0.11111 as a fraction

2 min read 15-01-2025

Meta Description: Learn how to convert the repeating decimal 0.11111... into a fraction. This guide provides a clear, step-by-step process and explains the underlying math. Discover the simple solution to this common math problem and master the technique for similar repeating decimals.

The decimal 0.11111... (where the 1s repeat infinitely) might seem tricky to express as a fraction, but it's simpler than you think. This guide will walk you through the process, explaining each step clearly. Understanding this method will empower you to convert other repeating decimals to fractions.

Understanding Repeating Decimals

Before we begin, it's crucial to understand what a repeating decimal is. A repeating decimal is a decimal number where one or more digits repeat infinitely. In our case, the digit "1" repeats endlessly after the decimal point. We often represent this with a bar over the repeating digits: 0.¯¯1.

Converting 0.11111... to a Fraction

Here's a step-by-step method to convert the repeating decimal 0.¯¯1 into a fraction:

Step 1: Set up an Equation

Let's represent the repeating decimal with a variable, say 'x':

x = 0.11111...

Step 2: Multiply to Shift the Decimal

Multiply both sides of the equation by 10. This shifts the repeating part of the decimal one place to the left:

10x = 1.11111...

Step 3: Subtract the Original Equation

Now, subtract the original equation (x = 0.11111...) from the equation in Step 2 (10x = 1.11111...). Notice what happens to the repeating part:

10x - x = 1.11111... - 0.11111...

This simplifies to:

9x = 1

Step 4: Solve for x

Solve for 'x' by dividing both sides of the equation by 9:

x = 1/9

Therefore, the fraction equivalent of the repeating decimal 0.11111... is 1/9.

Why This 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 we can easily solve to find the fractional representation.

Practicing with Other Repeating Decimals

This same method can be used to convert other repeating decimals into fractions. The key is to multiply by a power of 10 that shifts the repeating part to align perfectly when you subtract. For instance, for a decimal with two repeating digits, you'd multiply by 100.

Common Mistakes to Avoid

Incorrect Multiplication: Make sure you're multiplying by the correct power of 10 to align the repeating digits properly.
Arithmetic Errors: Double-check your subtraction and division calculations.

By following these steps and understanding the underlying principles, converting repeating decimals into fractions becomes a manageable task. Remember to practice with various repeating decimals to solidify your understanding.