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{ "quality": "blurry", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": true, "concepts_used": [ "Real Number Comparison", "Fraction to Decimal Conversion", "Number Line Intervals" ], "new_concepts": [], "current_concepts": [ "Real Number Comparison", "Fraction to Decimal Conversion", "Number Line Intervals" ] } ## Convert the interval boundaries to decimals $$ \begin{aligned} &\frac{1}{2} = 0.5 \ &\frac{2}{3} \approx 0.667 \end{aligned} $$ The target interval is $(0.5, 0.667)$. ## Convert the options to decimals $$ \begin{aligned} &\text{Option A: } 2^{-2} = \frac{1}{4} = 0.25 \ &\text{Option B: } \sqrt{2} \approx 1.414 \ &\text{Option C: } 0.48 \ &\text{Option D: } \frac{4}{7} \approx 0.571 \end{aligned} $$ ## Compare values to the target interval $$ \begin{aligned} &0.25 < 0.5 \quad (\text{Outside}) \ &1.414 > 0.667 \quad (\text{Outside}) \ &0.48 < 0.5 \quad (\text{Outside}) \ &0.5 < 0.571 < 0.667 \quad (\text{Inside}) \end{aligned} $$

(A) $2^{-2}$

(B) $\sqrt{2}$

(C) $0.48$

(D) $\frac{4}{7}$

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Explanation:

Convert the interval boundaries to decimals

Convert the options to decimals

Compare values to the target interval

Answer: