What is 0.66666666666 as a fraction

What is 0.66666666666 as a fraction

The Correct Answer and Explanation is :

The correct answer is: (\frac{2}{3})

The decimal number (0.66666666666) can be expressed as the fraction (\frac{2}{3}). Here’s how we can derive that:

Step 1: Understanding the Decimal

The number (0.66666666666) is a repeating decimal, often written as (0.\overline{6}). This notation indicates that the digit 6 repeats indefinitely.

Step 2: Setting Up an Equation

Let’s represent (x) as the repeating decimal:
[
x = 0.66666666666\ldots
]

Step 3: Manipulating the Equation

To eliminate the repeating part, multiply both sides of the equation by 10:
[
10x = 6.66666666666\ldots
]

Now we have two equations:

1. (x = 0.66666666666\ldots)
2. (10x = 6.66666666666\ldots)

Step 4: Subtracting the Equations

Next, subtract the first equation from the second:
[
10x – x = 6.66666666666\ldots – 0.66666666666\ldots
]
This simplifies to:
[
9x = 6
]

Step 5: Solving for (x)

Now, divide both sides by 9:
[
x = \frac{6}{9}
]

Step 6: Simplifying the Fraction

The fraction (\frac{6}{9}) can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 3:
[
x = \frac{6 \div 3}{9 \div 3} = \frac{2}{3}
]

Conclusion

Thus, the decimal (0.66666666666) as a fraction is (\frac{2}{3}). Understanding how to convert repeating decimals into fractions can be useful in various mathematical applications, including calculations, approximations, and number theory. This method illustrates a systematic approach to handling infinite decimal expansions.