How To Write A Repeating Decimal As A Fraction: A Step-by-Step Guide

Converting repeating decimals to fractions can seem tricky at first. Don’t worry! It’s a manageable process that, once understood, will unlock a new level of mathematical understanding. This guide breaks down the process into clear, easy-to-follow steps, making it simple for anyone to convert these numbers with repeating digits into their fractional counterparts. We’ll cover the core principles, explore different scenarios, and provide plenty of examples to solidify your understanding.

Understanding Repeating Decimals: The Basics

Before diving into the conversion process, let’s define what we’re working with. A repeating decimal is a decimal number where one or more digits repeat infinitely after the decimal point. These repeating digits are often indicated with a bar over the repeating digits. For example, 0.333… is written as 0.3̅, and 0.142857142857… is written as 0.142857̅. Understanding this notation is crucial to correctly applying the conversion method.

Step 1: Setting Up the Equation

The first step involves setting up an equation. Let’s start with a simple example: 0.6̅.

  1. Assign a variable: Let x equal the repeating decimal. So, x = 0.6̅
  2. Identify the repeating block: In this case, the repeating block is simply the digit 6.

Step 2: Multiplying to Shift the Decimal

The next step is to multiply both sides of the equation by a power of 10. The power of 10 depends on the number of digits in the repeating block. Since our repeating block is just one digit long (the 6), we multiply by 10.

  1. Multiply by 10: 10x = 10 * 0.6̅. This gives us 10x = 6.6̅.

This multiplication shifts the decimal point to the right, placing the repeating block just after the decimal point.

Step 3: Subtracting to Eliminate the Repeating Part

Now, we subtract the original equation (x = 0.6̅) from the multiplied equation (10x = 6.6̅). This is the key to eliminating the repeating part.

  1. Subtract the equations:
    • 10x = 6.6̅
    • -x = 0.6̅
    • 9x = 6

Notice how the repeating decimals cancel each other out.

Step 4: Solving for x

The final step is to solve for x.

  1. Isolate x: Divide both sides of the equation by 9.

    • 9x / 9 = 6 / 9
    • x = 6/9

  2. Simplify the fraction: The fraction 6/9 can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3.

    • x = 2/3

Therefore, 0.6̅ is equal to the fraction 2/3.

Handling More Complex Repeating Decimals

The process slightly changes when dealing with more complex repeating decimals, such as those with multiple digits in the repeating block.

Dealing with Two Repeating Digits: 0.18̅

Let’s work through an example with two repeating digits: 0.18̅.

  1. Set up the equation: x = 0.18̅
  2. Identify the repeating block: The repeating block is 18 (two digits).
  3. Multiply by 100: Since there are two repeating digits, we multiply by 100 (10 to the power of 2): 100x = 18.18̅
  4. Subtract:
    • 100x = 18.18̅
    • -x = 0.18̅
    • 99x = 18
  5. Solve for x: x = 18/99. Simplify to x = 2/11

So, 0.18̅ is equal to 2/11.

Handling Repeating Decimals with Non-Repeating Digits

Sometimes, you’ll encounter repeating decimals that have non-repeating digits before the repeating part. For example, 0.16̅. Here’s how to handle these.

Converting 0.16̅

  1. Set up the equation: x = 0.16̅
  2. Multiply to shift the repeating block: We have one digit before the repeating block. Multiply by 10: 10x = 1.6̅
  3. Identify the repeating block: The repeating block is 6 (one digit).
  4. Multiply again to shift the repeating digit: Multiply by 10 again: 100x = 16.6̅
  5. Subtract:
    • 100x = 16.6̅
    • -10x = 1.6̅
    • 90x = 15
  6. Solve for x: x = 15/90. Simplify to x = 1/6

Therefore, 0.16̅ equals 1/6. This method requires two multiplications to isolate the repeating portion.

Working with Mixed Numbers and Repeating Decimals

Converting mixed numbers with repeating decimals combines the principles we’ve learned. First, convert the repeating decimal part to a fraction, then combine it with the whole number.

Example: Converting 2.3̅

  1. Focus on the decimal part: Let y = 0.3̅
  2. Convert the decimal to a fraction: Using the steps described earlier, 10y = 3.3̅, and subtracting y from both sides gives 9y = 3, which simplifies to y = 1/3.
  3. Combine with the whole number: 2.3̅ = 2 + 1/3.
  4. Convert to an improper fraction: 2 + 1/3 = 6/3 + 1/3 = 7/3

So, 2.3̅ is equal to 7/3.

Simplifying the Resulting Fractions

After converting a repeating decimal to a fraction, always simplify the resulting fraction to its lowest terms. This involves dividing both the numerator and the denominator by their greatest common divisor (GCD). For example, if you arrived at 4/6, the GCD is 2, and simplifying gives you 2/3.

Common Mistakes to Avoid

  • Incorrect Multiplication Factor: Ensure you multiply by the correct power of 10, based on the length of the repeating block.
  • Incorrect Subtraction: Carefully subtract the original equation from the multiplied equation, paying close attention to decimal placement.
  • Forgetting to Simplify: Always simplify your final fraction.

Practice Makes Perfect: Exercises and Examples

Practice is key to mastering this skill. Try converting these repeating decimals into fractions:

  • 0.7̅
  • 0.83̅
  • 0.27̅
  • 1.45̅
  • 0.09̅

Work through each step methodically, and then check your answers. The more you practice, the more comfortable and confident you’ll become.

Solutions to Practice Problems

  • 0.7̅ = 7/9
  • 0.83̅ = 5/6
  • 0.27̅ = 3/11
  • 1.45̅ = 13/9
  • 0.09̅ = 1/11

FAQs

What if the repeating block starts after a few non-repeating digits?

Follow the steps for handling non-repeating digits, multiplying by a power of 10 to move the decimal to the start of the repeating block, and then by a further power of 10 to capture the repeating part.

Is there a quick shortcut for simple cases?

For single-digit repeating decimals like 0.6̅, 0.7̅, etc., you can often directly write the repeating digit over 9 (e.g., 0.6̅ = 6/9, which simplifies to 2/3). However, always use the method for more complex scenarios to ensure accuracy.

Why does this method work?

The method works because we are essentially creating equations and subtracting them. This eliminates the repeating part, allowing us to isolate the variable and solve for the equivalent fraction.

Can all repeating decimals be written as fractions?

Yes, all repeating decimals, by definition, can be expressed as fractions. This is a fundamental property of rational numbers.

How do I know if a decimal is repeating?

Repeating decimals have a clear pattern of digits that repeat infinitely. This pattern can be indicated by a bar over the repeating block of digits. If the digits don’t form a clear, repeating pattern, the decimal is likely non-repeating (and may be irrational if the decimal continues infinitely without a clear pattern).

Conclusion

Converting repeating decimals to fractions is a valuable skill that strengthens your understanding of numbers and their relationships. By mastering the steps outlined in this guide – from setting up the equation and multiplying, to subtracting and simplifying – you can confidently convert any repeating decimal into its fractional form. Remember to practice regularly and apply the principles to various examples. This detailed approach, with clear explanations and helpful examples, will equip you with the knowledge and confidence to tackle any repeating decimal conversion challenge.