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Ratios, Rates & Proportions

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Table of Contents

Writing Ratios Equivalent Ratios Rates Proportions Application problems

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Sampling Scale Drawings Similar Figures Writing an Equivalent Rate

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Writing Ratios

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Ratios

What do you know about ratios? When have you seen or used ratios? Ratio - A comparison of two numbers by division Find the ratio of boys to girls in this class

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Ratios can be written three different ways: a to b a : b a b Each is read, "the ratio of a to b." Each ratio should be in simplest form.

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There are 48 animals in the field. Twenty are cows and the rest are horses. Write the ratio in three ways:

• a. The number of cows to the number of horses
• b. The number of horses to the number of animals in the field

Remember to write your ratios in simplest form!

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1 There are 27 cupcakes. 9 are chocolate, 7 are vanilla and the rest are strawberry. What is the ratio of vanilla cupcakes to strawberry cupcakes? A 7 : 9 B

7 27

C

7 11

D 1 : 3

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2 There are 27 cupcakes. 9 are chocolate, 7 are vanilla and the rest are strawberry. What is the ratio of chocolate & strawberry cupcakes to vanilla & chocolate cupcakes? A 20 16 B 11 7 C 5 4 D 16 20

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3 There are 27 cupcakes. 9 are chocolate, 7 are vanilla and the rest are strawberry. What is the ratio of chocolate cupcakes to total cupcakes? A 7 9 B

7 27

C

9 27

D 1 3

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4 There are 27 cupcakes. 9 are chocolate, 7 are vanilla and the rest are strawberry. What is the ratio of total cupcakes to vanilla cupcakes? A 27 to 9 B 7 to 27 C 27 to 7 D 11 to 27

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Equivalent Ratios

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Equivalent ratios have the same value 3 : 2 is equivalent to 6: 4 1 to 3 is equivalent to 9 to 27 5 35 6 is equivalent to 42

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4 12 5 15 x 3 Since the numerator and denominator were multiplied by the same value, the ratios are equivalent There are two ways to determine if ratios are equivalent. 1. 4 12 5 15 x 3 4 12 5 15 Since the cross products are equal, the ratios are equivalent. 4 x 15 = 5 x 12 60 = 60

• 2. Cross Products

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5 4 is equivalent to 8 9 18 True False

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6 5 is equivalent to 30 9 54 True False

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7 18:12 is equivalent to 9, which is equivalent to 36 6 24 True False

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8 2 is equivalent to 10 , which is equivalent to 40 24 120 480 True False

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9 1:7 is equivalent to 10 , which is equivalent to 5 to 65 70 True False

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Rates

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Rate: a ratio of two quantities measured in different units Examples of rates: 4 participants/2 teams 5 gallons/3 rooms 8 burgers/2 tomatoes Unit rate: Rate with a denominator of one Often expressed with the word "per" Examples of unit rates: 34 miles/gallon 2 cookies per person 62 words/minute

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Finding a Unit Rate

Six friends have pizza together. The bill is $63. What is the cost per person? Hint: Since the question asks for cost per person, the cost should be first, or in the numerator. $63 6 people Since unit rates always have a denominator of one, rewrite the rate so that the denominator is one. $63 6 6 people 6 $10.50 1 person The cost of pizza is $10.50 per person

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10 Sixty cupcakes are at a party for twenty children. How many cupcakes per person?

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11 John's car can travel 94.5 miles on 3 gallons of gas. How many miles per gallon can the car travel?

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12 The snake can slither 24 feet in half a day. How many feet can the snake move in an hour?

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13 There are six chaperones at the dance of 100 students. How many students per chaperone are there?

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14 The recipe calls for 6 cups of flour for every four eggs. How many cups of flour are needed for one egg?

