Free Pre-Algebra Lesson 37! page 1 Lesson 37 Scale and Proportion Ratios and rates are a powerful way to compare data. Comparing and calculating with ratios and rates is one of the most common and useful ways we think mathematically every day. When ratios are equal, we say they are in proportion. The most visual way to think about proportion is seeing the same shape in different sizes. Similar Figures Scale models are interesting because they look just like the real thing except for the size. The way to achieve that is to keep the proportions the ratios of measurements the same in the model as in the original item. Fascination with Scale People seem to enjoy things in unusual sizes. Consider the drawings on graph paper below. 1 2 2 4 These figures are similar. Each of the sides in the larger figure is twice the length of the corresponding side in the smaller figure. The ratio 1/2 is the same as the ratio 2/4. Check that this is true for any two sides. The shapes of the figures look the same. Only the size is different. These figures are not similar. You can see that the shapes are slightly different. The larger figure seems to have a relatively shorter, thinner base compared to the smaller figure. The corresponding sides are not in proportion, since the ratio 1/2 is not the same as the ratio 2/3.
Free Pre-Algebra Lesson 37! page 2 Photographs and photocopies are easily re-sized with lenses or digital computations. When you re-size an image with a computer program, you are usually given a default choice to maintain aspect ratio. That means that when you resize one side, the other stays in the same proportion as in the original. If you do not choose this option, the photo will appear distorted. Art Glossary and Vocabulary: Grid Enlarging from http://www.bluemoonwebdesign.com/art-glossary-2.asp A grid is made over the smaller image. By copying the parts of the picture square by square into a larger grid, the picture is enlarged (by hand) while keeping the proportions constant. Largest carp ever caught! 256 pounds. Photo: width 1.875 inches x length 3 inches. Ratio of width to length 1.875 / 3 = 0.625. The photo was reduced in size but the ratio of width to length is still the same. Only the width of this photo was reduced, causing the fish to appear relatively longer and thinner. Width 1 inch x Length 1.6 inches. Ratio of width to length is 1 / 1.6 = 0.625. Width 1 inch x Length 3 inches. Ratio of width to length is 1 / 3 = 0.333 Example: Find the ratio of the shorter side to the longer side for each rectangle. Which rectangles are similar? This rectangle is similar to the first rectangle. 1.3 inches by 1.95 inches 1.5 inches x 1.5 inches 0.8 inches by 1.2 inches 0.77 inches x 1.5 inches Ratio approx 0.67 Ratio 1 Ratio approx 0.67 Ratio approx 0.51
Free Pre-Algebra Lesson 37! page 3 (Almost) Every Which Way An interesting thing about equal ratios is that you can set them up and compare in a variety of ways. Suppose we have the two similar rectangles below. You can organize the information about the rectangles into a table, which turns out to be very helpful. CM BLUE PINK LENGTH 2 3 WIDTH 1 1.5 To compare the length and width of each rectangle, you use the columns of the table: length width = 2 1 = 3 1.5 = 2 The length is twice the width in each rectangle. width length = 1 2 = 1.5 3 = 0.5 The width is 1/2 the length in each rectangle. The rectangles are similar so the column ratios and their reciprocals are equal. The ratio of length to width for both rectangles is 2, and the ratio of width to length is the same ratio upside down, the reciprocal, 1/2 = 0.5. These ratios compare the length to the width within each rectangle. The rectangles are similar so it is also true that the ratios of corresponding sides of the rectangles are equal. The ratios are in the rows of the table. blue pink = 2 3 = 1 1.5 = 0.6 The sides of the blue rectangle are 2/3 of the corresponding sides of the pink rectangle. pink blue = 3 2 = 1.5 1 = 1.5 The sides of the pink rectangle are 1.5 times the corresponding sides of the blue rectangle. These ratios compare the side of one triangle to the side of the other. The sides of the pink rectangle are 1.5 times the sides of the blue rectangle. However, be careful about saying that the pink rectangle is 1.5 times as big as the blue rectangle, because the ratio of areas is NOT the same as the ratio of sides. So you can compare the ratios of rows or columns in the table. How about along the diagonal? The numbers in the diagonal do not form equal ratios. Instead they form equal products, called cross-products. 2 1.5 = 3 1 = 3 You can see that the cross products of all the different ratios for the table are the same. People often use the cross product to check that two ratios are equal without doing division.
