Question

Convert 0.09 repeating
to a rational number in simplest form.

Show your work.

give me a few words for explanation .

Answer 1

To convert \( 0.09\overline{09} \) (where "09" repeats) to a rational number, we can use algebra.

Let \( x = 0.09\overline{09} \).

### Step 1: Set up the equation
This means:
\[ x = 0.09090909\ldots \]

### Step 2: Multiply to shift the decimal
To eliminate the repeating part, we can multiply \( x \) by 100, since the repeating block has 2 digits:
\[ 100x = 9.09090909\ldots \]

### Step 3: Set up a second equation
Now we have two equations:
1. \( x = 0.09090909\ldots \)
2. \( 100x = 9.09090909\ldots \)

### Step 4: Subtract the first equation from the second
Now we'll subtract the first equation from the second:
\[ 100x - x = 9.09090909\ldots - 0.09090909\ldots \]
\[ 99x = 9 \]

### Step 5: Solve for \( x \)
Now divide both sides by 99:
\[ x = \frac{9}{99} \]

### Step 6: Simplify the fraction
Now we simplify \( \frac{9}{99} \):
\[
\frac{9 \div 9}{99 \div 9} = \frac{1}{11}
\]

### Final Result
Thus, \( 0.09\overline{09} \) as a rational number in simplest form is:
\[
\frac{1}{11}
\]

### Explanation
By using algebra to represent the repeating decimal, we were able to find a fraction that represents the number exactly. This method involved setting up an equation, manipulating it to eliminate the repeating part, and simplifying the resulting fraction.