Express The Repeating Decimal As The Ratio Of Two Integers

Let's unravel the mystery of repeating decimals and transform them into elegant fractions, expressing them as the ratio of two integers.

Understanding Repeating Decimals

Repeating decimals, also known as recurring decimals, are decimal numbers that have a digit or a sequence of digits that repeats indefinitely. This repetition continues without end. These decimals are a common sight in mathematics and often arise when converting fractions to decimal form. Understanding their nature is the first step in transforming them into fractions.

• Examples of Repeating Decimals:
  • 0.3333... (repeating digit is 3)
  • 0.142857142857... (repeating sequence is 142857)
  • 1.6666... (repeating digit is 6)
  • 2.090909... (repeating sequence is 09)

Notation

To simplify the representation of repeating decimals, a special notation is used. A bar (vinculum) is placed over the repeating digit or sequence of digits. For example:

• 0. $\overline{3}$ represents 0.3333...
• 0. $\overline{142857}$ represents 0.142857142857...
• 1. $\overline{6}$ represents 1.6666...
• 2. $\overline{09}$ represents 2.090909...

This notation clearly indicates which digits are repeating, allowing for a concise and unambiguous representation of the repeating decimal.

The Algebraic Method: Converting Repeating Decimals to Fractions

The algebraic method is a powerful and reliable technique for converting repeating decimals into fractions. This method involves setting up an equation, manipulating it to eliminate the repeating part, and then solving for the unknown variable.

Steps to Convert Repeating Decimals to Fractions

1. Assign a variable: Let x equal the repeating decimal.
2. Multiply by a power of 10: Choose a power of 10 (10, 100, 1000, etc.) such that multiplying x by this power shifts the decimal point to the right, aligning the repeating part.
3. Subtract the original equation: Subtract the original equation (x = repeating decimal) from the new equation (10x, 100x, etc.). This eliminates the repeating part of the decimal.
4. Solve for x: Solve the resulting equation for x. This will give you x as a fraction.
5. Simplify the fraction: Simplify the fraction to its lowest terms, if possible.

Examples

Let's illustrate this method with several examples:

Example 1: Convert 0.$\overline{3}$ to a fraction

1. Let x = 0.$\overline{3}$ = 0.3333...
2. Multiply by 10: 10x = 3.3333...
3. Subtract the original equation:
   • 10x = 3.3333...
   • − x = 0.3333...
   • 9x = 3
4. Solve for x:
   • x = 3/9
5. Simplify:
   • x = 1/3

Therefore, 0.$\overline{3}$ = 1/3.

Example 2: Convert 0.$\overline{142857}$ to a fraction

1. Let x = 0.$\overline{142857}$ = 0.142857142857...
2. Multiply by 1,000,000: 1,000,000x = 142857.142857...
3. Subtract the original equation:
   • 1,000,000x = 142857.142857...
   • − x = 0.142857...
   • 999,999x = 142857
4. Solve for x:
   • x = 142857/999999
5. Simplify:
   • x = 1/7

Therefore, 0.$\overline{142857}$ = 1/7.

Example 3: Convert 1.$\overline{6}$ to a fraction

1. Let x = 1.$\overline{6}$ = 1.6666...
2. Multiply by 10: 10x = 16.6666...
3. Subtract the original equation:
   • 10x = 16.6666...
   • − x = 1.6666...
   • 9x = 15
4. Solve for x:
   • x = 15/9
5. Simplify:
   • x = 5/3

Therefore, 1.$\overline{6}$ = 5/3.

Example 4: Convert 2.$\overline{09}$ to a fraction

1. Let x = 2.$\overline{09}$ = 2.090909...
2. Multiply by 100: 100x = 209.0909...
3. Subtract the original equation:
   • 100x = 209.0909...
   • − x = 2.0909...
   • 99x = 207
4. Solve for x:
   • x = 207/99
5. Simplify:
   • x = 23/11

Therefore, 2.$\overline{09}$ = 23/11.

Understanding the Math Behind the Method

The algebraic method works because multiplying by a power of 10 and then subtracting the original number effectively eliminates the infinitely repeating part of the decimal. This allows us to create a simple equation that can be solved for x, which represents the fraction we are trying to find.

Why Does It Work?

Consider the repeating decimal 0.$\overline{3}$. When we multiply it by 10, we get 3.$\overline{3}$. Subtracting the original number (0.$\overline{3}$) from this result leaves us with exactly 3. The infinitely repeating parts cancel each other out:

3. 3333... − 0.3333... = 3.0000...

