First, find the actual distance: 3.2 inches × 25 miles/inch = 80 miles. On the new map with scale 1 inch : 40 miles, the distance will be 80 miles ÷ 40 miles/inch = 2.0 inches. Choice B incorrectly uses the ratio 3.2 ÷ 1.28. Choice C incorrectly multiplies 3.2 × (40/25). Choice D uses an incorrect proportion setup.

When you encounter scale problems, you need to work through them step-by-step: first convert the blueprint measurement to actual size, then apply any additional modifications.

Start by finding the actual diameter of the window shown on the blueprint. The scale tells you that $$\frac{1}{4}$$ inch represents 2 feet. The window's diameter is $$\frac{3}{8}$$ inch on the blueprint. To convert this, set up a proportion: $$\frac{1/4 \text{ inch}}{2 \text{ feet}} = \frac{3/8 \text{ inch}}{x \text{ feet}}$$

Cross-multiply: $$\frac{1}{4} \cdot x = 2 \cdot \frac{3}{8}$$, so $$\frac{x}{4} = \frac{6}{8} = \frac{3}{4}$$

Therefore: $$x = 3$$ feet. This is the actual diameter of the window shown on the blueprint.

Now apply the contractor's modification. The installed window will have 1.5 times this actual diameter: $$3 \times 1.5 = 4.5$$ feet.

Looking at the wrong answers: Choice A (9 feet) incorrectly multiplies the blueprint diameter by 24 instead of finding the scale conversion first. Choice B (6 feet) makes an error by multiplying $$\frac{3}{8} \times 2 \times 8$$ without properly handling the scale. Choice C (3 feet) gives you the actual size from the blueprint but forgets to apply the 1.5 multiplier.

The correct answer is D (4.5 feet).

Study tip: In multi-step scale problems, always convert to real-world measurements first, then apply any additional changes. Don't try to do everything at once—you'll lose track of which measurements are scaled and which are actual.

First find the actual width: Area = length × width, so 192 = 16 × width, giving width = 12 feet. The scale is 0.5 inch : 8 feet, or 1 inch : 16 feet. On the floor plan: length = 16 feet ÷ 16 feet/inch = 1 inch; width = 12 feet ÷ 16 feet/inch = 0.75 inches. Choice A incorrectly calculates the width. Choice B doubles both dimensions. Choice D uses the given scale ratio directly without conversion.

This question tests scale drawing problems: finding actual lengths/areas from scaled drawings (multiply by scale, use scale² for areas) and reproducing drawings at different scales. Scale 1:50,000 means 1 cm on the drawing equals 50,000 cm actual (multiply drawing by 50,000 for actual, divide actual by 50,000 for drawing). Example: map scale 1:50,000, 4 cm on map represents 4×50,000=200,000 cm=2 km actual (multiply drawing by scale factor). Areas scale by factor²: drawing 2 cm × 3 cm = 6 cm² at 1:100 scale represents actual 200 cm × 300 cm = 60,000 cm² (or 6×100²=60,000), factor squared because area is length×width both scaled. Reproducing: original 6 cm at 1:50 (actual: 6×50=300 cm), new scale 1:25 (new drawing: 300÷25=12 cm—half the scale factor means twice the drawing size). In this case, the map distance is 4 cm at 1:50,000, computing actual 200,000 cm=2 km (since 200,000 cm ÷ 100,000 cm/km = 2 km). The correct calculation is multiplying the drawing length by the scale factor and converting units properly.

This problem tests scale drawing problems by finding the actual length from a scaled map, where you multiply the map distance by the scale factor to get the real distance. A scale of 1:n means 1 unit on the drawing equals n units in actual, so multiply drawing by n for actual, or divide actual by n for drawing. For this map scale of 1:50,000, 4 cm on the map represents 4 × 50,000 = 200,000 cm actual. To convert, 200,000 cm = 2,000 m = 2 km, since there are 100 cm in a meter and 1,000 m in a km. A common error is dividing instead of multiplying, like 4 ÷ 50,000 = 0.00008, claiming that's the actual in km. Steps: (1) identify scale 1:50,000 and drawing 4 cm, (2) determine direction (drawing to actual: multiply), (3) calculate 4 × 50,000 = 200,000 cm, (4) convert to km (200,000 / 100,000 = 2 km). Common mistakes include wrong direction (dividing for actual), inconsistent units (not converting cm to km), or misinterpreting scale backwards.

