Which option correctly describes the nature of (0.\overline{123})?
Answer and explanation
Correct answer: Rational number
Let \(x=0.\overline{123}\). Then \(1000x=123.\overline{123}\). Subtracting gives \(999x=123\), so \(x=\dfrac{123}{999}=\dfrac{41}{333}\). Since it equals a ratio of integers, it is rational. Option B is wrong because the decimal is repeating and converts to a fraction; C is wrong because the number is a real decimal; D is wrong because the value lies between 0 and 1, not an integer. Exam tip: For a repeating block of length n, use n nines in the denominator (e.g., 3-digit repeat → denominator 999).
Frequently asked questions
What is the correct answer to this question?
Rational number
Why is this the correct answer?
Let \(x=0.\overline{123}\). Then \(1000x=123.\overline{123}\). Subtracting gives \(999x=123\), so \(x=\dfrac{123}{999}=\dfrac{41}{333}\). Since it equals a ratio of integers, it is rational. Option B is wrong because the decimal is repeating and converts to a fraction; C is wrong because the number is a real decimal; D is wrong because the value lies between 0 and 1, not an integer. Exam tip: For a repeating block of length n, use n nines in the denominator (e.g., 3-digit repeat → denominator 999).
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Irrational numbers and real numbers.
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