Ratio and Proportions Unit 6

Transcription

1 Ratio and Proportions Unit 6 The ratio of circles to triangles is 3:2 Name Date Period 1

2 Lesson 1: Equal Ratios and Proportions Vocabulary: 1. Ratio: A comparison of two quantities by division. Can be written as b a, a : b, or a to b. (b = 0) 2. Proportion: An equation that shows that two ratios are equivalent (equal). 3. Cross Product: In two ratios, the cross products are found by multiplying the denominator of one ratio by the numerator of the other. If you cross multiply and the products are the same then it is proportional. *If two ratios form a proportion, if the cross products are equal. Three ways to write a ratio: 5 to 7 5 : 7 Understanding Ratios Nicholas surveyed members of his class if they go on Facebook on the weekends. The table below shows their answers. Use the table to answer the following questions. Sometimes on Facebook Always on Facebook Never on Facebook What is the ratio of the number of students who always go on Facebook to the number who never go on Facebook? 2. What is the ratio of the number of students who sometimes go on Facebook to the total number of students surveyed? 3. What does the ratio 30:7 represent? Equivalent Ratios 1. The ratio of the number of wrestlers to the number of football players at Sachem is 16 to 48. Represent this ratio as a fraction.. You can use what you know about fractions to find equal ratios and write the ratio in simplest form. Write an equivalent ratio. 2. Write three ratios that are equal to the given ratio 12: 21. (Express this ratio as a fraction to help you.) equivalent ratio 1 equivalent ratio 2 equivalent ratio 3 Check if each pair of ratios form a proportion. Write = if the ratios are proportional. If they are not write =. 1) ) 6:33 2:11 3) 8 to to 9 2

3 What are the 2 steps in solving proportions? 1) 2) Practice: 1) ) n to 2 and 9 to 1 3) 32 : y and 8 : 9 4) 12 n 2x ) 11: 5 and x : 10 6) x ) 12 to 16 and 18 to n 8) 10 n ) y ) 1.7 : 2.5 and 3.4 : d 11) 15 x 60 12) 5 4 x ) There are 250 people waiting on line to ride a roller coaster and the wait time is 20 minutes. How long is the waiting time when 240 people are on line? 14) If Sarah can type 2 pages in 10 minutes, if she types for 30 minutes how many pages will she be able to type? 3

4 Lesson 1 Homework: Proportions 1. A basket of fruit has 2 apples, 1 orange, and 1 banana. Write a ratio for each comparison in three ways. (Look at the classwork if you are not sure how to write a ratio in 3 ways.) a. oranges to apples b. bananas to oranges c. apples to all pieces of fruit Do these form a true proportion? Write yes or no ) 7 : 12 and 21 : 35 3) 1) 9 Blue 4) 3 to 8 and 12 to 32, 36 Blue Red 68 Red Solve the following proportions for the given variable ) 2 to 15 and x to 45 7) x : 8 and 2 : 4 5) 3 x 8) 40 x ) 18 to 12 and 6 and x 10) 7 10 x ) 5x ) 5 3 x ) 24 : x and 3 : 5 6 Write a proportion for each of the following then SOLVE! 14) 6 Earth-pounds equals 1 moon-pound. How many moon-pounds would 96 Earth-pounds equal? Earth pounds Moon pounds = 15) About 4 out of every 5 people are right-handed. If there are 30 students in a class, how many would you expect to be right-handed? = 4

5 Lesson 1: Proportions extra practice When multiplying a Monomial (one term) times a Binomial (two terms), you MUST: *Put parentheses around the binomial and distribute!! Example: 2x Step 1: Cross Multiply 5(2x + 4) = 3(10) Step 2: Simplify 10x + 20 = Step 3: Solve 10x = x = 1 Simplify. 4 1) 5x ) 10 x ) 3 3x Solve the following proportions for the given variable: 5) 1 4 6) 3 x ) 7 4 x 2 8 2x Review: Simplify ) x 6 2 x 2 12) 5x 4(2x + 3) Solve. 13) John and his friends go to a carnival and there is an admission fee of $15. Each ride cost $5 to go on and he can spend at most $50. How many rides can John go on? Solve and graph. 14). 5x 55 15) 30 4x 14 16) 8x 64 5

