Which kinds of numbers are rational? Select all that apply. $\newline$ Multi-select Choices: $\newline$ (A) integers $\newline$ (B) nonterminating, nonrepeating decimals $\newline$ (C) fractions of whole numbers $\newline$ (D) terminating decimals $\newline$ (E) nonterminating, repeating decimals

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Grade 8

Checkpoint: Rational and irrational numbers

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Q. Which kinds of numbers are rational? Select all that apply.

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Multi-select Choices:

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(A) integers

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(B) nonterminating, nonrepeating decimals

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(D) terminating decimals

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(E) nonterminating, repeating decimals

Define rational numbers: Step $1$ : Define rational numbers. Rational numbers are numbers that can be expressed as a fraction where both the numerator and the denominator are integers, and the denominator is not zero.
Integers: Step $2$ : Analyze choice (A) - integers. $\newline$ Integers (like $-1$ , $0$ , $1$ , $2$ ) can be expressed as fractions (e.g., $1$ can be written as $\frac{1}{1}$ ). Therefore, integers are rational numbers.
Nonterminating decimals: Step $3$ : Analyze choice (B) - nonterminating, nonrepeating decimals. $\newline$ Nonterminating, nonrepeating decimals (like $\pi$ or the square root of $2$ ) cannot be expressed as fractions of whole numbers. Hence, they are not rational numbers.
Fractions of whole numbers: Step $4$ : Analyze choice (C) - fractions of whole numbers. $\newline$ Fractions of whole numbers (like $\frac{1}{2}$ , $\frac{3}{4}$ ) are the very definition of rational numbers, as they can be expressed as a ratio of two integers.
Terminating decimals: Step $5$ : Analyze choice (D) - terminating decimals. $\newline$ Terminating decimals (like $0.5$ , $0.75$ ) can be expressed as fractions ( $0.5$ as $\frac{1}{2}$ , $0.75$ as $\frac{3}{4}$ ), making them rational numbers.
Nonterminating, repeating decimals: Step $6$ : Analyze choice (E) - nonterminating, repeating decimals. $\newline$ Nonterminating, repeating decimals (like $0.333\ldots$ , $0.666\ldots$ ) can be expressed as fractions ( $0.333\ldots$ as $\frac{1}{3}$ , $0.666\ldots$ as $\frac{2}{3}$ ), so they are rational numbers.