0.27777 as a fraction

Daniel Parker

2 min read

·

25-02-2025

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Converting repeating decimals, like 0.27777..., into fractions might seem daunting, but it's a straightforward process. This guide will walk you through converting 0.27777... (or 0.27̅) into its fractional equivalent. We'll break down the steps and explain the reasoning behind each one.

Understanding Repeating Decimals

The bar over the 7 (0.27̅) indicates that the digit 7 repeats infinitely. This is crucial because it differentiates it from a terminating decimal (like 0.27) which has a finite number of digits. Understanding this repetition is key to converting it to a fraction.

Steps to Convert 0.27777... to a Fraction

Here's how to convert the repeating decimal 0.27777... into a fraction:

1. Set up an Equation:

Let 'x' represent the repeating decimal:

x = 0.27777...

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 = 2.77777...

3. Multiply Again to Align Repeating Parts:

Now, multiply the original equation (x = 0.27777...) by 100:

100x = 27.77777...

4. Subtract to Eliminate the Repeating Part:

Subtract the equation from step 2 (10x = 2.77777...) from the equation in step 3 (100x = 27.77777...):

100x - 10x = 27.77777... - 2.77777...

This simplifies to:

90x = 25

5. Solve for x:

Divide both sides by 90 to isolate 'x':

x = 25/90

6. Simplify the Fraction:

Reduce the fraction to its simplest form by finding the greatest common divisor (GCD) of 25 and 90. The GCD is 5.

x = (25 ÷ 5) / (90 ÷ 5) = 5/18

Therefore, 0.27777... expressed as a fraction is 5/18.

Verifying the Result

You can verify this by performing long division: dividing 5 by 18 will yield the repeating decimal 0.27777...

Other Repeating Decimals

This method can be applied to any repeating decimal. The key is to multiply by powers of 10 to align the repeating part and then subtract to eliminate the repeating portion. The more digits in the repeating part, the higher the power of 10 you'll need to use.

Conclusion

Converting a repeating decimal like 0.27777... to a fraction involves a series of algebraic manipulations. By following these steps, you can confidently convert any repeating decimal into its fractional equivalent. Remember, practice makes perfect! Try converting other repeating decimals using this method to solidify your understanding.