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Population Density Population Density: A unit rate of people per square mile This data is compiled by the US Census Bureau every 10 years and is used when determining the number of Representatives each state gets in the House of Representatives. Look at the following Population Densities from 2009... New Jersey: 1,184 people per square mile Montana: 7 people per square mile California: 237 people per square mile

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15 The population of Newark, NJ is 278,980 people in 24.14 square miles. What is its population density?

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16 The population of Moorestown, NJ is 19,509 people in 15 square miles. What is its population density?

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17 The population of Waco, TX is 124,009 people in 75.8 square

• miles. What is its population density?

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Writing an Equivalent Rate

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To write equivalent rates, conversion factors must be used. Conversion factors are used to convert from one unit to another. Conversion factors must be equal to 1 . Some examples of conversion factors: 1 pound or 16 ounces 16 ounces 1 pound 12 inches or 1 foot 1 foot 12 inches 3 feet or 1 yard 1 yard 3 feet 1 day or 24 hours 24 hours 1 day Create 5 conversion factors of your own!

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Identify the conversion factor that results in the desired unit. Find a conversion factor that converts minutes to seconds. minutes 60 seconds 1 minute seconds

• r

1 minute 60 seconds Hint: You want the rate of minute to cancel, so that you are left with the rate of seconds PULL

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Identify the conversion factor that results in the desired unit. Find a conversion factor that converts 12 feet to yards. 12 feet 3 feet 1 yard ? yards

• r

1 yard 3 feet Hint: You want the rate of feet to cancel, so that you are left with the rate of yards. PULL PULL

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Identify the conversion factor that results in the desired unit. Find a conversion factor that converts miles to feet. 5 miles 5280 feet 1 mile ? feet

• r

1 mile 5280 feet Hint: You want the rate of miles to cancel, so that you are left with the rate of feet PULL PULL

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To write equivalent rates, conversion factors must be used. Example 1: 2 inches ? inches 1 hour 1 day 2 inches 24 hours 48 inches 1 hour 1 day 1 day 5 feet ? feet 1 sec 1 hour 5 feet 60 sec 300 feet 1 sec 1 hour 1 hour Example 2:

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18 Write the equivalent rate. 40 mi ? mi 1 min 1 h

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19 Write the equivalent rate. 54 inches ? inches 1 year 1 month

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20 Write the equivalent rate. 1 day 1week $75 ? dollars

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21 Write the equivalent rate. 30 sec 1min 425 mi ? miles

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22 Write the equivalent rate. 40 feet inches 3 hrs hr Hint: Find the equivalent rate and then determine the unit rate

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23 Write the equivalent rate. 20,000 feet ? feet 4 seconds minute Hint: Find the equivalent rate and then determine the unit rate

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24 Write the equivalent rate. 1200 people ? people 6 days hr Hint: Find the equivalent rate and then determine the unit rate

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Proportions

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A proportion is an equation that states that two ratios are equivalent. Example: 2 12 3 18

5 15 9 27

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If one of the numbers in a proportion is unknown, mental math can be used to find an equivalent ratio. Example 1: 2 6 3 x x 3 2 6 3 x Hint: To find the value of x, multiply 3 by 3 also. 2 6 3 9 x 3

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If one of the numbers in a proportion is unknown, mental math can be used to find an equivalent ratio. Example: 28 7 32 x 4 Hint: To find the value of x, divide 32 by 4 also. 28 7 32 x 28 7 32 8 4

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25 Solve the proportion using equivalent ratios 2 8 5 x

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26 Solve the proportion using equivalent ratios 4 x 9 36

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27 Solve the proportion using equivalent ratios 7 35 2 x

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28 Solve the proportion using equivalent ratios x 4 60 12

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29 Solve the proportion using equivalent ratios 3 21 x 28

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In a proportion, the cross products are equal. 5 30 2 12 5 12 2 30 60 60

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Proportions can also be solved using cross products. 4 12 5 x 4x = 5 12 4x = 60 x = 15 Cross multiply Solve for x 7 x 8 48 8x = 7 48 8x = 336 x = 42 Example 2 Cross multiply Solve for x