Free Pre-Algebra Lesson 37! page 4 Example: Organize the information from the triangles into a table, check that the cross-products are equal, and write the requested ratios. a. Organize the information into a table. CM GREEN ORANGE SIDE A 3 0.6 SIDE C 5 1 b. Check that the cross products are equal. Are the triangles similar? c. What is the ratio of side a to side c in each triangle? 3 5 = 0.6 0.6 1 = 0.6 d. What is the ratio of the sides of the green triangle to the corresponding sides of the orange triangle? 3 1 = 3 5 0.6 = 3 Cross products are equal, therefore the triangles are similar. 3 0.6 = 5 5 1 = 5 This means that the sides of the green triangle are 5 times as long as the sides of the orange triangle. Since we know the triangles are similar, we can find the length of a missing side from the ratios. Example: Find the length of side b of the orange triangle in the problem above. 1 Make a table that includes the missing side (use a variable). CM GREEN ORANGE SIDE B 4 b SIDE C 5 1 2 Use the cross-products to write an equation. 4 1= 5 b 3 Simplify and solve the equation. 5b = 4 5b / 5 = 4 / 5 b = 4 5 = 0.8 Side b of the orange triangle is 0.8 cm long. Example: The rectangles shown are similar. Fill in the table and find the length of the larger rectangle. a. Fill in the table. Use a variable for the unknown side. LENGTH L 1 1 /4 WIDTH 2 3/4 b. Find the length of the larger rectangle. 3 4 L = 5 2 3 4 L = 2 11 4 = 2 1 5 = 5 4 2 2 2 4 3 3 4 L = 5 2 4 3 L = 10 3 = 3 1 3 inches
Free Pre-Algebra Lesson 37! page 5 Scale Models are usually made according to a scale, which is a ratio that compares the sizes of the model to the original. For example, the O Scale for model railroads is written 1:48, read 1 to 48. (The colon : is in place of the fraction bar / as an alternative way to write a ratio.) The ratio means that 1 inch in the model corresponds to 48 inches on the original train. Model makers often need to figure out measurements in their scale corresponding to measurements of the original object using the scale, but the scales are usually designed to make such conversions very easy. For example, dollhouses often come in a 1:12 scale, meaning that 1 inch in the dollhouse corresponds to 12 inches (1 foot) in the original house. A scale is written as a ratio of like units, so the scale 1:12 means that something measuring 1 inch in the dollhouse measures 12 inches in a real house. Since 12 inches is 1 foot, the choice of scale allows us to simply change the unit of measurement to find the corresponding length. If a real piano keyboard is 2 1 / 2 feet above the floor, the dollhouse piano keyboard will be 2 1 / 2 inches above the floor. If the dollhouse bookshelf is 6 inches tall, the corresponding real bookshelf is 6 feet tall. Architects working in metric units may use a scale 1:100 for plans, because there are 100 cm in 1 m. That way distances measuring 1 m on the construction site will be represented by 1 cm on the architectural plans, and conversions consist only of changing the name of the unit of measurement. Sometimes the scale is not exactly a change of unit, but requires slightly more work. Example: A model airplane is made in a 1:48 scale. If the original aircraft had a wing span of 37 feet, what is the wingspan of the model? The 1:48 scale represents a ratio of like units. We could make a table: MEASUREMENT MODEL 1 x feet ORIGINAL 48 37 feet Solving the cross-product equation: 1 37 = 48x 48x = 37 48x / 48 = 37 / 48 x! 0.77 feet As a model-builder, this answer might not be very helpful. It would be better in inches. 0.77 feet 12 inches = 9.25 inches 1 1 foot If you re doing many of these conversions, you d probably set it up with different units in the scale. Since 48 inches is the same as 4 feet, you could write model measurements in inches and original measurements in feet. Then solving the crossproduct equation would give your answer directly in inches. MEASUREMENT MODEL 1 inch x inches ORIGINAL 4 feet 37 feet 1 37 = 4x 4x = 37 4x / 4 = 37 / 4 x = 9.25 inches
Free Pre-Algebra Lesson 37! page 6 Maps and Scale To the right, see part of a map of Yosemite National Park. The map has a scale, showing distances on the map corresponding to real distances that visitors would hike or drive. The ruler below shows that 2 inches on the map (at the size shown) corresponds to 5 miles of real distance. To estimate distances on the map, people usually match some distance on the scale to a length on a finger, then use that length to measure out curvy roads or rivers. To be more precise, they might calculate with the units included, as the model builders did above. Hodgdon Meadow 120 Information Station Big Oak Flat Entrance W Big Old Big Oak South closed to vehicle traffic Oak Flat TUOLUMNE GROVE Flat Road Crane Flat Road Fork Tuolumne Tioga Road closed November to May east of this point River However, to a geographer, the scale of the map should properly be written in like units (so that the units cancel from the ratio). So to find the scale of the map, we d convert 5 miles to inches. 0 1 0 1 MERCED GROVE 5 Kilometers 5 Miles 5 miles 1 5280 feet 1 mile 12 inches = 316,800 inches 1 foot The ratio of map distance to real distance is 1 2 inches 316,800 inches = 1 158,400 158,400 The scale is 1:158,400, meaning that distances in the park are 158,400 times as great as they are on the map. Map Scales from a Geography Syllabus http://krygier.owu.edu/krygier_html/geog_222/geog_222_lo/geog_2 22_lo04.html Example: Show that the ratio of inches to miles and cm to km for the first map are the same as the representative fraction given for that map. a. Convert 1,485 miles to inches. (63,360 inches = 1 mile) 1,485 miles 1 63,360 inches 1 mile = 94,089,600 inches When rounded, the ratio is about 1 / 94,000,000. b. Convert 940 km to cm. (100,000 cm = 1 km) 940 km 1 The ratio is 1 / 94,000,000. 100,000 cm 1 km = 94,000,000 cm Note that it s far easier to convert the version in metric units, since in that system conversions are based on powers of ten.!
Free Pre-Algebra Lesson 37! page 7 Lesson 37: Scale and Proportion Worksheet Name 1. Fill in the table with the information from the rectangles. 2. Use the cross-products to check whether or not the rectangles are similar. LENGTH WIDTH 3. What is the ratio of length to width in the two rectangles? 4. What is the ratio of the side of the large rectangle to the corresponding side of the small rectangle? Fill in the blank: The length is times the width in both rectangles. Fill in the blank: The sides of the large rectangle are times the corresponding sides of the small rectangle. 5. Use the steps following to verify that the triangles are similar, and to find side c of the larger triangle. a. Fill in the table for sides a and b and check cross products in order to verify that the triangles are similar. SIDE A SIDE B b. Fill in the table for sides a and c and use cross products to write an equation for the missing side. c. Solve the equation to find the side c. SIDE A SIDE C
SIDE A SIDE SIDE B A Free Pre-Algebra SIDE C Lesson 37! page 8 6. A typical H0 (1:87) model railroad engine is 1.97 inches tall, and 3.94 to 11.81 inches long. SIDE A 1.25 0.5 (http://en.wikipedia.org/wiki/rail_transport_modelling) SIDE SIDE B A 1.25 3 1.2 0.5 a. How tall is the real engine being modeled? b. To what real length does 3.94 inches correspond? SIDE C C 1.3 CONVERSION SIDE A MODEL 1 CONVERSION 1.97 SIDE C MODEL REAL 87 1 1.97 REAL 87 SIDE A 1.25 0.5 CONVERSION SIDE C C 1.3 MODEL 1 3.94 c. How many feet of track correspond to 5280 feet (1 mile) for real REAL trains? 87 CONVERSION MODEL 1 CONVERSION 1.97 REAL MODEL 871 REAL 87 5280 CONVERSION MODEL 1 3.94 REAL 87 CONVERSION MODEL 1 3.94 REAL 87 HO Scale GE 44-ton switcher made by Bachmann, shown with a pencil for size. CONVERSION 7. a. A map MODEL legend shows 1 a line 7/8 inch long to represent 2000 feet. How many feet are represented by a line 1 inch long? REAL 87 5280 CONVERSION MAP 7/8 inch 1 inch REAL 2000 feet x feet b. What is the scale of the map as a ratio of like units? c. The real distances are about times as far as the distances shown on the map. (Round to the nearest thousand.)
Free Pre-Algebra Lesson 37! page 9 8. A Google map showing part of Pleasant Hill, California. The map scale is in the lower left hand corner. a. Measure the scale with your finger and estimate the length and width of the (real) Diablo Valley College campus. b. What is the approximate real length of Taylor Blvd from Pleasant Hill Road to Contra Costa Blvd? c. If the length on the map representing 2000 feet is 7/8 inch long, how many inches on the map represents a distance of 5280 feet (1 mile)? CONVERSION MAP 7/8 inch x inches REAL 2000 feet 5280 feet
Free Pre-Algebra Lesson 37! page 10 Lesson 37: Scale and Proportion Homework 37A c Name Formulas for Circles and Spheres: C = 2!r A =!r 2 V = 4 3!r 3 1. Find the circumference and area of a circle with radius 12.5 miles. Round to the nearest tenth. 2. Find the volume of a sphere with radius 12.5 miles. Round to the nearest whole number. 3. Solve the equation. 0.7x! 0.5 = 1.95 4. If a circle has circumference 8.75 inches, what is the radius, to the nearest tenth? 5. Use the table to make the requested comparisons. b. Write one or more sentences comparing the data. 1965 2005 U.S. Population 195 million 296 million Number of Smokers 48 million 50 million a. Find the rate comparing smokers to total population for 1965 and 2005. Write as a decimal rounded to the nearest hundredth. 6. Change the numbers to scientific notation. a. 78,999,000,000,000 b. 715,000,000,000,000,000 8. a. Write the number in standard form. They were charging $2.8 million for that house. 7. Change to standard notation. a. 2.13 x 10 14 b. Write with a decimal point and place value name. This house was only $1,400,000. b. 2.7 x 10 8
Free Pre-Algebra Lesson 37! page 11 9. The rectangles are similar. Find the width of the smaller rectangle. 10. a. What is the ratio of length to width for each of the rectangles in #9? L W b. Fill in the blank: The length is times the width in both rectangles. c. What is the ratio of the sides CONV. of the large rectangle to the corresponding sides of the small rectangle? MODEL 1 x feet REAL 180 311.5 feet 11. Pictured below is the Revell 1:180 USS Lionfish Submarine model. d. Fill in the blank: CONV. The sides MODEL of the larger 1 rectangle are times the corresponding sides of the smaller rectangle. REAL 180 12. A map legend shows that a length of 1 1 / 4 inches on the map corresponds to a distance of CONVERSION 500 miles. The distance MAP between 5/4 inches Oakland, California 4 5 /16 inches and Chicago, Illinois is about 4 5 / 16 inches on the map. About how many miles apart REAL are the 500 cities? miles x miles CONVERSION a. The actual length of the submarine is 311.5 feet. What is the length of the model, in feet? MAP REAL 5/4 inches 500 miles MODEL 1 REAL 180 CONV. b. What is the length of the model in inches?
Free Pre-Algebra Lesson 37! page 12 Lesson 37: Scale and Proportion Homework 37A Answers Formulas for Circles and Spheres: C = 2!r A =!r 2 V = 4 3!r 3 1. Find the circumference and area of a circle with radius 12.5 miles. Round to the nearest tenth. ( ) " 78.5 miles ( ) 2 " 490.9 square miles C = 2! 12.5 miles A =! 12.5 miles 2. Find the volume of a sphere with radius 12.5 miles. Round to the nearest whole number. V = 4! ( 12.5 miles) 3 " 8181 cubic miles 3 3. Solve the equation. 0.7x! 0.5 = 1.95 0.7x! 0.5 = 1.95 0.7x! 0.5 + 0.5 = 1.95 + 0.5 0.7x = 2.45 0.7x / 0.7 = 2.45 / 0.7 x = 3.5 4. If a circle has circumference 8.75 inches, what is the radius, to the nearest tenth? 8.75 inches = 2!r 2!r = 8.75 2!r / (2!) = 8.75 / (2!) r " 1.4 inches 5. Use the table to make the requested comparisons. 1965 2005 U.S. Population 195 million 296 million Number of Smokers 48 million 50 million a. Find the rate comparing smokers to total population for 1965 and 2005. Write as a decimal rounded to the nearest hundredth. 1965: 48/195 is about 0.25 2005: 50/296 is about 0.17 6. Change the numbers to scientific notation. a. 78,999,000,000,000 7.8999 x 10 13 b. 715,000,000,000,000,000 7.15 x 10 17 b. Write one or more sentences comparing the data. Although the number of smokers has increased since 1965, the total population has increased even more. Smokers are a smaller ratio of total population in 2005 than they were in 1965. 8. a. Write the number in standard form. They were charging $2.8 million for that house. $2,800,000 7. Change to standard notation. a. 2.13 x 10 14 213,000,000,000,000 b. 2.7 x 10 8 270,000,000 b. Write with a decimal point and place value name. This house was only $1,400,000. $1.4 million
Free Pre-Algebra Lesson 37! page 13 9. The rectangles are similar. Find the width of the smaller rectangle. 3.75W = (1.5)(1.25) 3.75W = 1.875 3.75W / 3.75 = 1.875 / 3.75 W = 0.5 The width is 0.5 feet. 11. Pictured below is the Revell 1:180 USS Lionfish Submarine model. a. The actual length of the submarine is 311.5 feet. What is the length of the model, in feet? 180x = 311.5 180x / 180 = 311.5 / 180 x! 1.73 feet L 3.75 1.5 W 1.25 W CONV. MODEL 1 x feet REAL 180 311.5 feet 10. a. What is the ratio of length to width for each of the rectangles in #9? b. Fill in the blank: 3.75 1.25 = 3 1.5 0.5 = 3 The length is 3 times the width in both rectangles. c. What is the ratio of the sides of the large rectangle to the corresponding sides of the small rectangle? d. Fill in the blank: 3.75 1.5 = 2.5 1.25 0.5 = 2.5 The sides of the larger rectangle are 2.5 times the corresponding sides of the smaller rectangle. 12. A map legend shows that a length of 1 1 / 4 inches on the map corresponds to a distance of 500 miles. The distance between Oakland, California and Chicago, Illinois is about 4 5 / 16 inches on the map. About how many miles apart are the cities? 5 4 x =! 69 $ " # 16% & 500 5 4 x = 8625 4 x = 1725 ( ) = 8625 CONVERSION MAP 5/4 inches 4 5 /16 inches REAL 500 miles x miles 4 4 5 5 4 x = 8625 4 4 5 b. What is the length of the model in inches? 1.73 feet 12 inches = 20.76 inches 1 1 foot Oakland and Chicago are about 1,725 miles apart.
Free Pre-Algebra Lesson 37: Scale and Proportion Homework 37B Lesson 37! page 14a Name Formulas for Circles and Spheres: C = 2!r A =!r 2 V = 4 3!r 3 1. Find the circumference and area of a circle with radius 7.3 m. Round to the nearest tenth. 2. Find the volume of a sphere with radius 7.3 m. Round to the nearest whole number. 3. Solve the equation. 1.7x + 8.1= 16.6 4. If a circle has circumference 45 inches, what is the radius, to the nearest tenth? 5. Use the table to make the requested comparisons. Median Home Prices by state, Adjusted for inflation to equivalent 2000 dollars. 1980 2000 CALIFORNIA $249,800 $211,500 COLORADO $105,700 $166,600 http://www.census.gov/hhes/www/housing/census/historic/val ues.html Find the difference in home prices from 1980 to 2000 in California and in Colorado, and compare in one or more sentences. 6. Change the numbers to scientific notation. a. 35,087,000,000 b. 882,993,000,000,000 7. Change to standard notation. a. 4.09 x 10 10 8. a. Write the number in standard form. The cost of the cleanup is estimated at $7.2 billion. b. Write with a decimal point and place value name. There are over 9,500,000,000,000 reasons why I shouldn t do that. b. 1.15 x 10 8
Free Pre-Algebra Lesson 37! page 15a 9. The rectangles are similar. Find the width of the smaller rectangle. 10. a. What is the ratio of length to width for each of the rectangles in #9? L W b. Fill in the blank: The length is times the width in both rectangles. c. What is the ratio of the sides of the large rectangle to the corresponding sides of the small rectangle? d. Fill in the blank: The sides of the larger rectangle are about times the corresponding sides of the smaller rectangle. 11. Pictured below is the Revell 1:535 USS Missouri Battleship model. 12. A map legend shows that a length of 3 / 4 inches on the map corresponds to a distance of 10 miles. The distance between Pleasant Hill, California and Davis, CA is about 3 3 / 4 inches on the map. About how many miles apart are the cities? CONVERSION MAP a. The actual length of the battleship is 887.2 feet. What is the length of the model, in feet? REAL MODEL REAL CONVERSION b. What is the length of the model in inches? SIDE A 1.3 2.86 SIDE B 2.1 4.62