0.6666 as a fraction

2 min read 15-01-2025

The repeating decimal 0.6666... (often shortened to 0.6 recurring or 0.6̅) might seem tricky, but converting it to a fraction is easier than you think. This article will guide you through the process, explaining the method and why it works. We'll also explore related concepts and show you how to tackle similar repeating decimals.

Understanding Repeating Decimals

Repeating decimals, also known as recurring decimals, are numbers where one or more digits repeat infinitely after the decimal point. 0.6666... is a classic example, with the digit "6" repeating endlessly. Understanding this infinite repetition is key to converting it into a fraction.

Converting 0.6666... to a Fraction: The Method

Here's a step-by-step approach:

1. Set up an equation:

Let x = 0.6666...

2. Multiply to shift the decimal:

Multiply both sides of the equation by 10:

10x = 6.6666...

3. Subtract the original equation:

Subtract the original equation (x = 0.6666...) from the new equation (10x = 6.6666...):

10x - x = 6.6666... - 0.6666...

This simplifies to:

9x = 6

4. Solve for x:

Divide both sides by 9:

x = 6/9

5. Simplify the fraction:

Both 6 and 9 are divisible by 3:

x = 2/3

Therefore, 0.6666... is equal to the fraction 2/3.

Why This Method Works

The method works because multiplying by a power of 10 (in this case, 10) shifts the repeating decimal portion. Subtracting the original equation cancels out the repeating part, leaving a simple equation that can be solved to find the fraction.

Other Repeating Decimals

This method can be adapted for other repeating decimals. The key is to multiply by the appropriate power of 10 to align the repeating part before subtraction. For example, to convert 0.333... to a fraction, you would follow the same steps, and find that it equals 1/3.

Practical Applications

Understanding how to convert repeating decimals to fractions is crucial in various fields, including:

Mathematics: Simplifying expressions and solving equations.
Science: Precise measurements and calculations.
Engineering: Designing and building accurate systems.

Conclusion

Converting 0.6666... to a fraction (2/3) is a straightforward process. By understanding the method of shifting and subtracting, you can easily convert other repeating decimals into their fractional equivalents. This knowledge is beneficial for various mathematical and scientific applications. Remember the simple steps outlined above to tackle these seemingly complex numbers with confidence. Now you can confidently tackle any repeating decimal you encounter!