two solutions of different concentrations of acid are mixed creating 40 ml of a solution that is 32% acid. one-quarter of the solution is made up of a 20% acid solution. the remaining three-quarters is made up of a solution of unknown concentration, c.

which equation can be used to determine c, the unknown concentration?

$30c + 10(0.2) = 40(0.32)$

$\frac{3}{4}(c) +
\frac{1}{4}(0.2) = 40(0.32)$

$\frac{3}{4}(c)(\frac{1}{4}(0.2)) = 40(0.32)$

$30(c)(10(0.2)) = 40(0.32)$

Question

two solutions of different concentrations of acid are mixed creating 40 ml of a solution that is 32% acid. one-quarter of the solution is made up of a 20% acid solution. the remaining three-quarters is made up of a solution of unknown concentration, c.

which equation can be used to determine c, the unknown concentration?

$30c + 10(0.2) = 40(0.32)$

$\frac{3}{4}(c) +
\frac{1}{4}(0.2) = 40(0.32)$

$\frac{3}{4}(c)(\frac{1}{4}(0.2)) = 40(0.32)$

$30(c)(10(0.2)) = 40(0.32)$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Algebraic Modeling" ], "new_concepts": [ "Mixture Word Problems" ], "current_concepts": [ "Algebraic Modeling", "Mixture Word Problems" ] } ## Identify given values and volumes Using the *Algebraic Modeling* knowledge point $$ \begin{aligned} &V_{\text{total}} = 40\text{ mL}\ &C_{\text{total}} = 32\% = 0.32\ &V_1 = \frac{1}{4} \times 40\text{ mL} = 10\text{ mL}\ &C_1 = 20\% = 0.2\ &V_2 = \frac{3}{4} \times 40\text{ mL} = 30\text{ mL}\ &C_2 = c \end{aligned} $$ ## Formulate the mixture equation Using the *Algebraic Modeling* knowledge point $$ \begin{aligned} &\text{Amount of pure acid in solution 1} + \text{Amount of pure acid in solution 2} = \text{Total pure acid}\ &V_2 \cdot C_2 + V_1 \cdot C_1 = V_{\text{total}} \cdot C_{\text{total}}\ &30c + 10(0.2) = 40(0.32) \end{aligned} $$ ## Compare with the given options Using the *Algebraic Modeling* knowledge point $$ \begin{aligned} &\text{Option 1: } 30c + 10(0.2) = 40(0.32) \quad (\text{Matches derived equation})\ &\text{Option 2: } \frac{3}{4}(c) + \frac{1}{4}(0.2) = 40(0.32) \quad (\text{Incorrect: mixes fractions of volume with total volume on RHS})\ &\text{Option 3: } \frac{3}{4}(c)(\frac{1}{4}(0.2)) = 40(0.32) \quad (\text{Incorrect: multiplies the components instead of adding})\ &\text{Option 4: } 30(c)(10(0.2)) = 40(0.32) \quad (\text{Incorrect: multiplies the components instead of adding}) \end{aligned} $$

(A) $30c + 10(0.2) = 40(0.32)$

(B) $\frac{3}{4}(c) + \frac{1}{4}(0.2) = 40(0.32)$

(C) $\frac{3}{4}(c)(\frac{1}{4}(0.2)) = 40(0.32)$

(D) $30(c)(10(0.2)) = 40(0.32)$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Algebra", "Algebraic Modeling" ] }

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Question

Explanation:

Identify given values and volumes

Formulate the mixture equation

Compare with the given options

Answer: