Express 0.123̅ in the form of a rational number.

To express \( 0.123\overline{3} \) as a rational number, let's follow these steps:

Let:
\[
x = 0.123\overline{3}
\]
This means the decimal repeats indefinitely with 3s.

### Step 1: Eliminate the repeating decimal
Multiply \( x \) by 10 to shift the decimal point one place to the right:
\[
10x = 1.23\overline{3}
\]

Now subtract the original \( x \) from this equation to eliminate the repeating part:
\[
10x - x = (1.23\overline{3}) - (0.123\overline{3})
\]

This simplifies to:
\[
9x = 1.11
\]

### Step 2: Solve for \( x \)
Now, divide both sides of the equation by 9:
\[
x = \frac{1.11}{9}
\]

### Step 3: Convert \( 1.11 \) to a fraction
Notice that \( 1.11 \) is a terminating decimal, so we can express it as a fraction:
\[
1.11 = \frac{111}{100}
\]

### Step 4: Substitute back into the equation
Substitute \( \frac{111}{100} \) back into the equation for \( x \):
\[
x = \frac{\frac{111}{100}}{9} = \frac{111}{900}
\]

### Step 5: Simplify the fraction
Now, simplify the fraction \( \frac{111}{900} \). The greatest common divisor (GCD) of 111 and 900 is 3, so divide both numerator and denominator by 3:
\[
x = \frac{111 \div 3}{900 \div 3} = \frac{37}{300}
\]

Thus, \( 0.123\overline{3} \) as a rational number is:
\[
x = \frac{37}{300}
\]

Would you like more details on this solution or have any further questions?

Here are 5 related questions:
1. How do you convert other repeating decimals into fractions?
2. What is the significance of rational numbers in number theory?
3. Can non-repeating decimals be expressed as fractions?
4. How does the method differ for repeating patterns longer than one digit?
5. What are examples of repeating decimals that represent simple fractions?

**Tip:** When converting repeating decimals to fractions, it's useful to know the general formula \( x = \frac{n}{d} \), where \( n \) is the repeating part's sum and \( d \) is the number of repeating digits times 9.

Learn how to convert the repeating decimal 0.123̅ into a rational number with step-by-step instructions. This problem involves algebraic manipulation and the method for converting repeating decimals into fractions. Suitable for Grades 7-9.

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