Converting Repeating Decimals to Fractions: Step-by-Step Guide

Converting a repeating decimal like 0.129129129... into a fraction involves a simple algebraic method. The goal is to eliminate the repeating part of the decimal.

Method to Convert Repeating Decimals to Fractions

Follow these steps to convert a repeating decimal to a fraction:

1. Set the decimal number equal to a variable (e.g., \( x \)).
2. Identify the repeating part of the decimal.
3. Count the number of digits in the repeating part. Let this number be \( n \).
4. Multiply the equation from step 1 by \( 10^n \).
5. Subtract the original equation (from step 1) from the new equation (from step 4). This step eliminates the repeating decimal part.
6. Solve the resulting equation for the variable \( x \).
7. Simplify the resulting fraction to its lowest terms.

Applying the Method to 0.129129129...

Let's apply the steps to convert 0.129129129... to a fraction.

1. Set \( x \) equal to the decimal:

   \( x = 0.129129129... \)

2. Identify the repeating part: The repeating part is "129".
3. Count the number of repeating digits: There are 3 repeating digits. So, \( n = 3 \).
4. Multiply the equation by \( 10^n = 10^3 = 1000 \):

   \( 1000x = 129.129129... \)

5. Subtract the original equation (\( x = 0.129129129... \)) from the new equation (\( 1000x = 129.129129... \)):

   \( 1000x - x = (129.129129...) - (0.129129129...) \)

   \( 999x = 129 \)

6. Solve for \( x \):

   \( x = \frac{129}{999} \)

7. Simplify the fraction: Both the numerator (129) and the denominator (999) are divisible by 3.

   Divide the numerator by 3: \( 129 \div 3 = 43 \)

   Divide the denominator by 3: \( 999 \div 3 = 333 \)

   The simplified fraction is \( \frac{43}{333} \).

Comparing the Result with Options

The fraction we obtained is \( \frac{43}{333} \). Let's compare this with the given options:

• Option 1: 43/999
• Option 2: 129/333
• Option 3: 43/333
• Option 4: 129/666

Our result, \( \frac{43}{333} \), matches Option 3.

Revision Table: Steps for Converting Repeating Decimals

Additional Information: Exploring Rational Numbers

A repeating decimal is a characteristic feature of rational numbers. A rational number is any number that can be expressed as the quotient or fraction \( \frac{p}{q} \) of two integers, a numerator \( p \) and a non-zero denominator \( q \).

Properties of Rational Numbers and Decimals

• Every rational number can be written as either a terminating decimal (like \( \frac{1}{4} = 0.25 \)) or a repeating decimal (like \( \frac{1}{3} = 0.333... \) or \( \frac{43}{333} = 0.129129... \)).
• Conversely, every terminating decimal and every repeating decimal represents a rational number. The method demonstrated here is proof of this fact for repeating decimals.
• Irrational numbers, like \( \pi \) or \( \sqrt{2} \), have decimal representations that are non-terminating and non-repeating.

Understanding how to convert repeating decimals to fractions is important for working with rational numbers and solving various mathematical problems.