What Is 1.66666 As A Fraction

What is 1.66666... as a Fraction? Understanding Repeating Decimals

The seemingly simple question, "What is 1.66666... as a fraction?" opens a door to a fascinating world of mathematical concepts, including repeating decimals, algebraic manipulation, and the elegance of representing numbers in different forms. This article will not only answer this question definitively but also explore the underlying principles and provide you with the tools to convert other repeating decimals into fractions.

Understanding Repeating Decimals

Before diving into the conversion process, let's clarify what a repeating decimal is. A repeating decimal is a decimal number where one or more digits repeat infinitely. In our case, 1.66666... has the digit "6" repeating endlessly. We represent repeating decimals using a bar above the repeating digits. Thus, 1.66666... is written as 1.6̅. The bar indicates that the digit 6 repeats infinitely.

It's crucial to differentiate between terminating decimals (like 0.75) and repeating decimals. Terminating decimals can be easily converted to fractions by placing the digits over the appropriate power of 10. However, converting repeating decimals requires a different approach.

Converting 1.66666... (1.6̅) to a Fraction: The Algebraic Method

The most reliable method for converting repeating decimals to fractions is using algebra. Here's a step-by-step guide:

Step 1: Assign a Variable

Let's represent the repeating decimal with a variable, say 'x':

x = 1.6̅

Step 2: Multiply to Shift the Decimal

We need to manipulate the equation to isolate the repeating part. Multiply both sides of the equation by a power of 10 that shifts the decimal point to the right, aligning the repeating portion. Since only the "6" repeats, multiplying by 10 will suffice:

10x = 16.6̅

Step 3: Subtract the Original Equation

Subtracting the original equation (x = 1.6̅) from the modified equation (10x = 16.6̅) eliminates the repeating part:

10x - x = 16.6̅ - 1.6̅

This simplifies to:

9x = 15

Step 4: Solve for x

Divide both sides by 9 to solve for x:

x = 15/9

Step 5: Simplify the Fraction

Finally, simplify the fraction by finding the greatest common divisor (GCD) of the numerator and denominator. The GCD of 15 and 9 is 3. Dividing both the numerator and denominator by 3 gives us the simplified fraction:

x = 5/3

Therefore, 1.66666... (or 1.6̅) is equal to 5/3.

Alternative Method: Using the Geometric Series

Another approach involves understanding repeating decimals as infinite geometric series. The decimal 1.6̅ can be written as:

1 + 0.6 + 0.06 + 0.006 + ...

This is a geometric series with the first term (a) = 0.6 and the common ratio (r) = 0.1. The sum of an infinite geometric series is given by the formula:

Sum = a / (1 - r) (where |r| < 1)

Plugging in our values:

Sum = 0.6 / (1 - 0.1) = 0.6 / 0.9 = 2/3

Adding the whole number part (1) back in, we get:

1 + 2/3 = 5/3

This method confirms our previous result: 1.6̅ = 5/3.

Converting Other Repeating Decimals to Fractions

The algebraic method described above is versatile and can be applied to other repeating decimals. Let's illustrate with another example:

Example: Convert 0.3̅ to a fraction

1. Let x = 0.3̅
2. Multiply by 10: 10x = 3.3̅
3. Subtract the original equation: 10x - x = 3.3̅ - 0.3̅ => 9x = 3
4. Solve for x: x = 3/9
5. Simplify: x = 1/3

Therefore, 0.3̅ = 1/3.

Dealing with Repeating Decimals with Multiple Repeating Digits

The process is slightly more complex when multiple digits repeat. For instance, let's convert 0.12̅ to a fraction:

1. Let x = 0.12̅
2. Multiply by 100 (because two digits repeat): 100x = 12.12̅
3. Subtract the original equation: 100x - x = 12.12̅ - 0.12̅ => 99x = 12
4. Solve for x: x = 12/99
5. Simplify: x = 4/33

Thus, 0.12̅ = 4/33.

Practical Applications and Importance

The ability to convert repeating decimals to fractions has practical applications in various fields:

• Mathematics: It's fundamental for understanding number systems and performing calculations involving rational numbers.
• Engineering and Physics: Accurate representation of numbers is crucial in these fields, where fractions often provide a more precise value than rounded decimal approximations.
• Computer Science: Understanding decimal-to-fraction conversion is important for data representation and algorithm design.
• Finance: Accurate calculations in financial modeling and accounting require the precision offered by fractions.

Common Mistakes and How to Avoid Them

• Incorrect multiplication: Ensure you multiply by the correct power of 10 to align the repeating digits.
• Simplification errors: Always simplify the fraction to its lowest terms by finding the greatest common divisor.
• Ignoring the whole number part: Remember to add the whole number part back after converting the decimal part.

Conclusion: Mastering Decimal-to-Fraction Conversions

Converting repeating decimals to fractions is a fundamental skill in mathematics with far-reaching applications. By mastering the algebraic method and understanding the underlying principles of geometric series, you can confidently tackle a wide range of decimal-to-fraction conversion problems. The ability to represent numbers in both decimal and fractional forms enhances your mathematical fluency and opens doors to a deeper understanding of numerical concepts. Remember to practice regularly to solidify your understanding and develop proficiency in this essential mathematical skill. The more you practice, the easier and more intuitive this process will become.