close
close
6.6 repeating as a fraction

6.6 repeating as a fraction

2 min read 05-02-2025
6.6 repeating as a fraction

Meta Description: Learn how to convert the repeating decimal 6.666... into a fraction. This easy-to-follow guide breaks down the process step-by-step, perfect for students and anyone curious about math! Master this common fraction conversion technique and understand the underlying logic.

Understanding Repeating Decimals

Before diving into the conversion, 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, we're dealing with 6.666..., where the digit "6" repeats endlessly. This is often represented as 6.6̅ or 6.(6).

Converting 6.6 Repeating to a Fraction

The method for converting repeating decimals to fractions involves algebra. Here’s how to convert 6.6̅:

Step 1: Set up an equation.

Let's represent the repeating decimal as 'x':

x = 6.666...

Step 2: Multiply to shift the decimal.

Multiply both sides of the equation by 10 to shift the repeating digits to the left of the decimal point:

10x = 66.666...

Step 3: Subtract the original equation.

Now, subtract the original equation (x = 6.666...) from the equation we just created (10x = 66.666...):

10x - x = 66.666... - 6.666...

This simplifies to:

9x = 60

Step 4: Solve for x.

Divide both sides of the equation by 9 to isolate 'x':

x = 60/9

Step 5: Simplify the fraction.

Finally, simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3:

x = 20/3

Therefore, the fraction equivalent of 6.6 repeating is 20/3.

Why This Method Works

This method works because the multiplication and subtraction steps effectively eliminate the repeating part of the decimal. By shifting the decimal point and subtracting, we're left with a whole number that can easily be converted into a fraction.

Checking Your Work

To verify the accuracy of our conversion, you can divide 20 by 3. You'll find that the result is indeed 6.666..., confirming that 20/3 is the correct fractional representation.

Other Examples of Repeating Decimal Conversions

This method can be applied to any repeating decimal. For example, to convert 0.333... (0.3̅) to a fraction:

  1. x = 0.333...
  2. 10x = 3.333...
  3. 10x - x = 3.333... - 0.333... => 9x = 3
  4. x = 3/9 = 1/3

Conclusion

Converting repeating decimals to fractions might seem daunting at first, but with this step-by-step approach, it becomes straightforward. Remember the key steps: set up an equation, multiply to shift the decimal, subtract, solve for x, and simplify. This method provides a powerful tool for understanding the relationship between decimals and fractions. Now you can confidently tackle similar conversions!

Related Posts