If The Following Fractions Were Converted To Decimals, Which One Would Result In A Repeating Decimal?A) \[$\frac{3}{7}\$\]B) \[$\frac{5}{11}\$\]C) \[$\frac{3}{4}\$\]D) \[$\frac{1}{9}\$\]

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When dealing with fractions, it's essential to understand the concept of repeating decimals. A repeating decimal is a decimal representation of a number where a finite block of digits repeats indefinitely. In this article, we'll explore which of the given fractions would result in a repeating decimal when converted to decimals.

What are Repeating Decimals?

Repeating decimals are decimals that have a finite block of digits that repeat indefinitely. For example, the decimal representation of the fraction 1/3 is 0.333..., where the digit 3 repeats indefinitely. Repeating decimals can be represented in a variety of ways, including:

  • Repeating blocks: A finite block of digits that repeats indefinitely, such as 0.123123...
  • Periodic decimals: Decimals that have a repeating block of digits, such as 0.142857142857...

Why Do Fractions Result in Repeating Decimals?

Fractions result in repeating decimals because of the way they are represented in decimal form. When a fraction is converted to a decimal, the decimal representation is obtained by dividing the numerator by the denominator. If the denominator is not a power of 10, the decimal representation will have a repeating block of digits.

How to Determine if a Fraction Results in a Repeating Decimal

To determine if a fraction results in a repeating decimal, we can use the following criteria:

  • Denominator is not a power of 10: If the denominator is not a power of 10, the fraction will result in a repeating decimal.
  • Denominator has prime factors other than 2 and 5: If the denominator has prime factors other than 2 and 5, the fraction will result in a repeating decimal.

Analyzing the Given Fractions

Now that we understand the concept of repeating decimals and how to determine if a fraction results in a repeating decimal, let's analyze the given fractions:

A) 37\frac{3}{7}

The denominator of this fraction is 7, which is not a power of 10. Additionally, the prime factorization of 7 is simply 7, which means it has no prime factors other than 7. Therefore, this fraction will result in a repeating decimal.

B) 511\frac{5}{11}

The denominator of this fraction is 11, which is not a power of 10. Additionally, the prime factorization of 11 is simply 11, which means it has no prime factors other than 11. Therefore, this fraction will result in a repeating decimal.

C) 34\frac{3}{4}

The denominator of this fraction is 4, which is a power of 10 (2^2). Therefore, this fraction will not result in a repeating decimal.

D) 19\frac{1}{9}

The denominator of this fraction is 9, which is not a power of 10. Additionally, the prime factorization of 9 is 3^2, which means it has a prime factor other than 2 and 5. Therefore, this fraction will result in a repeating decimal.

Conclusion

In conclusion, the fractions that would result in a repeating decimal when converted to decimals are:

  • 37\frac{3}{7}
  • 511\frac{5}{11}
  • 19\frac{1}{9}

These fractions have denominators that are not powers of 10 and have prime factors other than 2 and 5, which means they will result in repeating decimals.

Final Answer

The correct answer is:

  • A) 37\frac{3}{7}
  • B) 511\frac{5}{11}
  • D) 19\frac{1}{9}