Proportional Relationships

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1 Proportional Relationships A Ratios, Rates, and Proportions - Ratios - Rates - Identifying and Writing Proportions - Solving Proportions - Customary Measurements LAB Generate Formulas to Convert Units B Proportions in Geometry LAB Make Similar Figures - Similar Figures and Proportions - Using Similar Figures - Scale Drawings and Scale Models KEYWORD: MSCA Ch You can use ratios and proportions to describe the relationship between this binoculars-shaped building and an actual pair of binoculars. Main Street Venice, California Chapter

2 Vocabulary Choose the best term from the list to complete each sentence.. A(n)? is a number in the form a, where b. b. A closed figure with three sides is called a(n)?.. Two fractions are? if they represent the same number.. One way to compare two fractions is to first find a(n)?. common denominator equivalent fraction quadrilateral triangle Complete these exercises to review skills you will need for this chapter. Write Equivalent Fractions Find two fractions that are equivalent to each fraction Compare Fractions Compare. Write or Solve Multiplication Equations Solve each equation.. x. t. y. m. c. f. n. y Multiply Fractions Solve. Write each answer in simplest form Proportional Relationships

3 The information below unpacks the standards. The Academic Vocabulary is highlighted and defined to help you understand the language of the standards. Refer to the lessons listed after each standard for help with the math terms and phrases. The Chapter Concept shows how the standard is applied in this chapter. California Standard NS. Interpret and use ratios in different contexts (e.g., batting averages, miles per hour) to show the relative sizes of two quantities, using appropriate notations ( ab, a to b, a:b). (Lessons -, -, -) NS. Use proportions to solve problems (e.g., determine the value of N if N, 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. Academic Vocabulary Chapter Concept interpret to understand and explain the meaning of You understand and can explain the meaning of ratios. You write context in this case, a real-world situation ratios in different forms. relative sizes sizes that are compared to each other Example: A recipe calls for Example: A distance of mile is large relative to a oranges and lemons. distance of inch. The ratio of oranges to lemons notation a way of writing or representing something can be written as, to, Example: The ratio to can also be written using or :. the notation :. polygon a closed plane figure formed by line segments that meet only at their endpoints Examples: triangles, squares, rectangles known polygon a polygon whose side lengths and angle measures are known You solve proportions by finding the value of a variable that makes the proportion true. Example: N N N N You use proportions to find unknown side lengths in similar figures. (Lessons -, -, -, -, -) (Lab -) AF. Demonstrate an understanding that rate is a measure of one quantity per unit value of another quantity. (Lessons -, -, -) unit value unit of a unit of measurement Examples: The unit value of distances measured in feet is foot. The unit value of time measured in seconds is second. You determine rates and use rates to solve problems. Example: A printer prints a -page report in minutes. pages pages minutes minute The printer prints at a rate of pages per minute. AF. Solve problems involving rates, average speed, distance, and time. (Lessons -, -) average speed the distance traveled by an object divided by the time taken to travel that distance Example: A car travels miles in hours. miles miles hours hour The average speed is mi/h. Standard AF. is also covered in this chapter. To see this standard unpacked, go to Chapter, p.. Chapter

4 Writing Strategy: Use Your Own Words Using your own words to explain a concept can help you understand the concept. For example, learning how to solve equations might seem difficult if the textbook does not explain solving equations in the same way that you would. As you work through each lesson: Identify the important ideas from the explanation in the book. Use your own words to explain these ideas. California Standards English-Language Arts Reading.. What Sara Reads An equation is a mathematical statement that two expressions are equal in value. When an equation contains a variable, a value of the variable that makes the statement true is called a solution of the equation. If a variable is multiplied by a number, you can often use division to isolate the variable. Divide both sides of the equation by the number. What Sara Writes An equation has an equal sign to show that two expressions are equal to each other. The solution of an equation that has a variable in it is a number that the variable is equal to. When the variable is multiplied by a number, you can undo the multiplication and get the variable alone by dividing both sides of the equation by the number. Try This Rewrite each sentence in your own words.. When solving equations containing addition and integers, isolate the variable by adding opposites.. When you solve equations that have one operation, you use an inverse operation to isolate the variable. Proportional Relationships

5 - Ratios California Standards NS. Interpret and use ratios in different contexts (e.g., batting averages, miles per hour) to show the relative sizes of two quantities, using appropriate notations ( a, b a to b, a:b). Vocabulary ratio Who uses this? Basketball players can use ratios to compare the number of baskets they make to the number they attempt. In basketball practice, Kathlene made baskets in attempts. She compared the number of baskets she made to the total number of attempts she made by using the ratio. A ratio is a comparison of two numbers or quantities. Kathlene can write her ratio of baskets made to attempts in three different ways. to EXAMPLE Writing Ratios A basket of fruit contains apples, bananas, and oranges. Write each ratio in all three forms. bananas to apples number of bananas number of apples There are bananas and apples. The ratio of bananas to apples can be written as, to, or :. bananas and apples to oranges number of bananas and apples number of oranges The ratio of bananas and apples to oranges can be written, to, or :. as oranges to total pieces of fruit number of oranges number of total pieces of fruit The ratio of oranges to total pieces of fruit can be written as, to, or :. Chapter Proportional Relationships

6 Sometimes a ratio can be simplified. To simplify a ratio, first write it in fraction form and then simplify the fraction. EXAMPLE Writing Ratios in Simplest Form At Franklin Middle School, there are students in the sixth grade and sixth-grade teachers. Write the ratio of students to teachers in simplest form. A fraction is in simplest form when the GCD of the numerator and denominator is. students teachers See Lesson -, p. Write the ratio as a fraction. Simplify. For every students, there is teacher. The ratio of students to teachers is to. To compare ratios, write them as fractions with common denominators. Then compare the numerators. EXAMPLE Reasoning Comparing Ratios Tell whether the wallet size photo or the portrait size photo has the greater ratio of width to length. Width (in.) Length (in.). Personal Desk Portrait Wallet Wallet: width (in.). length (in.) Portrait: width (in.) length (in.) Write the ratios as fractions with common denominators. Because. and the denominators are the same, the portrait size photo has the greater ratio of width to length. Think and Discuss. Explain why you think the ratio in Example B is not written as a mixed number.. Tell how to simplify a ratio.. Explain how to compare two ratios. - Ratios

7 - Exercises California Standards Practice NS. KEYWORD: MSCA - KEYWORD: MSCA Parent See Example See Example GUIDED PRACTICE Sun-Li has blue marbles, red marbles, and white marbles. Write each ratio in all three forms.. blue marbles to red marbles. red marbles to total marbles. In a -gallon aquarium, there are neon tetras and zebra danio fish. Write the ratio of neon tetras to zebra danio fish in simplest form. See Example. Tell whose DVD collection has the greater ratio of comedy movies to adventure movies. Joseph Yolanda Comedy Adventure See Example INDEPENDENT PRACTICE A soccer league has sixth-graders, seventh-graders, and eighth-graders. Write each ratio in all three forms.. th-graders to th-graders. th-graders to total students. th-graders to th-graders. th- and th-graders to th-graders See Example. Thirty-six people auditioned for a play, and people got roles. Write the ratio in simplest form of the number of people who auditioned to the number of people who got roles. See Example. Tell whose bag of nut mix has the greater ratio of peanuts to total nuts. Dina Don Almonds Cashews Peanuts Extra Practice See page EP. PRACTICE AND PROBLEM SOLVING Use the table for Exercises.. Tell whether group or group has the Opinions on Open-Campus Lunch greater ratio of the number of people For Group Group Group for an open-campus Against lunch to the number of No Opinion people with no opinion.. Which group has the least ratio of the number of people against an open-campus lunch to the total number of survey responses?. Estimation For each group, is the ratio of the number of people for an opencampus lunch to the number of people against it less than or greater than? Chapter Proportional Relationships

