which of the following numbers would fall between $\frac{1}{3}$ and $\frac{2}{3}$ on a number line?

select all answers that apply.

a $2^{-3}$

b $\sqrt{2}$

c $0.48$

d $\frac{4}{9}$

Question

which of the following numbers would fall between $\frac{1}{3}$ and $\frac{2}{3}$ on a number line?

select all answers that apply.

a $2^{-3}$

b $\sqrt{2}$

c $0.48$

d $\frac{4}{9}$

Accepted Answer

{ "quality": "clear", "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": [ "Negative Exponents", "Irrational Number Estimation" ], "current_concepts": [ "Real Number Comparison", "Fraction to Decimal Conversion", "Number Line Intervals", "Negative Exponents", "Irrational Number Estimation" ] } ## Convert the interval boundaries to decimals $$ \begin{aligned} &\frac{1}{3} \approx 0.333\ &\frac{2}{3} \approx 0.667\ &\text{Interval: } (0.333, 0.667) \end{aligned} $$ ## Evaluate each option as a decimal $$ \begin{aligned} &\text{A: } 2^{-3} = \frac{1}{2^3} = \frac{1}{8} = 0.125\ &\text{B: } \sqrt{2} \approx 1.414\ &\text{C: } 0.48\ &\text{D: } \frac{4}{9} \approx 0.444 \end{aligned} $$ ## Compare values to the interval $$ \begin{aligned} &0.125 < 0.333 \quad (\text{A is outside})\ &1.414 > 0.667 \quad (\text{B is outside})\ &0.333 < 0.48 < 0.667 \quad (\text{C is inside})\ &0.333 < 0.444 < 0.667 \quad (\text{D is inside}) \end{aligned} $$

(A) $2^{-3}$

(B) $\sqrt{2}$

(C) $0.48$

(D) $\frac{4}{9}$

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QUESTION IMAGE

Question

Explanation:

Convert the interval boundaries to decimals

Evaluate each option as a decimal

Compare values to the interval

Answer: