Converting Repeating Decimals to Fractions: A Comprehensive Guide

Understanding how to convert repeating decimals into fractions is a fundamental skill in mathematics. This guide will walk you through the process, using examples and explanations to make this concept clear and engaging.

Introduction to Repeating Decimals

A repeating decimal is a decimal number that has digits that infinitely repeat. For example, the decimal 0.1515151515... is a repeating decimal where the digits 15 repeat indefinitely. In mathematics, these kinds of decimals can be expressed as fractions, which is the subject of this article.

Steps to Convert a Repeating Decimal to a Fraction

Step 1: Let x be the repeating decimal.

Let us start by denoting the repeating decimal as x 0.1515151515...

Step 2: Multiply both sides by a power of 10.

Since the repeating part has 2 digits, we multiply both sides by 100:

10 15.1515151515...

Step 3: Write down another equation with the same repeating decimal but shifted.

We can also write:

x 0.1515151515...

Step 4: Subtract the two equations.

By subtracting the second equation from the first, we eliminate the repeating part:

10 - x 15.1515151515... - 0.1515151515...

This simplifies to:

99x 15

Step 5: Solve for x.

To find x, we divide both sides by 99:

x frac{15}{99}

Step 6: Reduce the fraction.

We can simplify this fraction by dividing the numerator and the denominator by their greatest common divisor (GCD), which is 3:

x frac{15 div 3}{99 div 3} frac{5}{33}

Generalizing the Process

The method we used can be generalized for any repeating decimal with a certain length of repetition. For a repeating decimal such as 0.ABCABCABC..., ABC is the repeating part. Here are the steps in a concise form:

Step 1: Let x be the repeating decimal.

x 0.ABCABCABC...

Step 2: Multiply both sides by a power of 10.

Since the repeating part has 3 digits, we multiply both sides by 1000:

100 ABC. ABCABCABC...

Step 3: Write down another equation with the same repeating decimal but shifted.

We can also write:

x 0.ABCABCABC...

Step 4: Subtract the two equations.

By subtracting the second equation from the first, we eliminate the repeating part:

100 - x ABC. ABCABCABC... - 0.ABCABCABC...

This simplifies to:

999x ABC

Step 5: Solve for x.

To find x, we divide both sides by 999:

x frac{ABC}{999}

Conclusion

Converting repeating decimals to fractions is a crucial skill that simplifies complex numbers and makes calculations easier. By following the steps outlined in this guide, you can easily convert any repeating decimal into a reduced fraction. Practice with different repeating decimals to master this technique.

Understanding the concept of converting repeating decimals to fractions will not only help in mathematics but also in various real-world applications where precision and accuracy are essential.