8 Science The pressure of water at different depths can be measured in atmospheres, or atm. The water pressure on a scuba diver increases as the diver descends below the surface. Use the table for Exercises. Write each ratio in all three forms.. pressure at ft to pressure at surface. pressure at ft to pressure at surface. pressure at ft to pressure at surface. pressure at ft to pressure at ft. pressure at ft to pressure at ft Pressure Experienced by Diver Depth (ft) Pressure (atm). Tell whether the ratio of pressure at ft to pressure at ft is greater than or less than the ratio of pressure at ft to pressure at ft.. Challenge The ratio of the beginning pressure and the new pressure when a scuba diver goes from ft to ft is less than the ratio of pressures when the diver goes from the surface to ft. The ratio of pressures is even less when the diver goes from ft to ft. Explain why this is true. KEYWORD: MSCA Pressure NS., NS., NS., AF.. Multiple Choice Johnson Middle School has sixth-graders, seventh-graders, and eighth-graders. Which statement is NOT true? A B C D The ratio of sixth-graders to seventh-graders is to. The ratio of eighth-graders to seventh-graders is :. The ratio of sixth-graders to students in all three grades is :. The ratio of eighth-graders to students in all three grades is to.. Short Response A pancake recipe calls for cups of pancake mix for every cups of milk. A biscuit recipe calls for cups of biscuit mix for every cup of milk. Which recipe has a greater ratio of mix to milk? Explain. Solve. (Lesson -).. x...y.. v... How many -yard pieces can be cut from yards of string? (Lesson -) - Ratios

9 - Rates California Standards AF. Demonstrate an understanding that rate is a measure of one quantity per unit value of another quantity. Also covered: NS., AF. Vocabulary rate unit rate Why learn this? You can use rates to determine a driver s average speed. (See Example.) The Lawsons are driving miles to a campground. They would like to reach the campground in hours of driving. What should their average speed be in miles per hour? Recall that a ratio is a comparison of two numbers or quantities. A rate is a special type of ratio that compares two quantities measured in different units. In order to answer the question above, you need to find the family s rate of travel. Their rate is miles. hours A unit rate is a rate whose denominator is. To change a rate to a unit rate, divide both the numerator and denominator by the denominator. Dividing the numerator and denominator of a rate by the same number does not change the value of the rate. EXAMPLE Finding Unit Rates During exercise, Sonia s heart beats times in minutes. How many times does it beat per minute? beats Write a rate that compares heart beats and time. minutes beats Divide the numerator and denominator by to minutes get an equivalent rate. beats Simplify. minute Sonia s heart beats times per minute. Batting averages are usually written as decimals without a zero to the left of the decimal point. A batting average compares number of hits to number of times at bat. A baseball player has hits in times at bat. What is the player s batting average? hits Write a rate that compares hits and at bats. at bats hits Divide the numerator and denominator by at bats to get an equivalent rate.. hits Simplify. at bat The player s batting average is.. Chapter Proportional Relationships

10 An average rate of speed is the ratio of distance traveled to time. The ratio is a rate because the units in the numerator and denominator are different. Speed is usually expressed as a unit rate. EXAMPLE Finding Average Speed The Lawsons want to drive the miles to a campground in hours. What should their average speed be in miles per hour? miles Write the rate. hours mile hour s s miles hour Divide the numerator and denominator by to get an equivalent rate. Their average speed should be miles per hour. A unit price is the price of one unit of an item. The unit used depends on how the item is sold. The table shows some examples. Type of Item Liquid Solid Any item Examples of Units Fluid ounces, quarts, gallons, liters Ounces, pounds, grams, kilograms Bottle, container, carton EXAMPLE Consumer Math Application The Lawsons stop at a roadside farmers market. The market offers lemonade in three sizes. Which size lemonade has the lowest price per fluid ounce? Divide the price by the number of fluid ounces (fl oz) to find each unit price. Size fl oz fl oz fl oz Price $. $. $. $. $. $. fl oz fl oz $. $. fl oz fl oz $. fl oz fl oz Since $. $., the fl oz lemonade has the lowest price per fluid ounce. Think and Discuss. Explain how you can tell whether an expression represents a unit rate.. Suppose a store offers cereal with a price of $. per box. Another store offers cereal with a price of $. per box. Before determining which is the better buy, what variables must you consider? - Rates

11 - Exercises California Standards Practice NS., AF., AF. KEYWORD: MSCA - KEYWORD: MSCA Parent See Example GUIDED PRACTICE. A faucet leaks milliliters of water in minutes. How many milliliters of water does the faucet leak per minute?. A recipe for muffins calls for grams of oat flakes. How many grams of oat flakes are needed for each muffin? See Example See Example See Example. An airliner makes a,-mile flight in hours. What is the airliner s average rate of speed in miles per hour?. Consumer Math During a car trip, the Webers buy gasoline at three different stations. At the first station, they pay $. for gallons of gas. At the second, they pay $. for gallons. At the third, they pay $. for gallons. Which station offers the lowest price per gallon? INDEPENDENT PRACTICE. An after-school job pays $. for hours of work. How much money does the job pay per hour?. It took Samantha minutes to cook a turkey. If the turkey weighed pounds, how many minutes per pound did it take to cook the turkey? See Example See Example Extra Practice See page EP.. Sports The first Indianapolis auto race took place in. The winning car covered the miles in. hours. What was the winning car s average rate of speed in miles per hour?. Consumer Math A supermarket sells orange juice in three sizes. The fl oz container costs $., the fl oz container costs $., and the fl oz container costs $.. Which size orange juice has the lowest price per fluid ounce? PRACTICE AND PROBLEM SOLVING Find each unit rate. Round to the nearest hundredth, if necessary.. runs in games. $, for, ft. $, in mo. songs on CDs. mi on gal. words in min. hr for $. lb for $.. mi in trips. m in s.. mi in min., km in. hr. In Grant Middle School, each class has an equal number of students. There are classes and a total of, students. Write a rate that describes the distribution of students in the classes at Grant. What is the unit rate?. Estimation Use estimation to determine which is the better buy: minutes of phone time for $. or minutes for $.. Chapter Proportional Relationships

12 Find each unit price. Then decide which is the better buy.. $. or $.. $. oz oz y d or $. $.. yd. m or $.. m. Sports In the Summer Olympics, Justin Gatlin won the -meter race in. seconds. Shawn Crawford won the -meter race in. seconds. Which runner ran at a faster average rate?. Social Studies The population density of a country is the average number of people per unit of area. Write the population densities of the countries in the table as unit rates. Round your answers to the nearest person per square mile. Then rank the countries from least to greatest population density. Country Population Land Area (mi ) France,,, Germany,,, Poland,,,. Reasoning A store sells paper towels in packs of and packs of. Use this information to write a problem about comparing unit rates.. Write About It Michael Jordan has the highest scoring average in NBA history. He played in, games and scored, points. Explain how to find a unit rate to describe his scoring average. What is the unit rate?. Challenge Mike fills his car s gas tank with gallons of regular gas at $. per gallon. His car averages miles per gallon. Serena fills her car s tank with gallons of premium gas at $. per gallon. Her car averages miles per gallon. Compare the drivers unit costs of driving one mile. NS., NS., NS., AF., AF.. Multiple Choice What is the unit price of a -ounce box of cereal that sells for $.? A $. B $. C $. D $.. Short Response A carpenter needs minutes to make cuts in a board. If each cut takes the same length of time, at what rate is the carpenter cutting?. Julita s walking stick is feet long, and Toni s walking stick is feet long. Whose walking stick is longer and by how much? (Lesson -) Compare. Write,, or. (Lesson -). ml L.. mg. g. cm. mm - Rates

