The repeating decimal 0.111... is equivalent to

Rationale

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

The repeating decimal 0.111... represents the infinite sum of 1/10, 1/100, 1/1000, and so forth, which converges to the fraction 1/9. This relationship can be demonstrated through algebraic manipulation, confirming that 0.111... is indeed equal to 1/9.

A (1/7) YOUR ANSWER

The fraction 1/7 equals approximately 0.142857..., which is a non-repeating decimal. Since 0.111... continuously repeats and is less than 0.142857..., this option cannot represent the given repeating decimal.

B (1/9) CORRECT

This is the correct answer, as the decimal 0.111... can be expressed as the infinite series 1/10 + 1/100 + 1/1000 + ..., which sums to 1/9. This demonstrates that 0.111... is equivalent to the fraction 1/9, confirming its identity.

C (1/10) YOUR ANSWER

The fraction 1/10 is equivalent to the decimal 0.1. Since 0.111... has a different repeating structure, it cannot be equal to 1/10, which represents a single decimal place rather than an infinite series of repeating ones.

D (1/11) YOUR ANSWER

The fraction 1/11 is approximately equal to 0.090909..., representing a different repeating decimal. As such, it does not match the form of 0.111..., which consistently repeats the digit '1' rather than '09'.

Conclusion

The repeating decimal 0.111... is fundamentally equivalent to the fraction 1/9, as shown through mathematical derivation. Each incorrect choice—1/7, 1/10, and 1/11—demonstrates distinct decimal representations that do not align with the infinite repetition of '1' in 0.111.... Understanding this equivalence is crucial for grasping the properties of repeating decimals and their fractional counterparts.

The repeating decimal 0.111... is equivalent to

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

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