The repeating decimal 0.111... is equivalent to

Rationale

The repeating decimal 0.111... is equivalent to 1/9.

The repeating decimal 0.111... can be expressed as a fraction by recognizing the pattern that emerges when multiplying the decimal by 10. This manipulation reveals that the decimal is equal to 1/9, as it represents a perpetual sum of 1/10, 1/100, and so on, converging to this fraction.

A ([1/7]) YOUR ANSWER

The fraction 1/7 results in a repeating decimal of approximately 0.142857..., which does not match the repeating decimal 0.111.... Therefore, this choice is incorrect as it represents a different value altogether.

B ([1/9]) CORRECT

This is the correct answer. The decimal 0.111... can be converted into a fraction by understanding that it represents the sum of the infinite series 1/10 + 1/100 + 1/1000 + ..., which converges to 1/9. Thus, 0.111... is indeed equal to 1/9.

C ([1/10]) YOUR ANSWER

The fraction 1/10 equals 0.1, which clearly does not repeat and is distinctly different from 0.111.... Consequently, this choice does not represent the same value as the repeating decimal.

D ([1/11]) YOUR ANSWER

The fraction 1/11 results in a repeating decimal of approximately 0.090909..., which is also unrelated to the decimal 0.111.... Hence, this option is incorrect because it denotes a different numerical value.

Conclusion

The repeating decimal 0.111... is mathematically equivalent to the fraction 1/9, derived from the infinite series representation of the decimal. Other options, such as 1/7, 1/10, and 1/11, correspond to different repeating or terminating decimals, confirming that they do not equal 0.111.... Understanding these relationships is crucial for decimal-to-fraction conversions in mathematics.

The repeating decimal 0.111... is equivalent to

The repeating decimal 0.111... is equivalent to 1/9.

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