13 - Identifying and Writing Proportions California Standards NS. Interpret and use ratios in different contexts (e.g., batting averages, miles per hour) to show the relative sizes of two quantities, using appropriate notations ( a, b a to b, a:b). Vocabulary equivalent ratios proportion Why learn this? You can determine whether two ratios of length to width are equivalent. Students are measuring the width w and the length of their heads. The ratio of to w is inches to inches for Jean and centimeters to centimeters for Pat. Calipers have adjustable arms that are used to measure the thickness of objects. If two ratios are equivalent, they are said to be proportional to each other, or in proportion. and These ratios can be written as the fractions simplify to, they are equivalent. Equivalent ratios name the same comparison.. Since both ratios are ratios that An equation stating that two ratios are equivalent is called a proportion. The equation, or proportion, below states that the ratios and are equivalent. EXAMPLE Comparing Ratios in Simplest Form Determine whether the ratios are proportional., is already in simplest form. Simplify Since, the ratios are proportional., Simplify. Simplify. Since, the ratios are not proportional. Chapter Proportional Relationships

14 EXAMPLE Comparing Ratios Using a Common Denominator Use the data in the table to determine whether the ratios of oats to water are proportional for both servings of oatmeal. Write the ratios of oats to water for servings and for servings. Servings of Cups of Cups of Oatmeal Oats Water Ratio of oats to water, servings: Write the ratio as a fraction. Ratio of oats to water, servings: Write the ratios with a common denominator, such as. Write the ratio as a fraction. Since both ratios are equal to, they are proportional. You can find an equivalent ratio by multiplying or dividing the numerator and the denominator of a ratio by the same number. EXAMPLE Math Builders For more on proportions, see the Proportion Builder on page MB. Finding Equivalent Ratios and Writing Proportions Find a ratio equivalent to each ratio. Then use the ratios to write a proportion. Multiply both the numerator and denominator by any number, such as. Write a proportion. Divide both the numerator and denominator by a common factor, such as. Write a proportion. Think and Discuss. Explain why the ratios in Example B are not proportional.. Describe what it means for ratios to be proportional.. Give an example of a proportion. Then tell how you know it is a proportion. - Identifying and Writing Proportions

15 - Exercises California Standards Practice NS., AF. KEYWORD: MSCA - See Example See Example See Example See Example See Example See Example GUIDED PRACTICE INDEPENDENT PRACTICE KEYWORD: MSCA Parent Determine whether the ratios are proportional..,.,.,.,.,.,.,., Find a ratio equivalent to each ratio. Then use the ratios to write a proportion..... Determine whether the ratios are proportional..,.,.,.,.,.,.,., Find a ratio equivalent to each ratio. Then use the ratios to write a proportion Extra Practice See page EP. PRACTICE AND PROBLEM SOLVING Complete each table of equivalent ratios.. angelfish. squares tiger fish circles Find two ratios equivalent to each given ratio.. to. :.. :. to.. :. to. Ecology If you recycle one aluminum can, you save enough energy to run a TV for four hours. a. Write the ratio of cans to hours. b. Marti s class recycled enough aluminum cans to run a TV for, hours. Did the class recycle cans? Justify your answer using equivalent ratios.. Reasoning The ratio of girls to boys riding a bus is :. If the driver drops off the same number of girls as boys at the next stop, does the ratio of girls to boys remain :? Explain. Chapter Proportional Relationships

16 . Critical Thinking Write all possible proportions using only the numbers,, and.. School Last year in Kerry s school, the ratio of students to teachers was :. Write an equivalent ratio to show how many students and teachers there could have been at Kerry s school.. Science Students in a Number of Number biology class surveyed four Pond Salamanders of Frogs ponds to determine whether salamanders and frogs were inhabiting the area. a. What was the ratio of salamanders to frogs in Cypress Pond? Cypress Pond Mill Pond Clear Pond Gill Pond b. In which two ponds was the ratio of salamanders to frogs the same?. Marcus earned $ for hours of work. Phillip earned $ for hours of work. Are these pay rates proportional? Explain.. What s the Error? A student wrote the proportion student do wrong?. What did the. Write About It Explain two different ways to determine if two ratios are proportional.. Challenge A skydiver jumps out of an airplane. After. second, she has fallen feet. After. seconds, she has fallen feet. Is the rate (in feet per second) at which she falls the first feet proportional to the rate at which she falls the next feet? Explain. NS., NS., AF.. Multiple Choice Which ratio is NOT equivalent to? A B C D. Multiple Choice Which ratio can form a proportion with? A B C D Evaluate a b for each set of values. (Lesson -). a, b. a, b. a, b. A file drawer holds binders and, sheets of paper. Write the ratio of binders to sheets of paper in simplest form. (Lesson -) - Identifying and Writing Proportions

17 - Solving Proportions California Standards NS. Use proportions to solve problems (e.g., determine the value of N if N, find the length of a side of a polygon similar to a known polygon). Use crossmultiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse. Also covered: AF., AF. Vocabulary cross product A number multiplied by its multiplicative inverse is equal to. See Lesson -, p.. Who uses this? Bicyclists can solve proportions to find out how long it will take them to finish a race. (See Example.) For two ratios, the product of the numerator in one ratio and the denominator in the other is a cross product. If two ratios form a proportion, then the cross products are equal. a b c d Write a proportion, where a, b, c, and d are not equal to. d c, the multiplicative d c inverse of c. d a b d Simplify each side. c c d d c = a d b c The fraction a b d is equal to, so the c numerator must equal the denominator a b d c Multiply each side by d c Words In a proportion, the cross-products are equal. CROSS PRODUCT RULE Numbers Algebra If a b c d, where b and d, then a d b c. You can use the cross product rule to solve proportions with variables. EXAMPLE Solving Proportions Using Cross Products Use cross products to solve the proportion p. p p The cross products are equal. p Multiply. p Divide each side by. p Chapter Proportional Relationships

18 It is important to set up proportions correctly. Each ratio must compare corresponding quantities in the same order. Suppose a boat travels miles in hours and miles in x hours at the same speed. Either of these proportions could represent this situation. m h i r mi x hr mi hr mi x hr Trip Trip Trip Trip EXAMPLE California Sports The Tour of California is an approximately -mile bicycle race that takes place over an -day period. In, the winning time was hours, minutes, seconds. Sports Application The graph shows the time and distance Sunee rode her bike during training. She plans to enter a -mile race. If Sunee rides at the same rate she rode during training, how long will it take her to finish the race? The labeled point on the graph shows that Sunee rode miles in hours. Let t represent the time in hours it will take Sunee to finish the race. Method Set up a proportion in which each ratio compares distance to the time needed to ride that distance. m h i r mi t hr t The cross products are equal. t Multiply. t Divide each side by. t Method Set up a proportion in which one ratio compares distance and one ratio compares time. mi hr Training mi t hr Race t The cross products are equal. t Distance Time Multiply. t Divide each side by. t Both methods show that it will take Sunee hours to finish the race if she rides at her training rate. Distance (miles) Practice Ride.. Time (hours) (, ). - Solving Proportions

19 EXAMPLE PROBLEM SOLVING APPLICATION Density is the ratio of a substance s mass to its volume. The density of ice is. g/ml. What is the mass of ml of ice? Reasoning Understand the Problem Rewrite the question as a statement. Find the mass, in grams, of ml of ice. List the important information: mass (g) density volume (ml). g ml density of ice Make a Plan. Set up a proportion using the given information. Let m represent the mass of ml of ice.. g m ml ml mass volume Solve Solve the proportion.. m Write the proportion. m. The cross products are equal. m. Multiply. The mass of ml of ice is. g. Look Back Since the density of ice is. g/ml, each milliliter of ice has a mass of a little less than g. So ml of ice should have a mass of a little less than g. Since. is a little less than, the answer is reasonable. Think and Discuss. Explain how the term cross product can help you remember how to solve a proportion.. Describe the error in these steps: x ; x ; x.. Show how to use cross products to decide whether the ratios : and : are proportional. Chapter Proportional Relationships