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30 Use cross products to solve the proportion 9 = x 51 17

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31 x = 56 12 96 Use cross products to solve the proportion

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32 45 = _x 18 6 Use cross products to solve the proportion

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33 2 = _x 15 60 Use cross products to solve the proportion

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34 7 = _3 x 21 Use cross products to solve the proportion

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Application problems

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Chocolates at the candy store cost $5.99 per dozen. How much does one candy cost? Round your answer to the nearest cent. Solution: $5.99 1 dozen 1 dozen 12 $5.99 12 $0.50 per candy (Use equivalent rates)

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Example 2: There are 3 books per student. There are 570 students. How many books are there? Set up the proportion: Books Students 3 Where does the 570 go? 1 3 x 1 570 1x 3 570 x 1,710 books

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Example 3: The ratio of boys to girls is 4 to 5. There are 125 people on a

• team. How many are girls?

Set up the proportion: Girls People How did we determine this ratio? 5 Where does the 125 go? 9 5 x 9 125 9x 5 125 9x = 625 x = 69.44 70 girls = = =

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35 Cereal costs $3.99 for a one pound box. What is the price per ounce? Round your answer to the nearest penny.

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36

Which is the better buy? Brand A: $2.19 for 12 ounces Brand B: $2.49 for 16 ounces A

Brand A

B

Brand B

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37 There are 4 girls for every 10 boys at the party. There are 56 girls at the party. How many boys are there?

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38 The farmer has cows and chickens. He owns 5 chickens for every cow. He has a total of 96 animals. How many cows does he own?

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39 The auditorium can hold 1 person for every 5 square feet. It is 1210 square feet. How many people can the auditorium hold?

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40 The recipe for one serving calls for 4 oz of beef and 2 oz of bread crumbs. 50 people will be attending the dinner. How many lbs. of bread crumbs should be purchased?

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41 Mary received 4 votes for every vote that Jane received. 1250 people voted. How many votes did Jane receive?

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42 To make the desired shade of pink paint, Brandy uses 3 oz.

• f red paint for each oz. of white paint. She needs one

quart of pink paint. How many oz. of red paint will she need?

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43 In a sample of 50 randomly selected students at a school, 38 students eat breakfast every morning. There are 652 students in the school. Using these results, predict the number of students that eat breakfast. A 76 B 123 C 247 D 496

Question from ADP Algebra I End-of-Course Practice Test

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Sampling

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Your task is to count the number of whales in the ocean or the number of squirrels in a park. How could you do this? What problems might you face?

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How would you estimate the size of a crowd? What methods would you use? Could you use the same methods to estimate the number

• f wolves on a mountain?

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One way to estimate the number of wolves on a mountain is to use the CAPTURE - RECAPTURE METHOD.

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Suppose this represents all the wolves on the mountain.

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Wildlife biologists first find some wolves and tag them.

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Then they release them back onto the mountain.

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They wait until all the wolves have mixed together. Then they find a second group of wolves and count how many are tagged.

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Biologists use a proportion to estimate the total number of wolves on the mountain: tagged wolves on mountain tagged wolves in second group

total wolves on mountain total wolves in second group

For accuracy, they will often conduct more than one recapture.

=

8 2 w 9 2w = 72 w = 36

=

There are 36 wolves on the mountain

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Try This: Biologists are trying to determine how many fish are in the Rancocas Creek. They capture 27 fish, tag them and release them back into the Creek. 3 weeks later, they catch 45 fish. 7 of them are tagged. How many fish are in the creek? 27 7 f 45 27(45) = 7f 1215 = 7f 173.57 = f = There are 174 fish in the river

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A whole group is called a POPULATION. A part of a group is called a SAMPLE. When biologists study a group of wolves, they are choosing a

• sample. The population is all the wolves on the mountain.

