1.6 Repeating As A Fraction

Decoding the Mystery: 1.6 Repeating as a Fraction

Have you ever encountered the decimal 1.66666... and wondered how to express it as a fraction? This seemingly simple number holds a fascinating secret within its repeating digits. Understanding how to convert repeating decimals to fractions is a fundamental skill in mathematics, and this article will guide you through the process, exploring the underlying principles and offering various approaches to solve this specific problem and others like it. We'll delve into the why and how, ensuring you gain a complete understanding of this concept. This detailed explanation will cover different methods, addressing potential confusion and equipping you with the confidence to tackle similar problems independently.

Understanding Repeating Decimals

Before we dive into converting 1.6 repeating (which we can write as 1.6̅ or 1.6 with a bar over the 6), let's clarify what a repeating decimal is. A repeating decimal is a decimal number where one or more digits repeat infinitely. The repeating part is indicated by a bar placed over the repeating digits. For example:

• 0.3333... is written as 0.3̅
• 0.142857142857... is written as 0.142857̅
• 2.718281828... is not a repeating decimal because the repeating sequence is not consistent.

Method 1: Using Algebra to Solve for x

This is a classic and highly effective method for converting repeating decimals into fractions. Let's apply it to 1.6̅:

1. Assign a variable: Let x = 1.6̅

2. Multiply to shift the repeating digits: Multiply both sides of the equation by 10 (since only one digit is repeating). This gives us: 10x = 16.6̅

3. Subtract the original equation: Now subtract the original equation (x = 1.6̅) from the new equation (10x = 16.6̅):

   10x - x = 16.6̅ - 1.6̅

   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, 1.6̅ is equal to 5/3.

Method 2: Understanding Place Value and the Concept of Infinite Series

This method offers a deeper understanding of why the algebraic method works. It relies on the concept of an infinite geometric series.

1. Break down the decimal: We can express 1.6̅ as the sum of two parts: 1 + 0.6̅

2. Express the repeating part as a series: 0.6̅ can be written as an infinite geometric series: 6/10 + 6/100 + 6/1000 + ...

3. Apply the formula for an infinite geometric series: The formula for the sum of an infinite geometric series is: a / (1 - r), where 'a' is the first term and 'r' is the common ratio. In our case:

   • a = 6/10 = 3/5
   • r = 1/10

   Therefore, the sum of the series is: (3/5) / (1 - 1/10) = (3/5) / (9/10) = (3/5) * (10/9) = 2/3

4. Combine the parts: Add the whole number part (1) and the fractional part (2/3): 1 + 2/3 = 5/3

Again, we arrive at the fraction 5/3.

Method 3: Using a Calculator (for verification)

While not a method for deriving the fraction, using a calculator to convert the decimal to a fraction can serve as a useful verification. Many calculators have a function to convert decimals to fractions. Inputting 1.66666... (or as many 6s as your calculator allows) and using the fraction conversion function should yield 5/3 (or a close approximation). However, it's crucial to remember that the calculator's result is an approximation because it can't handle infinitely repeating digits perfectly. The algebraic method is the most precise for dealing with repeating decimals.

Explanation of the Result: 5/3

The fraction 5/3 is an improper fraction because the numerator (5) is larger than the denominator (3). This reflects the fact that the original decimal 1.6̅ is greater than 1. We can convert this improper fraction into a mixed number:

5 ÷ 3 = 1 with a remainder of 2. Therefore, 5/3 can be written as 1 2/3, which further clarifies the relationship between the fraction and the original decimal.

Dealing with More Complex Repeating Decimals

The methods described above can be adapted to handle more complex repeating decimals. For example, let's consider the decimal 0.142857̅:

1. Let x = 0.142857̅
2. Multiply to shift the repeating block: Since the repeating block has six digits, we multiply by 10⁶ (1,000,000): 1000000x = 142857.142857̅
3. Subtract the original equation: 1000000x - x = 142857.142857̅ - 0.142857̅ This simplifies to: 999999x = 142857
4. Solve for x: x = 142857 / 999999
5. Simplify the fraction: This fraction simplifies to 1/7.

This illustrates that the core principle remains the same; the key is to multiply by the appropriate power of 10 to align the repeating block and then subtract the original equation to eliminate the repeating part.

Frequently Asked Questions (FAQ)

Q: Why does this method work?

A: The method works because it exploits the properties of infinite geometric series and the concept of place value in our decimal system. By multiplying and subtracting, we effectively isolate the repeating part of the decimal and convert it into a fraction that can be simplified.

Q: Can this method be applied to all repeating decimals?

A: Yes, this algebraic method, along with the infinite geometric series approach, can be applied to all repeating decimals, regardless of the length or pattern of the repeating sequence.

Q: What if the repeating decimal has a non-repeating part before the repeating block?

A: Handle the non-repeating part separately. For instance, if you have 2.3̅, treat it as 2 + 0.3̅ and apply the method to 0.3̅. Then add the whole number part back in.

Q: Are there other methods to convert repeating decimals to fractions?

A: While the algebraic and geometric series methods are the most efficient and widely used, there are other less common approaches that might involve using continued fractions or other mathematical techniques. However, the methods described in this article are the most practical for most cases.

Conclusion

Converting repeating decimals to fractions is a fundamental skill in mathematics with practical applications in various fields. This article has explored three different methods to approach this problem, providing a clear understanding of both the procedure and the underlying mathematical principles. The algebraic method, using variable assignment and manipulation, offers a straightforward and reliable method. The geometric series method provides a deeper theoretical understanding. Mastering these techniques empowers you to confidently handle not only 1.6̅ but any repeating decimal you may encounter. Remember to always check your work by converting your final fraction back to a decimal to ensure accuracy. Practice makes perfect, so continue working through different examples to build your proficiency in this essential mathematical skill.