0.05555 as a fraction

2 min read 01-03-2025

Converting repeating decimals, like 0.05555..., into fractions can seem tricky, but it's a manageable process using a simple algebraic method. This guide will walk you through converting 0.05555... (or 0.05 repeating) into its fractional equivalent. We'll also explore some helpful tips for tackling similar problems.

Understanding Repeating Decimals

The number 0.05555... is a repeating decimal. The "…" indicates that the digit 5 repeats infinitely. This is different from a terminating decimal, which ends after a finite number of digits (like 0.25). We need a different approach to convert repeating decimals to fractions.

Converting 0.05555... to a Fraction

Here's how to convert the repeating decimal 0.05555... into a fraction:

1. Set up an equation:

Let x = 0.05555...

2. Multiply to shift the decimal:

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

10x = 0.55555...

3. Subtract the original equation:

Subtract the original equation (x = 0.05555...) from the equation in step 2:

10x - x = 0.55555... - 0.05555...

This simplifies to:

9x = 0.5

4. Solve for x:

Divide both sides by 9 to isolate x:

x = 0.5 / 9

5. Simplify the fraction:

To simplify the fraction, multiply the numerator and denominator by 2 to eliminate the decimal:

x = (0.5 * 2) / (9 * 2) = 1/18

Therefore, 0.05555... is equal to 1/18.

Why This Method Works

This method works because subtracting the original equation from the multiplied equation cancels out the infinitely repeating part of the decimal. This leaves us with a simple equation that can be easily solved to find the fractional equivalent.

Tips for Converting Repeating Decimals

Identify the repeating block: Determine the digits that repeat. In our example, it's "5".
Choose the appropriate multiplier: The multiplier (in this case, 10) should shift the repeating block to the left of the decimal point. If the repeating block had two digits, you'd multiply by 100, and so on.
Simplify the fraction: Always simplify your final fraction to its lowest terms.

Practicing with Other Repeating Decimals

Try converting other repeating decimals using this method. For instance, try converting 0.333... or 0.121212... into fractions. The more you practice, the more comfortable you'll become with this technique.

Conclusion

Converting a repeating decimal like 0.05555... into a fraction may seem challenging initially. However, by following these simple steps—setting up an equation, multiplying to shift the decimal, subtracting, solving, and simplifying—you can confidently transform repeating decimals into their fractional equivalents. Remember the key is to eliminate the repeating part of the decimal through subtraction. With practice, you'll master this valuable mathematical skill.