Population Sample

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Example: 860 out of 4,000 people surveyed watched Grey's Anatomy. How many people in the US watched if there are 93.1 million people? 860 x 4000 93,100,000 860(93,100,000) = 4000x 80,066,000,000 = 4000x 20,016,500 = x = 20,016,500 people watched

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Try This: 315 out of 600 people surveyed voted for Candidate A. How many votes can Candidate A expect in a town with a population of 1500?

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Margin of Error The results of sampling are estimates, which always contain some error. The margin of error estimates the interval that is most likely to include the exact result for the population. Margin of error is given as a percent in the problem. To find the interval using margin of error: · Find the percent of the population · Add/Subtract that amount from the answer to create an interval.

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860 out of 4,000 people surveyed watched Grey's Anatomy. How many people in the US watched if there are 93.1 million people? Estimate an interval with a 2% margin of error. Margin of Error = 2% This means 2% of the population! 2% of 93,100,000 (.02)(93,100,000) 1,862,000 So the interval is 20,016,500 + 1,862,000 18,154,500 to 21,818,500 R e c a l l R e c a l l

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Try This: 6 out of 150 tires need to be realigned. How many

• ut of 12,000 are going to need to be realigned?

Estimate an interval with a 3% margin of error. Estimate

M a r g i n

• f

E r r

• r

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44 You are an inspector. You find 3 faulty bulbs out of 50. Estimate the number of faulty bulbs in a lot of 2,000.

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45 You are an inspector. You find 3 faulty bulbs out of 50. Estimate the number of faulty bulbs in a lot of 2,000. Use a 2% margin of error. What is the amount you are going to + by?

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46 You are an inspector. You find 3 faulty bulbs out of 50. Estimate the number of faulty bulbs in a lot of 2,000. Use a 2% margin of error. What is the lower number in your interval?

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47 You are an inspector. You find 3 faulty bulbs out of 50. Estimate the number of faulty bulbs in a lot of 2,000. Use a 2% margin of error. What is the upper number in your interval?

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48 You survey 83 people leaving a voting site. 15 of them voted for Candidate A. If 3,000 people live in town, how many votes should Candidate A expect?

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49 You survey 83 people leaving a voting site. 15 of them voted for Candidate A. If 3,000 people live in town, how many votes should Candidate A expect? Find an interval using a 3% margin of error. What is the amount you are going to + by?

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50 You survey 83 people leaving a voting site. 15 of them voted for Candidate A. If 3,000 people live in town, how many votes should Candidate A expect? Find an interval using a 3% margin of error. What is the lower number in your interval?

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51 You survey 83 people leaving a voting site. 15 of them voted for Candidate A. If 3,000 people live in town, how many votes should Candidate A expect? Find an interval using a 3% margin of error. What is the upper number in your interval?

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Scale Drawings

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Scale drawings are used to represent objects that are either too large or too small for a life size drawing to be useful. Examples: A life size drawing of an ant or an atom would be too small to be useful. A life size drawing of the state of New Jersey or the Solar System would be too large to be useful.

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A scale is always provided with a scale drawing. The scale is the ratio: drawing real life (actual) When solving a problem involving scale drawings you should: · Write the scale as a ratio · Write the second ratio by putting the provided information in the correct location (drawing on top & real life on the bottom) · Solve the proportion

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Example: This drawing has a scale of "1:10", so anything drawn with the size of "1" would have a size of "10" in the real world, so a measurement of 150mm on the drawing would be 1500mm on the real horse.

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Example: The distance between Philadelphia and San Francisco is 2,950 miles. You look on a map and see the scale is 1 inch : 100 miles. What is the distance between the two cities on the map? 1 100 Write the scale as a ratio 1 x 100 2950 100x = 2950 x = 29.5 29.5 inches on the map =

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Try This: On a map, the distance between your town and Washington DC is 3.6 inches. The scale is 1 inch : 55 miles. What is the distance between the two cities?

