Which Fraction Has A Terminating Decimal As Its Decimal Expansion?A. $\frac{1}{3}$B. $\frac{1}{5}$C. $\frac{1}{7}$D. $\frac{1}{9}$

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Understanding Terminating and Non-Terminating Decimals

In mathematics, decimals can be classified into two categories: terminating and non-terminating decimals. A terminating decimal is a decimal that has a finite number of digits after the decimal point, whereas a non-terminating decimal has an infinite number of digits after the decimal point. For instance, the decimal 0.5 is a terminating decimal, whereas the decimal 0.333... is a non-terminating decimal.

Terminating Decimals and Fractional Forms

A terminating decimal can be expressed in fractional form as a ratio of two integers, where the denominator is a power of 10. For example, the decimal 0.5 can be expressed as the fraction 1/2, and the decimal 0.25 can be expressed as the fraction 1/4. On the other hand, a non-terminating decimal cannot be expressed in fractional form as a ratio of two integers.

Determining Terminating Decimals

To determine whether a fraction has a terminating decimal as its decimal expansion, we need to check if the denominator of the fraction is a power of 2 multiplied by a power of 5. If the denominator meets this condition, then the fraction has a terminating decimal as its decimal expansion.

Analyzing the Options

Let's analyze the options given in the problem:

A. 13\frac{1}{3}

The denominator of this fraction is 3, which is not a power of 2 multiplied by a power of 5. Therefore, this fraction does not have a terminating decimal as its decimal expansion.

B. 15\frac{1}{5}

The denominator of this fraction is 5, which is a power of 5. Therefore, this fraction has a terminating decimal as its decimal expansion.

C. 17\frac{1}{7}

The denominator of this fraction is 7, which is not a power of 2 multiplied by a power of 5. Therefore, this fraction does not have a terminating decimal as its decimal expansion.

D. 19\frac{1}{9}

The denominator of this fraction is 9, which is not a power of 2 multiplied by a power of 5. Therefore, this fraction does not have a terminating decimal as its decimal expansion.

Conclusion

Based on the analysis of the options, we can conclude that the fraction 15\frac{1}{5} has a terminating decimal as its decimal expansion.

Terminating Decimals and Fractional Forms: A Deeper Look

A terminating decimal can be expressed in fractional form as a ratio of two integers, where the denominator is a power of 10. For example, the decimal 0.5 can be expressed as the fraction 1/2, and the decimal 0.25 can be expressed as the fraction 1/4. On the other hand, a non-terminating decimal cannot be expressed in fractional form as a ratio of two integers.

The Role of Prime Factors

The prime factors of the denominator play a crucial role in determining whether a fraction has a terminating decimal as its decimal expansion. If the denominator has any prime factors other than 2 or 5, then the fraction does not have a terminating decimal as its decimal expansion.

Examples of Terminating Decimals

Here are some examples of fractions that have terminating decimals as their decimal expansions:

  • 12\frac{1}{2} = 0.5
  • 14\frac{1}{4} = 0.25
  • 15\frac{1}{5} = 0.2
  • 18\frac{1}{8} = 0.125

Examples of Non-Terminating Decimals

Here are some examples of fractions that do not have terminating decimals as their decimal expansions:

  • 13\frac{1}{3} = 0.333...
  • 17\frac{1}{7} = 0.142857...
  • 19\frac{1}{9} = 0.111111...

Conclusion

Q: What is the difference between a terminating and a non-terminating decimal?

A: A terminating decimal is a decimal that has a finite number of digits after the decimal point, whereas a non-terminating decimal has an infinite number of digits after the decimal point.

Q: How can I determine whether a fraction has a terminating decimal as its decimal expansion?

A: To determine whether a fraction has a terminating decimal as its decimal expansion, you need to check if the denominator of the fraction is a power of 2 multiplied by a power of 5. If the denominator meets this condition, then the fraction has a terminating decimal as its decimal expansion.

Q: What are some examples of fractions that have terminating decimals as their decimal expansions?

A: Here are some examples of fractions that have terminating decimals as their decimal expansions:

  • 12\frac{1}{2} = 0.5
  • 14\frac{1}{4} = 0.25
  • 15\frac{1}{5} = 0.2
  • 18\frac{1}{8} = 0.125

Q: What are some examples of fractions that do not have terminating decimals as their decimal expansions?

A: Here are some examples of fractions that do not have terminating decimals as their decimal expansions:

  • 13\frac{1}{3} = 0.333...
  • 17\frac{1}{7} = 0.142857...
  • 19\frac{1}{9} = 0.111111...

Q: Why is it important to understand the difference between terminating and non-terminating decimals?

A: Understanding the difference between terminating and non-terminating decimals is important because it helps you to determine whether a fraction can be expressed as a finite decimal or not. This is particularly useful in mathematics, science, and engineering, where decimal representations are often used to represent physical quantities.

Q: Can a non-terminating decimal be expressed in fractional form?

A: No, a non-terminating decimal cannot be expressed in fractional form as a ratio of two integers. This is because a non-terminating decimal has an infinite number of digits after the decimal point, which cannot be represented as a finite ratio of integers.

Q: Can a terminating decimal be expressed in fractional form?

A: Yes, a terminating decimal can be expressed in fractional form as a ratio of two integers, where the denominator is a power of 10. For example, the decimal 0.5 can be expressed as the fraction 1/2, and the decimal 0.25 can be expressed as the fraction 1/4.

Q: What is the role of prime factors in determining whether a fraction has a terminating decimal as its decimal expansion?

A: The prime factors of the denominator play a crucial role in determining whether a fraction has a terminating decimal as its decimal expansion. If the denominator has any prime factors other than 2 or 5, then the fraction does not have a terminating decimal as its decimal expansion.

Q: Can a fraction have both terminating and non-terminating decimals as its decimal expansions?

A: No, a fraction cannot have both terminating and non-terminating decimals as its decimal expansions. A fraction either has a terminating decimal as its decimal expansion or it does not.

Conclusion

In conclusion, understanding the difference between terminating and non-terminating decimals is crucial in mathematics, science, and engineering. By knowing how to determine whether a fraction has a terminating decimal as its decimal expansion, you can better understand and work with decimal representations in various contexts.