What Is .16666 As A Fraction

What is 0.16666... as a Fraction? A Deep Dive into Repeating Decimals

The seemingly simple question, "What is 0.16666... as a fraction?" opens a door to a fascinating world of mathematical concepts. While the immediate answer might seem elusive, understanding the process of converting repeating decimals to fractions reveals a powerful technique applicable to a wide range of decimal numbers. This comprehensive guide will not only answer the question but also delve into the underlying principles, offering you a robust understanding of the method and its implications.

Understanding Repeating Decimals

Before diving into the conversion process, let's clarify what we mean by a repeating decimal. A repeating decimal is a decimal number where one or more digits repeat infinitely. In our case, 0.16666..., the digit 6 repeats indefinitely. This is often represented using a bar over the repeating digit(s): 0.16̅. This notation clearly indicates that the 6 continues infinitely.

Converting Repeating Decimals to Fractions: A Step-by-Step Guide

The conversion of repeating decimals to fractions involves a clever algebraic manipulation. Let's apply this method to our example, 0.16666... or 0.16̅:

Step 1: Assign a Variable

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

x = 0.16̅

Step 2: Multiply to Shift the Repeating Part

We need to manipulate the equation to isolate the repeating part. Multiply both sides of the equation by a power of 10 that shifts the repeating digits to the left of the decimal point. Since the repeating part starts immediately after one digit after the decimal point, we multiply by 10:

10x = 1.66666...

Step 3: Subtract the Original Equation

Subtracting the original equation (x = 0.16̅) from the modified equation (10x = 1.66666...) eliminates the repeating part:

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

This simplifies to:

9x = 1.5

Step 4: Solve for x

Now, we solve for 'x' by dividing both sides by 9:

x = 1.5 / 9

Step 5: Simplify the Fraction

To express the answer in its simplest form, we can simplify the fraction:

x = 15/90

We can further simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 15:

x = 1/6

Therefore, 0.16666... is equivalent to the fraction 1/6.

Why This Method Works

The effectiveness of this method relies on the properties of infinite geometric series. A repeating decimal can be expressed as the sum of an infinite geometric series. For example, 0.16666... can be written as:

0.1 + 0.06 + 0.006 + 0.0006 + ...

This is a geometric series with the first term (a) = 0.06 and the common ratio (r) = 0.1. Since |r| < 1, the sum of this infinite series converges to a finite value, which is calculated using the formula:

Sum = a / (1 - r) = 0.06 / (1 - 0.1) = 0.06 / 0.9 = 1/15

Adding the 0.1 from the beginning, we get:

1/15 + 1/10 = 1/6

This confirms our result obtained through the algebraic manipulation.

Applying the Method to Other Repeating Decimals

This method isn't limited to 0.16666.... It can be applied to any repeating decimal. Let's consider another example: 0.3333... (0.3̅):

Step 1: x = 0.3̅

Step 2: 10x = 3.3̅

Step 3: 10x - x = 3.3̅ - 0.3̅ => 9x = 3

Step 4: x = 3/9

Step 5: x = 1/3

Therefore, 0.3333... is equivalent to 1/3.

Let's try a more complex example: 0.123123123... (0.123̅):

Step 1: x = 0.123̅

Step 2: 1000x = 123.123̅

Step 3: 1000x - x = 123.123̅ - 0.123̅ => 999x = 123

Step 4: x = 123/999

Step 5: x = 41/333

Therefore, 0.123123123... is equivalent to 41/333.

Dealing with Non-Repeating Parts

Some decimals have a non-repeating part followed by a repeating part. For example, 0.25777... (0.257̅). The process is similar, but we need to adjust the multiplication factor:

Step 1: x = 0.257̅

Step 2: 100x = 25.777...

Step 3: 1000x = 257.777...

Step 4: 1000x - 100x = 257.777... - 25.777... => 900x = 232

Step 5: x = 232/900

Step 6: Simplify: x = 58/225

Therefore, 0.25777... is equivalent to 58/225. The key is to multiply by a power of 10 that aligns the repeating part for subtraction.

Beyond the Basics: Exploring the Relationship Between Decimals and Fractions

The conversion of repeating decimals to fractions highlights a fundamental truth: every rational number (a number that can be expressed as a fraction of two integers) can be represented as either a terminating decimal or a repeating decimal. Irrational numbers, such as pi (π) or the square root of 2 (√2), cannot be expressed as fractions and have non-repeating, non-terminating decimal expansions.

Understanding this relationship between decimals and fractions provides a powerful tool for mathematical problem-solving and reinforces the interconnectedness of different number systems. It allows for seamless transitions between decimal and fractional representations, depending on the context and the need for precision.

Conclusion: Mastering the Art of Decimal-to-Fraction Conversion

Converting repeating decimals to fractions may seem challenging initially, but with a systematic approach and a grasp of the underlying principles, it becomes a straightforward process. The method described in this article, using algebraic manipulation, provides a reliable and efficient way to convert any repeating decimal into its equivalent fraction. This skill is not only valuable for mathematical calculations but also enhances understanding of the relationship between different number systems and their representations. Mastering this technique empowers you to tackle more complex mathematical problems and deepen your mathematical intuition.