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52 The distance between Moorestown, NJ and Duck, NC is 910 miles. What is the distance on a map with a scale of 1 inch to 110 miles?

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53 The distance between Philadelphia and Las Vegas is 8.5 inches on a map with a scale 1.5 in : 500 miles . What is the distance in miles?

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54 You are building a room that is 4.6 m long and 3.3 m wide. The scale on the architect's drawing is 1 cm : 2.5 m. What is the length of the room on the drawing?

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55 You are building a room that is 4.6 m long and 3.3 m wide. The scale on the architect's drawing is 1 cm : 2.5 m. What is the width of the room on the drawing?

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56 Find the length of a 72 inch wide door on a scale drawing with a scale 1 inch : 2 feet.

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57 You recently purchased a scale model of a car. The scale is 1cm : 24m. What is the length of the model car if the real car is 4m?

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58 You recently purchased a scale model of a car. The scale is 1cm : 24m. The length of the model's steering wheel is 2.25 cm. What is the actual length

• f the steering wheel?

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59

The figure is a scale of the east side of a house. In the drawing, the side of each square represents 4 feet. Find the width and height of the door. A 4 ft by 9 ft B 4 ft by 12 ft C 4 ft by 8 ft D 4 ft by 10 ft

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60

On a map, the scale is 1/2 inch= 300 miles. Find the actual distance between two stores that are 5 1/2 inches apart on the map. A 3000 miles B 2,727 miles C 3,300 miles D 1,650 miles

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61

On a map with a scale of 1 inch =100 miles, the distance between two cities is 7.5 inches. If a car travels 55 miles per hour, about how long will it take to get from one city to the other. A 13 hrs 45 min. B 14 hrs 30 min. C 12 hrs D 12 hrs 45 min.

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Similar Figures

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Two objects are similar if they are the same shape but different sizes. In similar objects: · corresponding angles are congruent · corresponding sides are proportional

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To check for similarity: · Check to see that corresponding angles are congruent · Check to see that corresponding sides are proportional (Cross products are equal)

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Example: Is the pair of polygons similar? Explain your answer. 4 3 6 4.5 4(4.5) = 6(3) 18 = 18 YES 4 yd 3 yd 6 yd 4.5 yd =

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Example: Is the pair of polygons similar? Explain your answer. 5 8 10 13 5(13) = 10(8) 65 = 80 NO 5 m 8 m 10 m 13 m =

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Example: Find the value of x in the pair of similar polygons. 15 6 x 10 15(10) = 6x 150 = 6x 25 cm = x 15 cm = x 6 cm 8 cm 10 cm

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Try This: Find the value of y in the pair of similar polygons.

y 15 in 5 in 7.5 in

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62 Are the polygons similar? You must be able to justify your answer. Yes No 15 ft 9 ft 21 ft 12 ft

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63 Are the polygons similar? You must be able to justify your answer. Yes No 10 m 8 m 2.5 m 2 m

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64 Are the polygons similar? You must be able to justify your answer. Yes No 15 yd 6 yd 15 yd 37.5 yd

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65 Find the measure of the missing value in the pair

• f similar polygons.

110

y

110 80 80

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66 Find the measure of the missing value in the pair

• f similar polygons.

25 ft 25 ft 18 ft 17.5 ft w

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67 Find the measure of the missing value in the pair

• f similar polygons.

17 m 4.25 m 4 m x

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68 Find the measure of the missing value in the pair

• f similar polygons.

11 mm 38.5 mm 6 mm

y

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69 Find the measure of the missing value in the pair

• f similar polygons.

119 m 7 m 63 m ? 13 m

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70 Find the measure of the missing value in the pair

• f similar polygons.

119 m 7 m 63 m 9 m 13 m ?

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71 Find the measure of the missing value in the pair

• f similar polygons.

5 mm 2 mm 27.5 mm x

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