What Is 1.6 Repeating As A Fraction

What is 1.6 Repeating as a Fraction? A Comprehensive Guide

The question of how to represent the repeating decimal 1.666... as a fraction is a common one in mathematics. Understanding this conversion not only solidifies your grasp of decimal-to-fraction conversions but also highlights fundamental concepts in number systems. This comprehensive guide will walk you through several methods to solve this, explaining the underlying principles and offering valuable insights into similar problems.

Understanding Repeating Decimals

Before diving into the conversion, let's clarify what a repeating decimal is. A repeating decimal, also known as a recurring decimal, is a decimal number that has a digit or a group of digits that repeat infinitely. In our case, 1.666..., the digit 6 repeats endlessly. We often represent this with a bar over the repeating digits: 1.$\overline{6}$. This notation clearly indicates the repeating pattern.

Method 1: Algebraic Manipulation

This is a classic and widely used method for converting repeating decimals to fractions. It involves assigning a variable to the repeating decimal and manipulating the equation to eliminate the repeating part. Let's apply it to 1.$\overline{6}$:

1. Assign a variable: Let x = 1.$\overline{6}$

2. Multiply to shift the decimal: Multiply both sides of the equation by 10 to shift the repeating part: 10x = 16.$\overline{6}$

3. Subtract the original equation: Subtract the original equation (x = 1.$\overline{6}$) from the new equation (10x = 16.$\overline{6}$):

   10x - x = 16.$\overline{6}$ - 1.$\overline{6}$

   This simplifies to: 9x = 15

4. Solve for x: Divide both sides by 9:

   x = 15/9

5. Simplify the fraction: Reduce the fraction to its simplest form by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 3:

   x = 5/3

Therefore, 1.$\overline{6}$ is equal to 5/3.

Method 2: Using the Geometric Series Formula

Repeating decimals can also be viewed as infinite geometric series. A geometric series is a series where each term is obtained by multiplying the previous term by a constant value (the common ratio). Let's explore this approach:

1. Express as a sum: We can express 1.$\overline{6}$ as the sum of a whole number and an infinite geometric series:

   1.$\overline{6}$ = 1 + 0.666... = 1 + 0.6 + 0.06 + 0.006 + ...

2. Identify the common ratio: The common ratio (r) in this geometric series is 0.1. Each term is obtained by multiplying the previous term by 0.1.

3. Apply the geometric series formula: The sum of an infinite geometric series is given by the formula:

   S = a / (1 - r)

   where 'a' is the first term and 'r' is the common ratio. In our case, a = 0.6 and r = 0.1. Substituting these values, we get:

   S = 0.6 / (1 - 0.1) = 0.6 / 0.9 = 6/9

4. Add the whole number and simplify: Remember that we initially separated the whole number 1. Now we add it back:

   1 + 6/9 = 1 + 2/3 = 5/3

Therefore, using the geometric series formula, we again arrive at 5/3.

Method 3: Understanding the Place Value System

This method provides a more intuitive understanding of the conversion. It focuses on interpreting the repeating decimal in terms of its place values.

1. Represent the repeating part: The repeating decimal 1.$\overline{6}$ can be written as 1 + 0.$\overline{6}$. The repeating part, 0.$\overline{6}$, represents an infinite sum of fractions:

   0.$\overline{6}$ = 6/10 + 6/100 + 6/1000 + ...

2. Recognize the geometric series (again): This is another geometric series with a = 6/10 and r = 1/10. Using the formula (though you may intuitively grasp the pattern):

   S = (6/10) / (1 - 1/10) = (6/10) / (9/10) = 6/9 = 2/3

3. Add the whole number part: Add the whole number part (1) to the result:

   1 + 2/3 = 5/3

Once more, we obtain 5/3.

Extending the Concept: Converting Other Repeating Decimals

The methods described above can be applied to convert any repeating decimal to a fraction. The key is to identify the repeating pattern and manipulate the equations appropriately. Let's consider an example: 0.$\overline{37}$

1. Assign a variable: Let x = 0.$\overline{37}$

2. Multiply to shift the decimal: Multiply by 100 (since two digits repeat): 100x = 37.$\overline{37}$

3. Subtract the original equation: 100x - x = 37.$\overline{37}$ - 0.$\overline{37}$ => 99x = 37

4. Solve for x: x = 37/99

Therefore, 0.$\overline{37}$ = 37/99

Practical Applications and Importance

Converting repeating decimals to fractions is a crucial skill in various fields:

• Engineering and Physics: Precise calculations often require fractional representations for accuracy.
• Computer Science: Understanding number systems is fundamental to programming and algorithms.
• Finance: Working with percentages and interest calculations frequently involves fraction manipulation.
• Mathematics: It reinforces the connection between different number systems and strengthens algebraic skills.

Conclusion: Mastering Decimal-to-Fraction Conversions

Converting repeating decimals like 1.6 repeating to fractions is a fundamental concept in mathematics. This guide has explored three distinct yet interconnected methods – algebraic manipulation, the geometric series formula, and a place value-based approach – each offering unique insights into the underlying principles. By understanding these methods, you can confidently convert any repeating decimal to its equivalent fractional representation and apply this knowledge across various mathematical and practical scenarios. Remember that the key lies in identifying the repeating pattern, strategically manipulating equations, and simplifying the resulting fraction to its lowest terms. Mastering this skill is crucial for solidifying your understanding of number systems and their interrelationships.