This problem tests scale drawing problems by finding the drawing length from an actual measurement, where you divide the actual by the scale factor to get the drawing size. A scale of 1:n means 1 unit on the drawing equals n units in actual, so multiply drawing by n for actual, or divide actual by n for drawing. For this floor plan scale of 1 inch:5 feet, a 15-foot wall on the plan is 15 ÷ 5 = 3 inches. This correct scaling calculation ensures the plan accurately represents the real dimensions in reduced form. A common error is multiplying instead of dividing, like 15 × 5 = 75 inches, which would be too large for the plan. Steps: (1) identify scale 1:5 (inches to feet) and actual 15 feet, (2) determine direction (actual to drawing: divide), (3) calculate 15 ÷ 5 = 3 inches. Common mistakes include wrong direction (multiplying for drawing size), units inconsistent (mixing feet and inches without conversion), or scale ratio interpreted backwards.

This question tests scale drawing problems: finding actual lengths/areas from scaled drawings (multiply by scale, use scale² for areas) and reproducing drawings at different scales. Scale 1:25 means 1 cm on the model equals 25 cm actual (multiply model by 25 for actual, divide actual by 25 for model). Example: model scale 1:25, 18 cm on model represents 18×25=450 cm=4.5 m actual (multiply model by scale factor). Areas scale by factor²: drawing 2 cm × 3 cm = 6 cm² at 1:100 scale represents actual 200 cm × 300 cm = 60,000 cm² (or 6×100²=60,000), factor squared because area is length×width both scaled. Reproducing: original 6 cm at 1:50 (actual: 6×50=300 cm), new scale 1:25 (new drawing: 300÷25=12 cm—half the scale factor means twice the drawing size). In this case, model length 18 cm at 1:25, computing actual 450 cm=4.5 m (450÷100=4.5). The correct calculation is multiplying by the scale and converting cm to m properly.

This question tests scale drawing problems: finding actual lengths/areas from scaled drawings (multiply by scale, use scale² for areas) and reproducing drawings at different scales. Scale 1 inch:4 miles means 1 inch on map = 4 miles actual (multiply map by 4 for actual, divide actual by 4 for map). Example: map scale 1 inch:4 miles, 3.5 inches on map represents 3.5×4=14 miles actual (multiply map by scale factor). Areas scale by factor²: drawing 2 cm × 3 cm = 6 cm² at 1:100 scale represents actual 200 cm × 300 cm = 60,000 cm² (or 6×100²=60,000), factor squared because area is length×width both scaled. Reproducing: original 6 cm at 1:50 (actual: 6×50=300 cm), new scale 1:25 (new drawing: 300÷25=12 cm—half the scale factor means twice the drawing size). In this case, map distance 3.5 inches at 1:4, computing actual 3.5×4=14 miles. The correct calculation is multiplying the map distance by the scale factor without unit conversion issues.

This problem tests scale drawing problems by finding actual area from a plan, using scale squared for areas. A scale of 1:n means 1 unit on the drawing equals n units in actual, so multiply drawing by n for lengths, and by n² for areas. For this playground 4 cm × 5 cm at 1:200, actual 800 cm × 1,000 cm, area 800,000 cm² = 80 m² (divide by 10,000). This correct scaling uses area factor 200² = 40,000, so 20 cm² × 40,000 = 800,000 cm² = 80 m². A common error is linear scale for area, like 20 × 200 = 4,000 cm². Steps: (1) identify scale 1:200 and drawing area 20 cm², (2) use scale² = 40,000, (3) calculate 20 × 40,000 = 800,000 cm², (4) convert to m² (800,000 / 10,000 = 80). Common mistakes include forgetting to square, wrong units, or misdirection.

This question tests scale drawing problems: finding actual lengths/areas from scaled drawings (multiply by scale, use scale² for areas) and reproducing drawings at different scales. Scale 1:8 (inches to feet) means 1 inch on the blueprint equals 8 feet actual (multiply blueprint by 8 for actual, divide actual by 8 for blueprint). Example: map scale 1:50,000, 4 cm on map represents 4×50,000=200,000 cm=2 km actual (multiply drawing by scale factor). Areas scale by factor²: blueprint 2.5 in × 1.5 in = 3.75 in² at 1:8 scale represents actual 20 ft × 12 ft = 240 ft² (or 3.75×8²=3.75×64=240), factor squared because area is length×width both scaled. Reproducing: original 6 cm at 1:50 (actual: 6×50=300 cm), new scale 1:25 (new drawing: 300÷25=12 cm—half the scale factor means twice the drawing size). In this case, blueprint area 3.75 in² at 1:8, computing actual 3.75×64=240 ft² using scale² factor. A common error is using linear scale for areas (3.75×8=30 not 3.75×64=240), or miscalculating dimensions (like 2.5×8=20, 1.5×8=12, but forgetting to multiply).