write each scale in simplest form and as an equation.
7. two cities that are actually 350 kilometers apart are 7 centimeters apart on a map. what is the scale of the map?
8. a drawing of a solar heating panel on a blueprint is 2 centimeters wide. the actual panel is 3 meters wide. what is the scale of the drawing?
9. the length of a room on a drawing is 3.25 inches. the actual room will be 15 feet $8\frac{1}{2}$ inches. what is the scale of the drawing?
10. challenge: what is the ratio in simplest form of 80 centimeters to 2 kilometers?

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Accepted Answer

{ "quality": "clear", "question_count": 4, "skills_matched": [ "step_cot", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Unit Conversion in Ratios", "Ratio Simplification", "Scale Ratios" ], "new_concepts": [], "current_concepts": [ "Unit Conversion in Ratios", "Ratio Simplification", "Scale Ratios" ] } ## Convert units and simplify scale for Question 7 $$ \begin{aligned} &\text{Map distance} = 7\text{ cm},\quad \text{Actual distance} = 350\text{ km} = 35,000,000\text{ cm}\ &\text{Ratio} = \frac{7}{35,000,000} = \frac{1}{5,000,000} \implies 1 : 5,000,000\ &\text{Equation: } d = 5,000,000c \quad (\text{or } c = \frac{1}{5,000,000}d) \end{aligned} $$ ## Convert units and simplify scale for Question 8 $$ \begin{aligned} &\text{Drawing width} = 2\text{ cm},\quad \text{Actual width} = 3\text{ m} = 300\text{ cm}\ &\text{Ratio} = \frac{2}{300} = \frac{1}{150} \implies 1 : 150\ &\text{Equation: } w_{\text{actual}} = 150 w_{\text{drawing}} \quad (\text{or } w_{\text{drawing}} = \frac{1}{150}w_{\text{actual}}) \end{aligned} $$ ## Convert units and simplify scale for Questions 9 and 10 $$ \begin{aligned} &\text{Q9: Drawing} = 3.25\text{ in},\quad \text{Actual} = 15\text{ ft } 8\frac{1}{2}\text{ in} = 15 \times 12 + 8.5 = 188.5\text{ in}\ &\text{Ratio} = \frac{3.25}{188.5} = \frac{325}{18850} = \frac{13}{754} = \frac{1}{58} \implies 1 : 58\ &\text{Equation: } L_{\text{actual}} = 58 L_{\text{drawing}}\ &\text{Q10: Ratio of } 80\text{ cm to } 2\text{ km} = 200,000\text{ cm}\ &\text{Ratio} = \frac{80}{200,000} = \frac{8}{20,000} = \frac{1}{2,500} \implies 1 : 2,500 \end{aligned} $$ ## Question 7 - Simplest form: $1 : 5,000,000$ - Equation: $d = 5,000,000c$ (where $c$ is map distance in cm, and $d$ is actual distance in cm) ## Question 8 - Simplest form: $1 : 150$ - Equation: $y = 150x$ (where $x$ is drawing width, and $y$ is actual width in the same units) ## Question 9 - Simplest form: $1 : 58$ - Equation: $y = 58x$ (where $x$ is drawing length, and $y$ is actual length in the same units) ## Question 10 - Simplest form: $1 : 2,500$ { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Algebra", "Scale Ratios" ] }

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Convert units and simplify scale for Question 7

Convert units and simplify scale for Question 8

Convert units and simplify scale for Questions 9 and 10

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Question 7

Question 8

Question 9

Question 10