1.66667 as a fraction

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17-01-2025

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Understanding how to convert decimals to fractions is a fundamental skill in mathematics. This article will guide you through converting the decimal 1.66667 into its fractional equivalent. While 1.66667 is an approximation, we'll explore how to handle this and find the closest fraction.

Understanding Repeating Decimals

The number 1.66667 is almost a repeating decimal. A true repeating decimal, like 1.666..., is represented by a bar over the repeating digit(s). In this case, it would be 1.6̅. The slight difference (the 7 at the end) means we're dealing with an approximation, not a pure repeating decimal.

Converting 1.666... (the repeating decimal) to a Fraction

Let's first address the repeating decimal 1.666..., as it's closer to our given number than 1.66667. Here's how we convert it:

1. Let x = 1.666... This assigns a variable to our repeating decimal.

2. Multiply by 10: 10x = 16.666...

3. Subtract the original equation: Subtracting x from 10x gives us:

   10x - x = 16.666... - 1.666...

   This simplifies to: 9x = 15

4. Solve for x: Divide both sides by 9: x = 15/9

5. Simplify: Reduce the fraction by dividing both the numerator and the denominator by their greatest common divisor (3): x = 5/3

Therefore, the exact fraction representation of the repeating decimal 1.666... is 5/3.

Converting 1.66667 to a Fraction (Approximation)

Since 1.66667 is an approximation of the repeating decimal, our result will also be an approximation. To convert this to a fraction, we follow these steps:

1. Write the decimal as a fraction with a denominator of 1: 1.66667/1

2. Multiply the numerator and denominator by 100000 (to move the decimal point five places to the right): 166667/100000

This fraction, 166667/100000, is the fractional representation of 1.66667. While not simplified, it is an accurate representation of the given number. It cannot be simplified further as 166667 and 100000 don't share any common factors other than 1.

Which Fraction to Use?

The choice between 5/3 and 166667/100000 depends on the context:

• If dealing with a purely mathematical problem involving the repeating decimal 1.666..., then 5/3 is the correct and exact answer.

• If 1.66667 is a measurement or result from an experiment, then 166667/100000 offers a more precise fractional representation of that specific approximation.

This highlights the importance of distinguishing between an approximation and a repeating decimal.

Conclusion

Converting decimals to fractions involves understanding the nature of the decimal. While 5/3 represents the fraction for the repeating decimal 1.666..., the approximation 1.66667 is best represented as 166667/100000. Remember to consider the context to determine which fractional representation is most appropriate.