{and} \quad r = 0.01
\]
Now we can substitute \( a \) and \( r \) into the formula:
\[
x = \frac{0.75}{1 - 0.01} = \frac{0.75}{0.99}
\]

### Step 5: Simplify the fraction
Now we need to simplify \( \frac{0.75}{0.99} \). To eliminate the decimals, we can multiply both the numerator and the denominator by 100:
\[
x = \frac{0.75 \times 100}{0.99 \times 100} = \frac{75}{99}
\]

Now, we need to reduce the fraction \( \frac{75}{99} \) to its simplest form.

### Step 6: Find the greatest common divisor (GCD)
To reduce \( \frac{75}{99} \):
1. Find the GCD of 75 and 99. The prime factorization of 75 is \( 3 \times 5^2 \) and for 99 is \( 3^2 \times 11 \).
2. The GCD is 3.

### Step 7: Divide numerator and denominator by their GCD
Now we divide each by their GCD:
\[
\frac{75 \div 3}{99 \div 3} = \frac{25}{33}
\]

### Final Result
Thus, the repeating decimal \( 0.757575\ldots \) as a fraction in reduced form is:
\[
\boxed{\frac{25}{33}}
\]