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California Standards

Learn how to use proportions and scale factors to solve problems involving scale models and scale drawings. Discover the relationship between dimensions of scale models and their actual objects. Explore examples and practice finding scale factors.

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California Standards

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  1. California Standards NS1.3 Use proportions to solve problems (e.g., determine the value of N if T = , find the length of a side of a polygon similar to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse. 4 7 N 21

  2. Vocabulary scale model scale factor scale scale drawing

  3. This HO gauge model train is a scale model of a historic train. A scale model is a proportional model of a three-dimensional object. Its dimensions are related to the dimensions of the actual object by a ratio called the scale factor. The scale factor of an HO gauge model train is . 1 87 1 87 This means that each dimension of the model is of the corresponding dimension of the actual train.

  4. A scale is the ratio between two sets of measurements. Scales can use the same units or different units. The photograph shows a scale drawing of the model train. A scale drawingis a proportional two-dimensional drawing of an object. Both scale drawings and scale models can be smaller or larger than the objects they represent.

  5. 13.5 108 18 144 1 8 The scale factor is . Additional Example 1: Finding a Scale Factor Identify the scale factor. You can use the lengths or widths to find the scale factor. 1 8 blueprint length room length = = Write a ratio. Then simplify. 1 8 blueprint width room width = = Write a ratio. Then simplify.

  6. 1 6 The scale factor is . Check It Out! Example 1 Identify the scale factor. blueprint length aircraft length 2 12 1 6 = Write a ratio. Then simplify. = blueprint wing span aircraft wing span 3 18 1 6 = Write a ratio. Then simplify. =

  7. Additional Example 2: Using Scale Factors to Find Unknown Lengths A photograph was enlarged and made into a poster. The poster is 20.5 inches by 36 inches. The scale factor is . Find the size of the original photograph to the nearest tenth of an inch. 5 1 poster photo 5 1 = Think: 36 l 5 1 Write a proportion to find the length l. = 5l = 36 Find the cross products. l= 7.2 Divide.

  8. Additional Example 2 Continued A photograph was enlarged and made into a poster. The poster is 20.5 inches by 36 inches. The scale factor is . Find the size of the photograph. 5 1 poster photo 5 1 = Think: 20.5 w 5 1 Write a proportion to find the width w. = 5w = 20.5 Find the cross products. w= 4.1 Divide. The photo is 7.2 in. long and 4.1 in. wide.

  9. l 45 1.5 1 = Check It Out! Example 2 Mary’s father made her a dollhouse which was modeled after the blueprint of their home. The blueprint is 24 inches by 45 inches. The scale factor is . Find the size of the dollhouse. 1.5 1 dollhouse blueprint 1.5 1 = Think: Write a proportion to find the length l. Find the cross products. l = 45 · 1.5 l= 67.5 Multiply.

  10. Check It Out! 2 Continued Mary’s father made her a dollhouse which was modeled after the blueprint of their home. The blueprint is 24 inches by 45 inches. The scale factor is . Find the size of the dollhouse. 1.5 1 dollhouse blueprint 1.5 1 = Think: Write a proportion to find the width w. w 24 1.5 1 = w = 24 · 1.5 Find the cross products. w= 36 Multiply. The dollhouse is 67.5 inches long and 36 inches wide.

  11. Additional Example 3: Measurement Application On a road map with a scale of 1.5 inches = 60 miles, the distance between Pittsburgh and Philadelphia is 7.5 inches. What is the actual distance between the cities? Let d be the actual distance between the cities. 1.5 60 7.5 d = Write a proportion. 1.5 · d = 60 · 7.5 Find the cross product. 1.5d = 450 Multiply. 450 1.5 1.5d 1.5 = Divide both sides by 1.5. d = 300 The distance between the cities is 300 miles.

  12. Check It Out! Example 3 On a road map with a scale of 1 inch = 50 kilometers, the distance between Dallas and Houston is 7 inches. What is the actual distance between the two cities? Let d be the actual distance between the cities. 1 50 7 d = Write a proportion. Find the cross products. 1 · d = 50 · 7 1d = 350 Multiply. d = 350 The distance between the cities is 350 kilometers.

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