.16666 as a fraction

Dou Vamel yi

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15-01-2025

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Many encounter the repeating decimal .16666... and wonder how to express it as a fraction. This seemingly simple number holds a fascinating mathematical truth, and understanding its fractional equivalent is a great way to strengthen your understanding of decimals and fractions. Let's explore how to convert this repeating decimal into its simplest fraction form.

Understanding Repeating Decimals

Before we dive into the conversion, let's clarify 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 "1". We often represent this with a bar over the repeating digits, like this: 0.16̅.

The Conversion Process: Step-by-Step

Here's a straightforward method to convert the repeating decimal 0.16666... into a fraction:

1. Set up an equation:

Let x = 0.16666...

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 = 1.66666...

3. Subtract the original equation:

Subtract the original equation (x = 0.16666...) from the new equation (10x = 1.66666...):

10x - x = 1.66666... - 0.16666...

This simplifies to:

9x = 1.5

4. Solve for x:

Divide both sides by 9:

x = 1.5 / 9

5. Simplify the fraction:

To simplify, we can multiply both the numerator and denominator by 2 to eliminate the decimal:

x = (1.5 * 2) / (9 * 2) = 3/18

Now, simplify the fraction by finding the greatest common divisor (GCD) of 3 and 18, which is 3. Divide both numerator and denominator by 3:

x = 3/18 = 1/6

Therefore, the fraction equivalent of the repeating decimal 0.16666... is 1/6.

Alternative Method: Recognizing the Pattern

Sometimes, recognizing patterns can speed up the conversion. Notice that 0.16666... is close to 1/6, which is approximately 0.166666... Calculating 1/6 directly will confirm this.

Why Understanding This Matters

Converting repeating decimals to fractions is a crucial skill in various fields including:

• Mathematics: It strengthens your understanding of number systems and their interrelationships.
• Science: Precise calculations often require fractional representations.
• Engineering: Fractions are fundamental to many engineering calculations and designs.

Conclusion: The Power of Fractions

By following the steps outlined above, you can confidently convert repeating decimals, like 0.16666..., into their equivalent fractions. This conversion demonstrates the elegant relationship between decimals and fractions, highlighting the power and versatility of both number systems. Remember 1/6 is the simplest and most accurate representation of 0.16666...