20 - Exercises California Standards Practice NS., AF., AF. KEYWORD: MSCA MS - - See Example GUIDED PRACTICE Use cross products to solve each proportion.. x.. p. m KEYWORD: MSCA MS Parent t.. See Example See Example. The graph shows the time and distance that a horse ran around a track. If the horse runs at that same speed, how long will it take the horse to run. miles?. A stack of, one-dollar bills weighs pounds. How much does a stack of, one-dollar bills weigh? Distance (mi)... Horse s Speed (,.) Time (s) See Example INDEPENDENT PRACTICE Use cross products to solve each proportion. x.. r. h. v. t x.. s. n See Example See Example. The graph shows the relationship between the weight and cost of peaches at a grocery store. At this rate, how much would. pounds of peaches cost?. There are. ounces of soup in a can. This is equivalent to grams. If Jenna has ounces of soup, how many grams does she have? Round to the nearest whole gram. Cost ($) Cost of Peaches (., )... Weight (lb).. Extra Practice See page EP. PRACTICE AND PROBLEM SOLVING Solve each proportion. Then find another equivalent ratio.. h x.. t... w. y.. x. m.. q. Sandra drove. miles in hours at a constant speed. Use a proportion to find how long it would take her to drive. miles at the same speed.. Multi-Step In June, a camp has campers and counselors. In July, campers leave and new campers arrive. How many counselors does the camp need in July to keep an equivalent ratio of campers to counselors? - Solving Proportions

21 . Science On Monday a marine biologist took a random sample of fish from a pond and tagged them. On Tuesday she took a new sample of fish. Among them were fish that had been tagged on Monday. a. What comparison does the ratio represent? b. What is the ratio of the number of fish tagged on Monday to n, the estimated total number of fish in the pond? c. Use a proportion to estimate the number of fish in the pond.. Chemistry The table shows the type and number of atoms in one molecule of citric acid. Use a proportion to find the number of oxygen atoms in molecules of citric acid. Composition of Citric Acid Type of Atom Number of Atoms Carbon Hydrogen Oxygen. Earth Science You can find your distance from a thunderstorm by counting the number of seconds between a lightning flash and the thunder. For example, if the time difference is s, then the storm is km away. How far away is a storm if the time difference is s?. Reasoning Use a multiplicative inverse to show that the cross product rule is true for the proportion r s.. What s the Question? There are grams of protein in ounces of sautéed fish. If the answer is ounces, what is the question?. Write About It Give an example from your own life that can be described using a ratio. Then tell how a proportion can give you additional information.. Challenge Determine whether the proportion a b c d is equivalent to the proportion a d b c, where b and d. Use the cross product rule to explain your answer. NS., NS., AF., AF.. Multiple Choice A jet traveled, miles in hours. At this rate, how long would it take the jet to travel miles? A hr B hr C hr D hr x. Gridded Response What value of x makes the proportion true? Find the greatest common divisor (GCD). (Lesson -).,.,,.,, Find each unit rate. (Lesson -). miles in hours. books in weeks. $ in hours Chapter Proportional Relationships

22 - Customary Measurements California Standards AF. Convert one unit of measurement to another (e.g., from feet to miles, from centimeters to inches). Also covered: NS. Why learn this? You can use customary measurements to describe lengths, weights, and capacity. Just fluid ounces of a king cobra s venom is enough to kill a -ton elephant. You can use the following benchmarks to help you understand fluid ounces, tons, and other customary units of measure. Customary Unit Benchmark Length Inch (in.) Length of a small paper clip Foot (ft) Mile (mi) Length of a standard sheet of paper Length of about football fields Weight Ounce (oz) Weight of a slice of bread Pound (lb) Ton Weight of apples Weight of a buffalo Capacity Fluid ounce (fl oz) Amount of water in tablespoons Cup (c) Gallon (gal) Capacity of a standard measuring cup Capacity of a large milk jug EXAMPLE Choosing the Appropriate Customary Unit Choose the most appropriate customary unit for each measurement. Justify your answer. the length of a rug Feet the length of a rug is similar to the length of several sheets of paper. the weight of a magazine Ounces the weight of a magazine is similar to the weight of several slices of bread. the capacity of an aquarium Gallons the capacity of an aquarium is similar to the capacity of several large milk jugs. - Customary Measurements

23 The following table shows some common equivalent customary units. You can use equivalent measures to convert units of measure. Length Weight inches (in.) foot (ft) ounces (oz) pound (lb) feet yard (yd), pounds ton, feet mile (mi) EXAMPLE Capacity fluid ounces (fl oz) cups pints quarts cup (c) pint (pt) quart (qt) gallon (gal) Converting Customary Units Convert c to fluid ounces. Method : Use a proportion. Write a proportion using a ratio of equivalent measures. fluid ounces cups x x x Method : Multiply by. Multiply by a ratio equal to, and cancel the units. c fl oz c c fl oz fl oz Nineteen cups is equal to fluid ounces. EXAMPLE Reasoning Converting Between Metric and Customary Units One inch is about. centimeters. A bookmark has a length of centimeters. What is the length of the bookmark in inches, rounded to the nearest inch? inches centimeters x.. x.x.x.. Write a proportion using in. 艐. cm. The cross products are equal. Multiply. Divide each side by.. x Round to the nearest whole number. The bookmark is about inches long. Think and Discuss. Describe an object that you would weigh in ounces.. Explain how to convert yards to feet and feet to yards. Chapter Proportional Relationships

24 - Exercises California Standards Practice NS., AF. KEYWORD: MSCA - KEYWORD: MSCA Parent See Example GUIDED PRACTICE Choose the most appropriate customary unit for each measurement. Justify your answer.. the width of a sidewalk. the amount of water in a pool. the weight of a truck. the distance across Lake Erie See Example Convert each measure.. gal to quarts. mi to feet. oz to pounds.. c to fluid ounces See Example See Example. One gallon is about. liters. A car has a -liter gas tank. What is the capacity of the tank in gallons, rounded to the nearest tenth of a gallon? INDEPENDENT PRACTICE Choose the most appropriate customary unit for each measurement. Justify your answer.. the weight of a watermelon. the wingspan of a sparrow. the capacity of a soup bowl. the height of an office building See Example Convert each measure.. pt to quarts., ft to miles.. tons to pounds. ft to inches See Example Extra Practice See page EP.. A -pound weight has a mass of about. kilogram. What is the mass in kilograms of a sculpture that weighs pounds? Round to the nearest tenth of a kilogram. PRACTICE AND PROBLEM SOLVING Compare. Write,,or.. yd ft. oz lb. in. ft. tons, lb. gal qt.. c fl oz., ft mi. pt c. gal c. Literature The novel Twenty Thousand Leagues Under the Sea was written by Jules Verne in. One league is approximately. miles. How many miles are in, leagues?. Earth Science One meter is about. feet. The average depth of the Pacific Ocean is, feet. How deep is this in meters, rounded to the nearest meter? - Customary Measurements

25 California Agriculture In, the Grand Champion pumpkin at California s Half Moon Bay Art and Pumpkin Festival weighed, pounds. Order each set of measures from least to greatest.. ft; yd; in.. qt; gal; pt; c. ton;, oz; lb.. mi;, ft;, yd. fl oz; c;. qt.. yd;. ft; in.. Agriculture In one year, the United States produced nearly million pounds of pumpkins. How many ounces were produced by the state with the lowest production shown in the table?. Multi-Step A marathon is a race that is miles yards long. What is the length of a marathon in yards? U.S. Pumpkin Production State Pumpkins (million pounds) California Illinois New York Pennsylvania. In, a,-gallon ice cream float was made in Atlanta, Georgia. How many -pint servings did the float contain?. Reasoning Explain why it makes sense to divide when you convert a measurement to a larger unit.. What s the Error? A student converted ft to inches as follows. What did the student do wrong? What is the correct answer? ft x in. ft. Write About It Explain how to convert. tons to ounces.. Challenge A dollar bill is. in. long. A radio station gives away a prize consisting of a mile-long string of dollar bills. What is the approximate value of the prize? NS., NS., NS., AF.. Multiple Choice Which measure is the same as qt? A pt B gal C c D fl oz. Multiple Choice One fluid ounce is about milliliters. A juice box holds milliliters. About how many fluid ounces does the box hold? A fl oz B fl oz C fl oz D fl oz. James used cup of white flour and cup of wheat flour for a muffin recipe. How many cups of flour did James use in all? (Lesson -) Determine whether the ratios are proportional. (Lesson -).,.,.,., Chapter Proportional Relationships