This principle holds true for any repeating decimal. The key is to choose the correct power of 10 so that the repeating parts align and cancel out when subtracted.

Handling More Complex Repeating Decimals

The algebraic method can also handle more complex repeating decimals, including those with non-repeating digits before the repeating part.

Example: Convert 0.1$\overline{6}$ to a fraction

1. Let x = 0.1$\overline{6}$ = 0.16666...
2. Multiply by 10: 10x = 1.6666...
3. Multiply by 100: 100x = 16.6666...
4. Subtract the equation in step 2 from the equation in step 3:
   • 100x = 16.6666...
   • − 10x = 1.6666...
   • 90x = 15
5. Solve for x:
   • x = 15/90
6. Simplify:
   • x = 1/6

Therefore, 0.1$\overline{6}$ = 1/6.

In this example, we multiplied by 10 and 100 to align the repeating part before subtracting. This is a common technique for handling repeating decimals with non-repeating digits.

General Approach

1. Let x = the repeating decimal.
2. Multiply x by a power of 10 to move the decimal point to the beginning of the repeating block (e.g., 10x, 100x, 1000x).
3. Multiply x by another power of 10 to move the decimal point to the end of the first repeating block.
4. Subtract the two equations to eliminate the repeating part.
5. Solve for x and simplify the fraction.

Common Mistakes to Avoid

When converting repeating decimals to fractions, it's easy to make mistakes. Here are some common errors to avoid:

• Incorrectly aligning the repeating parts: Make sure you multiply by the correct power of 10 so that the repeating parts align perfectly before subtracting.
• Not simplifying the fraction: Always simplify the resulting fraction to its lowest terms.
• Misunderstanding the notation: Be clear about which digits are repeating and use the correct notation.
• Arithmetic errors: Double-check your calculations to avoid simple arithmetic mistakes.

Real-World Applications

Converting repeating decimals to fractions has several practical applications:

• Computer Science: Computers often use fractions to represent numbers accurately. Converting repeating decimals to fractions ensures precision in calculations.
• Engineering: Engineers need accurate measurements and calculations. Converting repeating decimals to fractions can help avoid rounding errors.
• Finance: Financial calculations often involve decimals. Converting repeating decimals to fractions can improve the accuracy of financial models.
• Mathematics Education: Understanding how to convert repeating decimals to fractions is a fundamental concept in mathematics education.

Alternative Methods

While the algebraic method is the most common and reliable, there are alternative methods for converting repeating decimals to fractions.

Pattern Recognition

Sometimes, you can recognize patterns in the repeating decimal that correspond to specific fractions. For example, 0.$\overline{3}$ is immediately recognizable as 1/3, and 0.$\overline{1}$ is recognizable as 1/9. However, this method is not always applicable and requires a good understanding of common fractions.

Geometric Series

Repeating decimals can also be represented as infinite geometric series. The sum of an infinite geometric series can be calculated using the formula:

S = a / (1 - r)

Where:

• S is the sum of the series.
• a is the first term of the series.
• r is the common ratio between terms.

For example, 0.$\overline{3}$ can be represented as:

0. 3 + 0.03 + 0.003 + 0.0003 + ...

Here, a = 0.3 and r = 0.1. Using the formula, we get:

S = 0.3 / (1 - 0.1) = 0.3 / 0.9 = 1/3

This method is more complex than the algebraic method but provides a different perspective on repeating decimals.

Practice Problems

To solidify your understanding, try converting the following repeating decimals to fractions:

1. 0.$\overline{6}$
2. 1.$\overline{2}$
3. 0.$\overline{45}$
4. 2. $\overline{18}$
5. 0. 2$\overline{3}$

Solutions to Practice Problems

1. 0.$\overline{6}$ = 2/3
2. 1.$\overline{2}$ = 11/9
3. 0.$\overline{45}$ = 5/11
4. 2.$\overline{18}$ = 24/11
5. 0. 2$\overline{3}$ = 7/30

Conclusion

Converting repeating decimals to fractions is a valuable skill with practical applications in various fields. The algebraic method provides a systematic and reliable way to perform this conversion. By understanding the underlying math and practicing regularly, you can master this skill and confidently express repeating decimals as the ratio of two integers. Embrace the elegance and precision of fractions, and continue to explore the fascinating world of mathematics!