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Which of the following are rational numbers?\newlineMulti-select Choices:\newline(A) 12\frac{1}{2}\newline(B) 10-10\newline(C) 1.777...1.777...\newline(D) 7-7

Full solution

Q. Which of the following are rational numbers?\newlineMulti-select Choices:\newline(A) 12\frac{1}{2}\newline(B) 10-10\newline(C) 1.777...1.777...\newline(D) 7-7
  1. Definition of Rational Number: A rational number is a number that can be expressed as the quotient or fraction pq\frac{p}{q} of two integers, where pp is the numerator, qq is the denominator, and q0q \neq 0. Let's evaluate each option to determine if it is a rational number.
  2. Option (A) Evaluation: Option (A) is 12\frac{1}{2}. This is a fraction where the numerator is 11 and the denominator is 22. Since both 11 and 22 are integers and the denominator is not zero, 12\frac{1}{2} is a rational number.
  3. Option (B) Evaluation: Option (B) is 10-10. This can be expressed as a fraction 10/1-10/1, where the numerator is 10-10 and the denominator is 11. Since both 10-10 and 11 are integers and the denominator is not zero, 10-10 is a rational number.
  4. Option (C) Evaluation: Option (C) is 1.7771.777\ldots The ellipsis (\ldots) indicates that the number is repeating indefinitely. If 1.7771.777\ldots is a repeating decimal, then it can be expressed as a fraction, which would make it a rational number. However, we need to confirm if the decimal is indeed repeating. If it is not repeating and is instead a non-repeating, non-terminating decimal, then it would not be a rational number. Since the problem does not specify that 1.7771.777\ldots is repeating, we cannot assume it is rational without further information.
  5. Option (D) Evaluation: Option (D) is 7-7. This can be expressed as a fraction 71-\frac{7}{1}, where the numerator is 7-7 and the denominator is 11. Since both 7-7 and 11 are integers and the denominator is not zero, 7-7 is a rational number.

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