26 - Use with Lesson - Generate Formulas to Convert Units California Standards Practice Extension of AF. Convert one unit of measurement to another (e.g., from feet to miles, from centimeters to inches). KEYWORD: MSCA Lab Activity Publishers, editors, and graphic designers measure lengths in picas. Measure each of the following line segments to the nearest inch, and record your results in the table. Ratio of Length Length Picas to Segment (in.) (picas) Inches Think and Discuss. Make a conjecture about the relationship between picas and inches.. Use your conjecture to write a formula relating inches n to picas p.. How many picas wide is a sheet of paper that is in. wide? Try This Using inches for x-coordinates and picas for y-coordinates, write ordered pairs for the data in the table. Then plot the points and draw a graph.. What shape is the graph?. Use the graph to find the number of picas that is equal to inches.. Use the graph to find the number of inches that is equal to picas.. A designer is laying out a page in a magazine. The dimensions of a photo are picas by picas. She doubles the dimensions of the photo. What are the new dimensions of the photo in inches? - Hands-On Lab

27 Quiz for Lessons - Through - - Ratios A bouquet has red, pink, yellow, and white flowers. Write each ratio in all three forms.. pink flowers to yellow flowers. red flowers to total flowers. A concession stand sold strawberry, banana, grape, and orange fruit drinks during a game. Tell whether the ratio of strawberry to orange drinks or the ratio of banana to grape drinks is greater. - Rates. A -gallon jug is. pounds heavier when it is full of water than when it is empty. How much does the water weigh per gallon?. Shaunti drove miles in. hours. What was her average speed in miles per hour?. A grocery store sells a oz bag of raisins for $. and a oz bag of raisins for $.. Which size bag has the lowest price per ounce? - Identifying and Writing Proportions Determine whether the ratios are proportional..,.,.,., Find a ratio equivalent to each ratio. Then use the ratios to write a proportion Solving Proportions Use cross products to solve each proportion.. n.. t.. z. x. One human year is said to equal dog years. If Cliff s dog is. years old in human years, what is his dog s age in dog years?. If CDs take up inches of shelf space, how many CDs will fit on inches of shelf space? - Customary Measurements Convert each measure.. lb to ounces. qt to pints. mi to feet. fl oz to cups. ft to yards., lb to tons Chapter Proportional Relationships

28 Make a Plan California Standards MR. Note the method of deriving the solution and demonstrate a conceptual understanding of the derivation by solving similar problems. Also covered: NS., MR., MR. Choose a problem-solving strategy The following are strategies that you might choose to help you solve a problem: Make a table Draw a diagram Find a pattern Guess and check Make an organized list Solve a simpler problem Work backward Make a model Write an equation Tell which strategy from the list above you would use to solve each problem. Explain your choice. A recipe for blueberry muffins calls for cup of milk and. cups of blueberries. Ashley wants to make more muffins than the recipe yields. In Ashley s muffin batter, there are. cups of blueberries. If she is using the recipe as a guide, how many cups of milk will she need? The length of a rectangle is cm, and its width is cm less than its length. A larger rectangle with dimensions that are proportional to those of the first has a length of cm. What is the width of the larger rectangle? Each of four brothers gets an allowance for doing chores at home each week. The amount of money each boy receives depends on his age. Jeremy is years old, and he gets $.. His -year-old brother gets $., and his -year-old brother gets $.. Determine a possible relationship between the boys ages and their allowances, and use it to determine how much money Jeremy s -year-old brother gets. According to an article in a medical journal, a healthful diet should include a ratio of. servings of meat to servings of vegetables. If you eat servings of meat per week, how many servings of vegetables should you eat? Focus on Problem Solving

29 Make Similar Figures - Use with Lesson - KEYWORD: MSCA Lab Similar figures are figures that have the same shape but not necessarily the same size. You can make similar figures by increasing or decreasing both dimensions of a rectangle while keeping the ratios of the side lengths proportional. Modeling similar figures using square tiles can help you solve proportions. Activity A rectangle made of square tiles measures tiles long and tiles wide. What is the length of a similar rectangle whose width is tiles? California Standards NS. Use proportions to solve problems (e.g., determine the value of N if N, 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. Use tiles to make a rectangle. Add tiles to increase the width of the rectangle to tiles. Notice that there are now sets of tiles along the width of the rectangle because. The width of the new rectangle is three times greater than the width of the original rectangle. To keep the ratios of the side measures proportional, the length must also be three times greater than the length of the original rectangle. Add tiles to increase the length of the rectangle to tiles. The length of the similar rectangle is tiles. Chapter Proportional Relationships

30 To check your answer, you can use ratios.? Write ratios using the corresponding side lengths.? Simplify each ratio. Use square tiles to model similar figures with the given dimensions. Then find the missing dimension of each similar rectangle. a. The original rectangle is tiles wide by tiles long. The similar rectangle is tiles wide by x tiles long. b. The original rectangle is tiles wide by tiles long. The similar rectangle is x tiles wide by tiles long. c. The original rectangle is tiles wide by tiles long. The similar rectangle is tiles wide by x tiles long. Think and Discuss. Sarah wants to increase the size of her rectangular backyard patio. Why must she change both dimensions of the patio to create a patio similar to the original?. In a backyard, a plot of land that is yd yd is used to grow tomatoes. The homeowner wants to decrease this plot to yd yd. Will the new plot be similar to the original? Why or why not? Try This. A rectangle is feet long and feet wide. What is the width of a similar rectangle whose length is feet?. A rectangle is feet long and feet wide. What is the length of a similar rectangle whose width is feet? Use square tiles to model similar rectangles to solve each proportion.. x. h. y. t. m. p. r k. - Hands-On Lab

31 Similar Figures and Proportions - California Standards Preparation for NS. Use proportions to solve problems (e.g., determine the value N of N if, find the length of a side of a polygon similar to a known polygon). Use crossmultiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse. Why learn this? You can use proportions to determine whether two photographs are similar. (See Exercise.) Similar figures have the same shape but not necessarily the same size. The symbol means is similar to. Corresponding angles of two or more figures are in the same relative position. Corresponding sides of two or more figures are between corresponding angles. B Vocabulary similar corresponding sides corresponding angles A E Corresponding angles C D F Corresponding sides SIMILAR FIGURES Two figures are similar if the measures of their corresponding angles are equal. the ratios of the lengths of their corresponding sides are proportional. EXAMPLE Reasoning When naming similar figures, list the letters of the corresponding angles in the same order. In Example, 䉭DEF 䉭QRS. Determining Whether Two Triangles Are Similar S Tell whether the triangles are similar. The corresponding angles of the figures have equal measures. in. F D 苶E 苶 corresponds to Q 苶R 苶. in. E 苶F 苶 corresponds to R 苶S 苶. in. D 苶F 苶 corresponds to Q 苶S 苶. D E in. DE? EF? DF QR RS QS?? in. R in. Q Write ratios using the corresponding sides. Substitute the lengths of the sides. Simplify each ratio. Since the measures of the corresponding angles are equal and the ratios of the corresponding sides are equivalent, the triangles are similar. Chapter Proportional Relationships

32 For triangles, if the corresponding side lengths are all proportional, then the corresponding angles must have equal measures. For figures that have four or more sides, if the corresponding side lengths are all proportional, then the corresponding angles may or may not have equal angle measures. cm Q cm ABCD and QRST are similar. EXAMPLE X cm C cm cm W cm D S cm B cm cm T cm A R Z cm cm Y ABCD and WXYZ are not similar. Determining Whether Two Four-Sided Figures Are Similar Tell whether the figures are similar. Reasoning F ft E A side of a figure can be named by its endpoints, with a bar above. AB 苶 苶 Without the bar, the letters indicate the length of the side. ft G M ft N ft ft ft L ft O H ft The corresponding angles of the figures have equal measures. Write each set of corresponding sides as a ratio. EF LM GH NO 苶F苶 corresponds to 苶LM E 苶. 苶H G 苶 corresponds to 苶 NO 苶. FG MN EH LO F苶G 苶 corresponds to 苶 MN 苶. 苶H E 苶 corresponds to 苶LO 苶. Determine whether the ratios of the lengths of the corresponding sides are proportional. EF? FG? GH? EH LM MN NO LO??? Write ratios using the corresponding sides. Substitute the lengths of the sides. Write the ratios with common denominators. Since the measures of the corresponding angles are equal and the ratios of the corresponding sides are equivalent, EFGH LMNO. Think and Discuss. Identify the corresponding angles of 䉭JKL and 䉭UTS.. Explain whether all rectangles are similar. Give specific examples to justify your answer. - Similar Figures and Proportions

