1.6 Repeating As A Fraction

Decoding the Mystery: 1.6 Repeating as a Fraction

Understanding how to convert repeating decimals, like 1.6 repeating (often written as 1.6̅ or 1.666...), into fractions is a fundamental concept in mathematics. This seemingly simple task unlocks a deeper understanding of number systems and provides a practical skill applicable in various fields, from basic arithmetic to advanced calculus. This comprehensive guide will not only show you how to convert 1.6 repeating to a fraction but also why the method works, providing a robust foundation for tackling similar problems.

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, 1.6 repeating means the digit "6" continues indefinitely after the decimal point: 1.666666... The bar over the "6" (1.6̅) is a common notation to indicate this repetition. Understanding this infinite repetition is key to converting it into a fraction.

Method 1: Algebraic Approach - Solving for x

This method uses algebraic manipulation to solve for the value of the repeating decimal, expressed as 'x'. This is a powerful and generalizable method suitable for any repeating decimal.

1. Assign a variable: Let x = 1.6̅

2. Multiply to shift the decimal: Multiply both sides of the equation by 10 to shift the repeating part to the left of the decimal point: 10x = 16.6̅

3. Subtract the original equation: Subtracting the original equation (x = 1.6̅) from the modified equation (10x = 16.6̅) eliminates the repeating decimal:

   10x - x = 16.6̅ - 1.6̅

   9x = 15

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

   x = 15/9

5. Simplify the fraction: Simplify the fraction by finding the greatest common divisor (GCD) of 15 and 9, which is 3. Divide both the numerator and denominator by 3:

   x = 5/3

Therefore, 1.6 repeating is equal to 5/3.

Method 2: The Fraction Formation Approach

This approach is a slightly quicker alternative, though it's less versatile than the algebraic method. It focuses directly on identifying the repeating part and building a fraction accordingly.

1. Identify the repeating part: The repeating part of 1.6̅ is "6".

2. Express the non-repeating part: The non-repeating part is "1".

3. Construct the fraction: The numerator is obtained by subtracting the non-repeating part from the number formed by combining the non-repeating and repeating parts (16 - 1 = 15). The denominator is constructed using as many 9s as there are repeating digits, followed by as many 0s as there are non-repeating digits after the decimal. In this case, we have one repeating digit (6), and no non-repeating digits after the decimal point, giving us a denominator of 9.

   Thus, the fraction is 15/9.

4. Simplify the fraction: Simplifying 15/9 as before, we arrive at the final answer: 5/3.

Why These Methods Work: A Deeper Dive

The success of both methods hinges on the understanding of infinite geometric series. A repeating decimal can be represented as the sum of an infinite geometric series. Let's illustrate this for 1.6̅:

1.6̅ = 1 + 0.6 + 0.06 + 0.006 + ...

This is a geometric series with:

• First term (a): 0.6
• Common ratio (r): 0.1

The sum of an infinite geometric series is given by the formula: S = a / (1 - r), provided that |r| < 1 (the absolute value of the common ratio is less than 1).

In our case:

S = 0.6 / (1 - 0.1) = 0.6 / 0.9 = 6/9 = 2/3

Adding the non-repeating part (1), we get:

1 + 2/3 = 3/3 + 2/3 = 5/3

This confirms our previous results. The algebraic method cleverly bypasses the explicit use of the geometric series formula by manipulating the equation to eliminate the infinite repetition. The second method is a shortcut based on the same underlying principle.

Expanding the Concepts: Different Repeating Decimals

The methods described above are not limited to 1.6̅. Let's consider a few examples to solidify the understanding:

Example 1: 0.3̅

• Algebraic Method:

  • x = 0.3̅
  • 10x = 3.3̅
  • 10x - x = 3.3̅ - 0.3̅
  • 9x = 3
  • x = 3/9 = 1/3

• Fraction Formation Method: Numerator = 3, Denominator = 9. Simplified to 1/3.

Example 2: 2.14̅

• Algebraic Method:

  • x = 2.14̅
  • 10x = 21.4̅
  • 100x = 214.4̅
  • 100x - 10x = 214.4̅ - 21.4̅
  • 90x = 193
  • x = 193/90

• Fraction Formation Method: Numerator = 214 - 21 = 193. Denominator = 90 (one repeating digit, one non-repeating digit after the decimal). Result: 193/90

Example 3: 0.123̅

• Algebraic Method:

  • x = 0.123̅
  • 1000x = 123.123̅
  • 1000x - x = 123
  • 999x = 123
  • x = 123/999 = 41/333

• Fraction Formation Method: Numerator = 123. Denominator = 999 (three repeating digits). Simplified to 41/333.

Frequently Asked Questions (FAQ)

• Q: What if the repeating part starts after several digits?

  A: You'll need to adjust the multiplication factor in the algebraic method to shift the repeating block to the left of the decimal. The fraction formation method also requires a slight adjustment to account for the non-repeating digits before the repetition begins.

• Q: Can this method be used for repeating decimals with more than one repeating digit?

  A: Absolutely! Both methods can handle decimals with any number of repeating digits. Just ensure that you multiply by the appropriate power of 10 in the algebraic method and use the correct number of 9s in the denominator of the fraction formation method.

• Q: Are there any limitations to these methods?

  A: While these methods are generally effective, they are primarily designed for purely repeating decimals. Decimals with a non-repeating part followed by a repeating part are slightly more complex, but still solvable using adaptations of the same principles. Terminating decimals (decimals that end) are not repeating decimals and should be handled using a different approach, typically by writing the decimal as a fraction with a denominator that is a power of 10.

Conclusion

Converting repeating decimals to fractions might seem daunting at first, but with a clear understanding of the underlying principles and practice with different examples, it becomes a manageable and even enjoyable mathematical exercise. The algebraic method offers a robust and general approach, while the fraction formation method provides a helpful shortcut for specific cases. Mastering these methods not only improves your understanding of decimal representation but also provides a valuable tool for various mathematical applications. Remember, the key is to break down the problem systematically, understanding the logic behind each step. With consistent practice, you'll confidently convert any repeating decimal into its equivalent fraction.