1.66666 repeating as a fraction

Charlotte

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01-03-2025

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Have you ever wondered how to express the repeating decimal 1.66666... as a fraction? It might seem daunting at first, but it's a straightforward process using a bit of algebra. This article will guide you through the steps, explaining the method and showing you how to tackle similar repeating decimals. Understanding this process is a key concept in mathematics and a helpful tool for anyone working with numbers.

Understanding Repeating Decimals

A repeating decimal is a decimal number where one or more digits repeat infinitely. In our case, the digit "6" repeats endlessly after the decimal point. We represent this using a bar over the repeating digits: 1.6̅. This notation is important for clearly indicating the repetition.

Converting 1.66666... to a Fraction: The Steps

Here's the step-by-step method to convert 1.6̅ to a fraction:

1. Set up an equation:

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

x = 1.66666...

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:

10x = 16.66666...

3. Subtract the original equation:

Now, subtract the original equation (x = 1.66666...) from the multiplied equation (10x = 16.66666...):

10x - x = 16.66666... - 1.66666...

This simplifies to:

9x = 15

4. Solve for x:

Divide both sides of the equation by 9 to isolate 'x':

x = 15/9

5. Simplify the fraction:

Finally, simplify the fraction by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 3:

x = 5/3

Therefore, the fraction equivalent of the repeating decimal 1.66666... is 5/3.

Why This Method Works

The method works because multiplying by powers of 10 shifts the decimal point, allowing us to subtract the repeating part and solve for the original value. This leaves us with a simple algebraic equation to solve. This technique is applicable to any repeating decimal, regardless of the length of the repeating sequence.

Other Examples of Converting Repeating Decimals

Let's look at a couple of examples to solidify the concept:

Example 1: 0.3333...

1. x = 0.3333...
2. 10x = 3.3333...
3. 10x - x = 3.3333... - 0.3333... => 9x = 3
4. x = 3/9 = 1/3

Example 2: 0.142857142857...

This one is slightly more complex because the repeating block is longer. You would multiply by 1,000,000 (10⁶) to shift the entire repeating block. The process remains the same, leading to a simplified fraction.

Conclusion: Mastering Repeating Decimals

Converting repeating decimals to fractions is a fundamental skill in mathematics. By following the steps outlined above, you can confidently tackle any repeating decimal and express it as a simplified fraction. Remember the key is to use algebra to eliminate the repeating part and solve for the unknown value. Now you can approach repeating decimals with confidence and understanding!