33 - Exercises California Standards Practice Preparation for NS. KEYWORD: MSCA - See Example GUIDED PRACTICE Tell whether the triangles are similar.. B. E m m m m A D m m F C KEYWORD: MSCA Parent V R in. in. in. in. Q T in. S in. W See Example Tell whether the figures are similar.. m m. m m m m m m cm cm. cm cm cm cm cm. cm See Example INDEPENDENT PRACTICE Tell whether the triangles are similar.. Q. K cm cm cm cm J L P R cm cm D in. L in. in. in. C J in. E in. K See Example Tell whether the figures are similar.. ft. ft ft ft ft ft ft ft m m m m m m m m PRACTICE AND PROBLEM SOLVING Extra Practice See page EP.. Tell whether the four-sided figures could be similar. Explain your answer. Chapter Proportional Relationships

34 . Kia wants similar prints in small and large sizes of a favorite photo. The photo lab sells prints in these sizes: in. in., in. in., in. in., in. in., and in. in. Which could she order to get similar prints? Tell whether the triangles are similar... m L A C m D F G ft ft ft N m m m ft m M ft E H B ft C The figure shows a ft by ft rectangle divided into four ft rectangular parts. Explain whether the rectangles in each pair are similar.. rectangle A and the original rectangle A ft ft B. rectangle C and rectangle B. the original rectangle and rectangle D C D ft Reasoning For Exercises, justify your answers using words or drawings.. Are all squares similar?. Are all -sided figures similar?. Are all rectangles similar?. Are all -sided figures similar?. Choose a Strategy What number gives the same result when multiplied by as it does when is added to it?. Write About It Tell how to decide whether two figures are similar.. Challenge Two triangles are similar. The ratio of the lengths of the corresponding sides is. If the length of one side of the larger triangle is feet, what is the length of the corresponding side of the smaller triangle? NS., NS., NS. NS.. Multiple Choice Luis wants to make a deck that is similar to one that is feet long and feet wide. If Luis s deck must be feet long, what must its width be? A feet feet. feet. feet B. Short Response If a real dollar bill measures. in. by. in. and a play dollar bill measures. in. by. in., is the play money similar to the real money? Explain your answer. Multiply. Write each answer in simplest form. (Lesson -).... Tell whether : or : is a greater ratio. (Lesson -) C D - Similar Figures and Proportions

35 - Using Similar Figures California Standards NS. Use proportions to solve problems (e.g., determine the value of N if N, find the length of a side of a polygon similar to a known polygon). Use crossmultiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse. Vocabulary indirect measurement Why learn this? You can use similar figures to determine the heights of totem poles and other tall objects. Native Americans of the Northwest, such as the Tlingit tribe of Alaska, carved totem poles out of tree trunks. These poles, sometimes painted with bright colors, could stand up to feet tall. Measuring the heights of tall objects, like some totem poles, cannot be done by using a ruler or yardstick. Instead, you can use indirect measurement. Indirect measurement is a method of using proportions to find an unknown length or distance in similar figures. EXAMPLE Finding Unknown Lengths in Similar Figures ABC JKL. Find the unknown length. cm B cm K A cm C x cm L cm J A B B JK C KL Write a proportion using corresponding sides. Substitute the lengths of the sides. x x Find the cross products. x Multiply. x Divide each side by. x KL is centimeters. Chapter Proportional Relationships

36 EXAMPLE Measurement Application A volleyball court is a rectangle that is similar in shape to an Olympic-sized pool. Find the width of the pool. m? m Let w the width of the pool. Write a proportion using corresponding side w lengths. w Find the cross products. w Multiply. w Divide each side by. w The pool is meters wide. m EXAMPLE Estimating with Indirect Measurement Estimate the height of the birdhouse in Chantal s yard, shown at right. h.. h Write a proportion. Use compatible numbers to estimate. h h Simplify. h Multiply each side by. h The birdhouse is about feet tall. ft. ft. ft Think and Discuss. Write another proportion that could be used to find the value of x in Example.. Name two objects that it would make sense to measure using indirect measurement. - Using Similar Figures

37 - California Standards Practice NS. Exercises KEYWORD: MSCA - KEYWORD: MSCA Parent GUIDED PRACTICE See Example 䉭XYZ 䉭PQR in each pair. Find the unknown lengths.. See Example R Y cm X cm P X. The rectangular gardens at right are similar in shape. How wide is the smaller garden? Z y m m P Q cm m m cm Z Q Y. a cm R m ft? ft ft See Example. A palm tree casts a shadow that is inches long. A surfboard casts a shadow that is inches long. Estimate the height of the palm tree. h in. in. in. INDEPENDENT PRACTICE See Example 䉭ABC 䉭DEF in each pair. Find the unknown lengths... B in. C in. A D. ft in. x D See Example E in. in. A. ft E b ft. ft F ft B. The two rectangular windows at right are similar. What is the height of the bigger window? F?. A cactus casts a shadow that is ft in. long. A gate nearby casts a shadow that is ft long. Estimate the height of the cactus. x ft in. ft Chapter Proportional Relationships ft ft. ft See Example C ft in.

38 Extra Practice See page EP. PRACTICE AND PROBLEM SOLVING. A building with a height of m casts a shadow that is m long while a taller building casts a m long shadow. What is the height of the taller building?. Two common envelope sizes are in. in. and in. in. Are these envelopes similar? Explain.. Art An art class is painting a mural composed of brightly colored geometric shapes. The class has decided that all the right triangles in the design will be similar to the right triangle that will be painted fire red. Find the measures of the right triangles in the table. Round your answers to the nearest tenth. Triangle Color Length (in.) Height (in.) Fire Red Blazing Orange Grape Purple Dynamite Blue. Reasoning Write a problem that can be solved using indirect measurement.. Write About It Assume you know the side lengths of one triangle and the length of one side of a second similar triangle. Explain how to use the properties of similar figures to find the unknown lengths in the second triangle.. Challenge ABE ACD. What is the value of y in the diagram? y D (, ) E (, y) A B (, ) C (, ) O x NS., AF., AF.. Multiple Choice Find the unknown length in the similar figures. A B cm C cm cm D cm cm x. cm cm. Gridded Response A building casts a -foot shadow. A -foot man standing next to the building casts a.-foot shadow. What is the height, in feet, of the building? Write each phrase as an algebraic expression. (Lesson -). the product of and y. less than a number. divided by z Convert each measure. (Lesson -). feet to inches. ounces to pounds. quarts to cups - Using Similar Figures

39 Scale Drawings and Scale Models - California Standards NS. Use proportions to solve problems (e.g., determine N the value of N if, find the length of a side of a polygon similar to a known polygon). Use crossmultiplication as a method for solving such problems, understanding it as the multiplication of both sides of an equation by a multiplicative inverse. Vocabulary scale model scale factor scale scale drawing Who uses this? Model builders use scale factors to create realistic models. This HO gauge model train is a scale model of a historic train. A scale model is a proportional threedimensional model of an object. Its dimensions are related to the dimensions of the actual object by a ratio called the scale factor. The HO scale factor is. This means that each dimension of the model is of the corresponding dimension of the actual train. 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 drawing is a proportional two-dimensional drawing of an object. Both scale drawings and scale models can be smaller or larger than the objects they represent. EXAMPLE Finding a Scale Factor Identify the scale factor. Reasoning Race Car Model Length (in.) Height (in.). You can use the lengths or heights to find the scale factor. A scale factor is always the ratio of the model s dimensions to the actual object s dimensions. model length race car length Write a ratio. Then simplify. model height. race car height The scale factor is. This is reasonable because the length of the race car is. in. The length of the model is in., which is less than. in., and is less than. Chapter Proportional Relationships

