lesson 11: ratios, rates, and proportions

11. solve the proportion $\frac{2}{5} = \frac{x}{15}$.

12. solve the proportion $\frac{12}{x} = \frac{18}{6}$.

13. solve for $w$ in the proportion $\frac{2w + 3}{5} = \frac{w - 2}{3}$.

in the following problems, the ratios are directly proportional. find the missing variable.
14. if $y_1 = 4$, $x_2 = 6$, and $y_2 = 8$, what is the value of $x_1$?

15. if $x_1 = 440$, $y_1 = 220$, and $x_2 = 360$, determine the value of $y_2$.

in exercises 16 and 17, the ratios are inversely proportional. find the missing value.
16. if $r_1 = 6$, $r_2 = 8$, and $i_1 = 12$, what is the value of $i_2$?

Question

lesson 11: ratios, rates, and proportions

11. solve the proportion $\frac{2}{5} = \frac{x}{15}$.

12. solve the proportion $\frac{12}{x} = \frac{18}{6}$.

13. solve for $w$ in the proportion $\frac{2w + 3}{5} = \frac{w - 2}{3}$.

in the following problems, the ratios are directly proportional. find the missing variable.
14. if $y_1 = 4$, $x_2 = 6$, and $y_2 = 8$, what is the value of $x_1$?

15. if $x_1 = 440$, $y_1 = 220$, and $x_2 = 360$, determine the value of $y_2$.

in exercises 16 and 17, the ratios are inversely proportional. find the missing value.
16. if $r_1 = 6$, $r_2 = 8$, and $i_1 = 12$, what is the value of $i_2$?

Accepted Answer

{ "quality": "clear", "question_count": 6, "skills_matched": [ "step_cot", "table_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Solving Proportions", "Direct Proportion Equations", "Inverse Proportion Equations" ], "new_concepts": [], "current_concepts": [ "Solving Proportions", "Direct Proportion Equations", "Inverse Proportion Equations" ] } ## Solve the proportions in questions 11 to 13 $$ \begin{aligned} &\text{Question 11: } \frac{2}{5} = \frac{x}{15} \implies 5x = 30 \implies x = 6\ &\text{Question 12: } \frac{12}{x} = \frac{18}{6} \implies 18x = 72 \implies x = 4\ &\text{Question 13: } \frac{2w + 3}{5} = \frac{w - 2}{3} \implies 3(2w + 3) = 5(w - 2) \implies 6w + 9 = 5w - 10 \implies w = -19 \end{aligned} $$ ## Solve the direct proportions in questions 14 and 15 $$ \begin{aligned} &\text{Question 14: } \frac{y_1}{x_1} = \frac{y_2}{x_2} \implies \frac{4}{x_1} = \frac{8}{6} \implies 8x_1 = 24 \implies x_1 = 3\ &\text{Question 15: } \frac{y_1}{x_1} = \frac{y_2}{x_2} \implies \frac{220}{440} = \frac{y_2}{360} \implies \frac{1}{2} = \frac{y_2}{360} \implies y_2 = 180 \end{aligned} $$ ## Solve the inverse proportion in question 16 $$ \begin{aligned} &\text{Question 16: } R_1 I_1 = R_2 I_2 \implies 6 \cdot 12 = 8 \cdot I_2 \implies 72 = 8I_2 \implies I_2 = 9 \end{aligned} $$ | No. | Problem | Answer | |---|---|---| | 11 | Solve the proportion $\frac{2}{5} = \frac{x}{15}$ | $x = 6$ | | 12 | Solve the proportion $\frac{12}{x} = \frac{18}{6}$ | $x = 4$ | | 13 | Solve for $w$ in the proportion $\frac{2w + 3}{5} = \frac{w - 2}{3}$ | $w = -19$ | | 14 | If $y_1 = 4$, $x_2 = 6$, and $y_2 = 8$ (directly proportional), find $x_1$ | $x_1 = 3$ | | 15 | If $x_1 = 440$, $y_1 = 220$, and $x_2 = 360$ (directly proportional), find $y_2$ | $y_2 = 180$ | | 16 | If $R_1 = 6$, $R_2 = 8$, and $I_1 = 12$ (inversely proportional), find $I_2$ | $I_2 = 9$ | { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Solving Proportions" ] }

No.	Problem	Answer
12	Solve the proportion \(\frac{12}{x} = \frac{18}{6}\)	\(x = 4\)
13	Solve for \(w\) in the proportion \(\frac{2w + 3}{5} = \frac{w - 2}{3}\)	\(w = -19\)
14	If \(y_1 = 4\), \(x_2 = 6\), and \(y_2 = 8\) (directly proportional), find \(x_1\)	\(x_1 = 3\)
15	If \(x_1 = 440\), \(y_1 = 220\), and \(x_2 = 360\) (directly proportional), find \(y_2\)	\(y_2 = 180\)
16	If \(R_1 = 6\), \(R_2 = 8\), and \(I_1 = 12\) (inversely proportional), find \(I_2\)	\(I_2 = 9\)

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QUESTION IMAGE

Question

Explanation:

Solve the proportions in questions 11 to 13

Solve the direct proportions in questions 14 and 15

Solve the inverse proportion in question 16

Answer: