Pre-Algebra 20 - Converting Repeating Decimal Numbers to Fractions

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To convert repeating decimals into fractions, one multiplies the decimal by a power of ten corresponding to the number of repeating digits and then subtracts the original to eliminate the repeat. This method allows for the simplification of equations, resulting in fractions such as 5/9 for 0.555... and 1/3 for 0.333..., demonstrating a clear process for any repeating decimal.

Insights

  • To convert repeating decimals into fractions, a systematic approach involves multiplying the decimal by a power of ten that corresponds to the number of repeating digits, then subtracting the original decimal to eliminate the repeats. For instance, 0.555... becomes 5/9 through the equation 10x x = 5, while 0.333... simplifies to 1/3 using the same method.
  • The process varies slightly based on the length of the repeating sequence; for decimals with multiple repeating digits, such as 0.123123..., you multiply by a higher power of ten (like 1000 for three repeating digits) to shift the decimal point appropriately. This results in equations like 1000x x = 123, ultimately leading to a fraction that can be simplified, demonstrating how the method adapts to different repeating patterns.

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Recent questions

  • How do I convert decimals to fractions?

    Converting decimals to fractions involves a systematic approach. For a simple decimal, you can express it as a fraction by placing the decimal number over a power of ten, depending on the number of decimal places. For example, 0.75 can be written as 75/100, which simplifies to 3/4. However, for repeating decimals, the process is slightly different. You multiply the decimal by a power of ten that corresponds to the number of repeating digits, then subtract the original decimal to eliminate the repeating part. This method allows you to derive a fraction that accurately represents the repeating decimal.

  • What is a repeating decimal?

    A repeating decimal is a decimal fraction that eventually repeats a sequence of digits indefinitely. For instance, the decimal 0.333... has the digit '3' repeating endlessly. This type of decimal can be expressed as a fraction, which is a rational number. The repeating part can be one digit, like in 0.666..., or multiple digits, such as in 0.123123..., where '123' repeats. Understanding repeating decimals is crucial in mathematics, as they illustrate the relationship between fractions and decimal representations, highlighting how certain decimal forms can be converted back into simpler fractional forms.

  • Why do decimals repeat?

    Decimals repeat due to the nature of division in base ten. When a fraction is expressed as a decimal, if the denominator has prime factors other than 2 or 5, the decimal representation will be repeating. For example, the fraction 1/3 results in the decimal 0.333..., which repeats because the division of 1 by 3 does not terminate. This phenomenon occurs because the long division process eventually leads to the same remainder, causing the same digits to recur. Understanding why decimals repeat helps in grasping the concept of rational numbers and their decimal equivalents.

  • How to identify repeating decimals?

    Identifying repeating decimals involves looking for patterns in the digits after the decimal point. A decimal is considered repeating if a sequence of digits recurs indefinitely. For example, in the decimal 0.142857142857..., the sequence '142857' repeats. To determine if a decimal is repeating, you can perform long division or convert the decimal to a fraction. If the decimal does not terminate and you notice a recurring sequence, it is classified as a repeating decimal. Recognizing these patterns is essential for converting them into fractions and understanding their properties in mathematics.

  • What are the benefits of converting decimals?

    Converting decimals to fractions has several benefits, particularly in mathematical calculations and simplifications. Fractions can provide exact values, whereas decimals may lead to rounding errors in calculations. By converting repeating decimals into fractions, one can express them in a more manageable form, making it easier to perform arithmetic operations. Additionally, understanding the conversion process enhances numerical literacy, allowing individuals to work with different representations of numbers effectively. This skill is particularly useful in fields such as finance, engineering, and science, where precision is crucial.

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Summary

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Converting Repeating Decimals to Fractions

  • To convert a repeating decimal with one infinitely repeating digit into a fraction, multiply the decimal by ten to shift it left, then subtract the original number. For example, for the repeating decimal 0.555..., the equation becomes 10x - x = 5, simplifying to 9x = 5, leading to x = 5/9.
  • For a repeating decimal with a single repeating digit, such as 0.333..., multiply by ten and subtract the original number to eliminate the repeating digit. This results in the equation 10x - x = 3, simplifying to 9x = 3, which gives x = 3/9, or reduced to 1/3.
  • When dealing with a repeating decimal with three repeating digits, like 0.123123..., multiply by one thousand (10^3) to shift three columns left, then subtract the original number. This leads to the equation 1000x - x = 123, simplifying to 999x = 123, resulting in x = 123/999, which can be reduced to 123/999.
  • For a repeating decimal with six repeating digits, such as 0.142857142857..., multiply by one million (10^6) to shift six columns left, then subtract the original number. This gives the equation 1,000,000x - x = 142857, simplifying to 999999x = 142857, leading to x = 142857/999999, which can be reduced to 1/7.
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