40 EXAMPLE Using Scale Factors to Find Unknown Lengths A photograph of Vincent van Gogh s painting Still Life with Irises Against a Yellow Background has dimensions. cm and. cm. The scale factor is. Find the size of the actual painting, to the nearest tenth of a centimeter. photo Think: p ainting. Write a proportion to find the length.. Find the cross products.. cm Multiply and round to the nearest tenth. ẇ Write a proportion to find the width w. w. Find the cross products. w. cm Multiply and round to the nearest tenth. The painting is. cm long and. cm wide. EXAMPLE Measurement Application On a map of Florida, the distance between Hialeah and Tampa is. cm. What is the actual distance d between the cities if the map scale is cm mi? map distance Think: a ctual distance. Write a proportion. d d. Find the cross products. d d Divide both sides by. d mi The distance between the cities is miles. Think and Discuss. Explain how you can tell whether a model with a scale factor of is larger or smaller than the original object.. Describe how to find the scale factor if an antenna is feet long and a scale drawing shows the length as foot long. - Scale Drawings and Scale Models

41 - Exercises California Standards Practice NS. KEYWORD: MSCA - See Example GUIDED PRACTICE Identify the scale factor... Grizzly Bear Model Height (in.) KEYWORD: MSCA Parent Moray Eel Model Length (ft) See Example See Example. In a photograph, a sculpture is. cm tall and. cm wide. The scale factor is. Find the size of the actual sculpture.. Ms. Jackson is driving from South Bend to Indianapolis. She measures a distance of. cm between the cities on her Indiana road map. What is the actual distance between the cities if the map scale is cm mi? INDEPENDENT PRACTICE See Example Identify the scale factor... Eagle Model Dolphin Model Wingspan (in.) Length (cm) See Example See Example Extra Practice See page EP.. On a scale drawing, a tree is inches tall. The scale factor is. Find the height of the actual tree.. Measurement On a road map of Virginia, the distance from Alexandria to Roanoke is. cm. What is the actual distance between the cities if the map scale is cm mi? PRACTICE AND PROBLEM SOLVING The scale factor of each model is :. Find the missing dimensions. Item Actual Dimensions Model Dimensions... Lamp Height: Height: in. Couch Height: in. Height: Length: in. Length: Chair Height: in. Height:. A building shaped like a pair of binoculars is a scale model of an actual pair of binoculars. The scale is ft in. What is the height of the building if the height of the actual binoculars is inches?. Critical Thinking A countertop is ft long. How long is it on a scale drawing with the scale in. yd?. Write About It A scale for a scale drawing is cm mm. Which will be larger, the actual object or the scale drawing? Explain. Chapter Proportional Relationships

42 History Use the map for Exercises.. In, Confederate troops marched from Chambersburg to Gettysburg in search of badly needed shoes. Use the ruler and the scale of the map to estimate how far the Confederate soldiers, many of whom were barefoot, marched.. Before the Civil War, the Mason-Dixon Line was considered the dividing line between the North and the South. If Gettysburg is about. miles north of the Mason-Dixon Line, how far apart in inches are Gettysburg and the Mason-Dixon Line on the map?. Reasoning Toby is making a scale model of the battlefield at Fredericksburg. The area he wants to model measures about mi by. mi. He plans to put the model on a. ft by. ft square table. On each side of the model he wants to leave at least in. between the model and the table edges. What is the largest scale he can use?. Challenge A map of Vicksburg, Mississippi, has a scale of mile to the inch. The map has been reduced so that inches on the original map appears as. inches on the reduced map. If the distance between two points on the reduced map is. inches, what is the actual distance in miles? This painting by H.A. Ogden depicts General Robert E. Lee at Fredericksburg in. NS., NS., AF.. Multiple Choice On a scale model with a scale of, the height of a shed is inches. What is the approximate height of the actual shed? A feet feet feet feet B. Gridded Response On a map, the scale is cm mi. If the distance between two cities on the map is. cm, what is the distance between the actual cities in miles? Order the numbers from least to greatest. (Lesson -).,.,..,.,...,,..,,. Convert each measure. (Lesson -). g to kilograms.. L to milliliters. cm to meters C D - Scale Drawings and Scale Models

43 Quiz for Lessons - Through - - Similar Figures and Proportions. Tell whether the triangles are similar. M cm cm R cm cm L N Q S cm cm. Tell whether the figures are similar. ft ft ft ft ft ft ft ft - Using Similar Figures ABC XYZ in each pair. Find the unknown lengths.. Y. B in. m B m z. m A m C X m Z. Reynaldo drew a rectangular design that was in. wide and in. long. He used a copy machine to enlarge the rectangular design so that the width was in. What was the length of the enlarged design?. Redon is ft in. tall, and his shadow is ft in. long. At the same time a building casts a shadow that is ft in. long. Estimate the height of the building. - Scale Drawings and Scale Models in. in. in. in.. An actor is ft tall. On a billboard for a new movie, the actor s picture is enlarged so that his height is. ft. What is the scale factor?. On a scale drawing, a driveway is in. long. The scale factor is. Find the length of the actual driveway.. A map of Texas has a scale of in. mi. If the distance from Dallas to San Antonio is mi, what is the distance in inches between two cities on the map? A C Z t Y X Chapter Proportional Relationships

44 Bug Juice When campers get thirsty, out comes the well-known camp beverage bug juice! The recipes show how two camps, Camp Big Sky and Camp Wild Flowers, make their bug juice. Each camp has campers. During a typical day, each camper drinks two -ounce cups of bug juice.. How many ounces of bug juice are consumed at each camp each day?. How much does it cost to make two quarts of bug juice at each camp?. Each camp has budgeted $ per day for bug juice. Is $ a day enough? How do you know? Show your work.. Campers begin to complain. They want their bug juice buggier. How could each camp change its recipe, continue to serve campers two -ounce cups of bug juice daily, and not spend more than $ per day for bug juice? Explain your reasoning. Camp Big Sky Bug Juice Recipe ne oz packet of mix A Add tap water to make quarts of bug juice. Camp Wild Flowers Bug Juice Recipe ne. oz packet of mix B oz sugar Add tap water to make quarts of bug juice. Prices oz packet of mix A $.. oz packet of mix B $. lb of sugar $. Concept Connection

45 Water Works You have three glasses: a -ounce glass, a -ounce glass, and an -ounce glass. The -ounce glass is full of water, and the other two glasses are empty. By pouring water from one glass to another, how can you get exactly ounces of water in one of the glasses? The step-by-step solution is described below. Fill the oz glass using water from the oz glass. Fill the oz glass using water from the oz glass. Pour the water from the oz glass into the oz glass. You now have ounces of water in the -ounce glass. Start again, but this time try to get exactly ounces of water in one glass. (Hint: Find a way to get ounce of water. Start by pouring water into the -ounce glass.) Next, using -ounce, -ounce, and -ounce glasses, try to get exactly ounces of water in one glass. Start with the -ounce glass full of water. (Hint: Start by pouring water into the -ounce glass.) Look at the sizes of the glasses in each problem. The volume of the third glass is the sum of the volumes of the first two glasses: and. Using any amounts for the two smaller glasses, and starting with the largest glass full, you can get any multiple of the smaller glass s volume. Try it and see. Concentration Concentration Each card in a deck of cards has a ratio on one side. Place each card face down. Each player or team takes a turn flipping over two cards. If the ratios on the cards are equivalent, the player or team can keep the pair. If not, the next player or team flips two cards. After every card has been turned over, the player or team with the most pairs wins. A complete copy of the rules and the game pieces are available online. Chapter Proportional Relationships KEYWORD: MSCA Games

46 Materials paper plates scissors markers A PROJECT Paper Plate Proportions Serve up some proportions on this book made from paper plates. Fold one of the paper plates in half. Cut out a narrow rectangle along the folded edge. The rectangle should be as long as the diameter of plate s inner circle. When you open the plate, you will have a narrow window in the center. Figure A Fold the second paper plate in half and then unfold it. Cut slits on both sides of the crease beginning from the edge of the plate to the inner circle. Figure B B C Roll up the plate with the slits so that the two slits touch each other. Then slide this plate into the narrow window in the other plate. Figure C When the rolled-up plate is halfway through the window, unroll it so that the slits fit on the sides of the window. Figure D D Close the book so that all the plates are folded in half. Taking Note of the Math Write the number and name of the chapter on the cover of the book. Then review the chapter, using the inside pages to take notes on ratios, rates, proportions, and similar figures. It s in the Bag!

47 Vocabulary corresponding angles corresponding sides..... cross product equivalent ratios indirect measurement.. proportion rate ratio scale scale drawing scale factor scale model similar unit rate Complete the sentences below with vocabulary words from the list above..? figures have the same shape but not necessarily the same size.. A(n)? is a comparison of two numbers, and a(n)? is a ratio that compares two quantities measured in different units.. The ratio used to enlarge or reduce similar figures is a(n)?. - Ratios (pp. ) NS. EXAMPLE Write the ratio of servings of bread to servings of vegetables in all three forms. Write your answers in simplest form. Write the ratio to in simplest form., to, : EXERCISES There are red, blue, and yellow balloons.. Write the ratio of blue balloons to total balloons in all three forms. Write your answer in simplest form.. Tell whether the ratio of red to blue balloons or the ratio of yellow balloons to total balloons is greater. - Rates (pp. ) NS., AF., AF. EXAMPLE Find each unit price. Then decide which has the lowest price per ounce. $. o z or $. oz $. o z $. o z and $. $. oz oz Since.., $. has the lowest oz price per ounce. EXERCISES Find each average rate of speed.. ft in s. mi in hr Find each unit price. Then decide which is the better buy. $. or $.. $ gal gal g or $ g Chapter Proportional Relationships

48 - Identifying and Writing Proportions (pp. ) NS. EXAMPLES Determine whether and are proportional. is already in simplest form. Simplify. The ratios are not proportional. Find a ratio equivalent to. Then use the ratios to write a proportion. Write an equivalent ratio. Write a proportion. EXERCISES Determine whether the ratios are proportional..,.,.,.,.,., Find a ratio equivalent to each ratio. Then use the ratios to write a proportion Solving Proportions (pp. ) NS., AF., AF. EXAMPLE Use cross products to solve p. p p Multiply the cross p products. p Divide each side by. p, or EXERCISES Use cross products to solve each proportion.. n. a b... x. y. w - Customary Measurements (pp. ) NS., AF. EXAMPLES Choose the most appropriate customary unit for the weight of a mouse. Justify your answer. Ounces the weight of a mouse is similar to the weight of a slice of bread. Convert mi to feet. feet, x m iles x,, ft EXERCISES Choose the most appropriate customary unit for each measurement. Justify your answer.. the height of a giraffe. the capacity of a washing machine. the width of a cell phone Convert each measure.. fl oz to pints.. tons to pounds., ft to miles Study Guide: Review

49 - Similar Figures and Proportions (pp. ) Prep for NS. EXAMPLE EXERCISES Tell whether the figures are similar. Tell whether the figures are similar. The corresponding angles of the figures. ft ft have equal measures.??? cm ft ft ft ft cm cm ft ft cm. B m The ratios of the Y cm C. m corresponding cm m Z cm. m sides are equivalent. m m The figures are similar. cm A X - Using Similar Figures (pp. ) NS. EXAMPLE EXERCISES ABC LMN. Find the unknown JKL DEF. Find the unknown length. length.. F x D AB AC K L M L B N t in. in. ft ft ft ft A C J L in. ft t E M t. A tree casts a ft shadow at the time of day when a ft stake casts a t t in. ft shadow. Estimate the height of in. t L N the tree. in. - Scale Drawings and Scale Models (pp. ) NS. EXAMPLE A model boat is inches long. The scale factor is. How long is the actual boat? m odel boat n Write a proportion. n Find the cross products. n Solve. The boat is inches long. EXERCISES. The Wright brothers Flyer had a -inch wingspan. Carla bought a model of the plane with a scale factor of. What is the model s wingspan?. The distance from Austin to Houston on a map is. inches. The map scale is inch miles. What is the actual distance? Chapter Proportional Relationships

50 A soccer team has sixth-graders, seventh-graders, and eighth-graders. Write each ratio in all three forms.. seventh-graders to sixth-graders. eighth-graders to total team members. Stan found pennies, nickels, dimes, and quarters. Tell whether the ratio of pennies to quarters or the ratio of nickels to dimes is greater.. Lenny sold tacos in hours. What was Lenny s average rate of taco sales?. A store sells a lb box of detergent for $. and a lb box of detergent for $.. Which size box has the lowest price per pound? Find a ratio equivalent to each ratio. Then use the ratios to write a proportion..... Use cross products to solve each proportion.. m. x.. t p. A submarine travels miles in hours. At this rate, how many hours will it take the submarine to travel miles? Convert each measure.., ft to miles.. lb to ounces. qt to gallons Tell whether the figures are similar.. C. ft ft F B A ft ft ft E D ft WYZ MNO in each pair. Find the unknown lengths.. M. Y Y W in. in. in. in. Z O in. n N O S cm cm cm. cm N P R T cm cm cm cm M Q m m m m O M. An -foot flagpole casts a shadow that is feet long at the same time as a nearby tree casts a shadow that is feet long. How tall is the tree?. A scale model of a building is in. by in. If the scale is in. ft, what are the dimensions of the actual building?. The distance from Portland to Seaside is mi. What is the distance in inches between the two towns on a map if the scale is in. mi? W m x Z N Chapter Test

51 Gridded Response: Write Gridded Responses When responding to a test item that requires you to place your answer in a grid, you must fill in the grid on your answer sheet correctly, or the item will be marked as incorrect.. Gridded Response: Solve the equation. r.... r.. r. Using a pencil, write your answer in the answer boxes at the top of the grid. Put the first digit of your answer in the leftmost box, or put the last digit of your answer in the rightmost box. On some grids, the fraction bar and the decimal point have a designated box. Put only one digit or symbol in each box. Do not leave a blank box in the middle of an answer. Shade the bubble for each digit or symbol in the same column as in the answer box. / Gridded Response: Divide.. The answer simplifies to,, or.. Mixed numbers and repeating decimals cannot be gridded, so you must grid the answer as. Write your answer in the answer boxes at the top of the grid. Put only one digit or symbol in each box. Do not leave a blank box in the middle of an answer. Shade the bubble for each digit or symbol in the same column as in the answer box. Chapter Proportional Relationships

52 Grid formats may vary from test to test. The grid in this book is used often, but it is not used on every test that has griddedresponse questions. Always examine the grid when taking a standardized test to be sure you know how to fill it in correctly. Read each statement, and then answer the questions that follow. Sample A A student correctly solved an equation for x and got as a result. Then the student filled in the grid as shown. Sample C A student subtracted from and got an answer of. Then the student filled in the grid as shown.. What error did the student make when finding the answer?. Explain why you cannot fill in a negative number on a grid.. Explain how to fill in the answer to ( ) correctly.. What error did the student make when filling in the grid?. Explain a second method of filling in the answer correctly. Sample B A student correctly multiplied. and.. Then the student filled in the grid as shown... What error did the student make when filling in the grid?. Explain how to fill in the answer correctly. Sample D A student correctly added and got as a result. Then the student filled in the grid as shown. /. What answer is shown in the grid?. Explain why you cannot show a mixed number in a grid.. Write two equivalent forms of the answer that could be filled in the grid correctly. Strategies for Success

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