0.26666 as a fraction

2 min read 22-02-2025

Meta Description: Learn how to convert the repeating decimal 0.26666... into a fraction. This guide provides a clear, step-by-step method, perfect for students and anyone needing to understand decimal-to-fraction conversions. We'll cover the process and explain the underlying math.

Understanding Repeating Decimals

Before we dive into converting 0.26666..., let's quickly define what a repeating decimal is. A repeating decimal is a decimal number where one or more digits repeat infinitely. In our case, the digit 6 repeats endlessly after the initial 2. We often represent this with a bar over the repeating part: 0.2\overline{6}.

Converting 0.26666... to a Fraction

Here's how to convert the repeating decimal 0.26666... into a fraction:

Step 1: Set up an Equation

Let's represent the decimal as 'x':

x = 0.26666...

Step 2: Multiply to Shift the Repeating Part

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

10x = 2.66666...

Step 3: Multiply Again to Align Repeating Parts

Now, multiply both sides of the original equation (x = 0.26666...) by 100:

100x = 26.66666...

Step 4: Subtract Equations

Subtract the equation from Step 2 from the equation in Step 3:

100x - 10x = 26.66666... - 2.66666...

This simplifies to:

90x = 24

Step 5: Solve for x

Divide both sides by 90 to isolate 'x':

x = 24/90

Step 6: Simplify the Fraction

Now, simplify the fraction by finding the greatest common divisor (GCD) of 24 and 90. The GCD of 24 and 90 is 6. Divide both the numerator and the denominator by 6:

x = (24 ÷ 6) / (90 ÷ 6) = 4/15

Therefore, 0.26666... as a fraction is 4/15.

Verifying the Result

You can verify this by performing long division: dividing 4 by 15 will result in 0.26666...

Other Examples of Converting Repeating Decimals to Fractions

This method works for other repeating decimals. The key is to multiply by powers of 10 to align the repeating portion and then subtract to eliminate the repeating part.

For example: to convert 0.777... to a fraction:

x = 0.777...
10x = 7.777...
10x - x = 7.777... - 0.777...
9x = 7
x = 7/9

Conclusion

Converting repeating decimals like 0.26666... to fractions involves a systematic approach. By following these steps, you can confidently transform any repeating decimal into its fractional equivalent. Remember to always simplify the resulting fraction to its lowest terms. Understanding this process is fundamental for anyone working with numbers and fractions. Now you can confidently tackle similar problems!