6 Lesson 2: Solving Word Problem Proportions Vocabulary: Scale: A ratio that compares the length in a drawing to the length of an actual object Scale Factor: A ratio of the lengths of two corresponding sides of two similar polygons Ratio Example 1: Spoken, this is 10 pencils per 1 package or 10 pencils to 1 package. Ratio Example 2: Brian can run 50 yards in 1 minute. How far can she run in 3 minutes? Write the proportions for the following questions and SOLVE: Round to the nearest tenth if necessary. 1) There are 5 dogs for every 3 cats in the pet store. If there are 20 dogs at the store, how many cats are at the store? Dogs Cats 2) Mr. Green can grow 12 tomato plants in a 3 square foot area. How many tomatoes can he grow if he has a 15 square foot area? Plants Area 3) Nick takes 3 hours to read 240 pages. At this rate, how many pages can he read in 2 hours? Hours Pages 4) A burro is standing near a cactus. The burro is 60 inches tall. His shadow is 4 feet long. The shadow of the cactus is 7 feet long. How tall is the cactus? (Draw a picture) Tall Shadow 6

7 5) The ratio of boys to girls in the seventh grade is 2 : 3. a. If there are 24 boys, how many girls are there? b. If there are 80 students, how many are girls? c. If there are 75 students, how many more girls are there than boys? 6. There were 35 children and 10 adults at a cookout. a. What is the ratio of adults to children at the cookout? b. What is the ratio of children to total people at the cookout? c. Five more children came to the cookout. Now what is the ratio of children to total people? 7. Justin is making cookies, using a recipe in which the ratio of flour to chocolate chips to sugar is 4: 2: 1 for each batch. a. If he is using 8 cups of flour how many cups of sugar does he need? b. To make 3 batches of cookies, what is the ratio of flour to chocolate chips to sugar? 6) Nancy is 5 feet tall. At a certain time of day, she measures her shadow, and finds it is 9 feet long. She also measures the shadow of a building which is 200 feet long. How tall is the building? 7) Ryan can run 4 blocks in 2 minutes. How long does he take to run 8 blocks at the same speed? 7

8 Lesson 2 Homework: Word Problems For each of the following - set up a proportion WITH LABELS and then solve. 1. Jack and Jill went up the hill to pick apples and pears. Jack picked 10 apples and 15 pears. Jill picked 20 apples and some pears. The ratio of apples and pears picked by both Jack and Jill were the same. Determine how many pears Jill picked. 2. It takes 18 people in Mr. DeMeo s math class to pull a 35 ton bus. How many people would it take to pull a 140 on bus? 3. A sample of 96 light bulbs consisted of 4 defective ones. Assume that today s batch of 6,000 light bulbs has the same proportion of defective bulbs as the sample. Determine the total number of defective bulbs made today. 5. The ratio of green M & M s to yellow is 2 : 5. a. If there are only green and yellow M & M s in the bag, what is the smallest number of M & M s possible? b. If there are 84 M & M s in the bag all together, how many are green? c. If red M & M s were added to the bag in part b to get a total of 100, what is the ratio of green to yellow to red? 6. Morgan and Kira have a number of jelly beans in a ratio of 5 : 3. Kira and Mann have a number of jelly beans in a ratio of 6 : 1. a. What is the ratio of Morgan s jelly beans to Mann s? b. If Morgan and Kira have 64 jelly beans, how many does Kira have? c. How many does Mann have? 8

9 Lesson 3: Computing Unit Rate and Unit Price Warm Up: Express each ratio as a fraction in simplest form: rooms to 48 windows 2. 3 gallons to 15 quarts Unit Rate Example 1) There are 12 flowers in 2 vases. How many flowers per vase? Unit Rate = Example 2) There are 18 chairs at 3 tables. How many chairs per table? Unit Rate = What is the definition of UNIT RATE? Practice: Find the unit rate of each: meters in 8 seconds how many meters for 1 second? 2. Express the ratio of $10 for 8 fish as a unit rate (1 fish). 3. $2,702 for 28 people = how much money for 1 person? 9

10 Unit Price - Example 1) The cost for a case of ketchup bottles (12 units per case) is $12. What is the cost of 1 bottle of ketchup? Unit Price = Example 2) A mother and daughter went shopping for their own Sunday football parties. Afterwards they compared how much they spent for the same items at two different stores. Which store had the better prices? King Kullen Deli Italian Hero $90 for 6 ft. Buffalo Wings - $25 for a platter of 50 wings Pasta Salad - $6.25 for 5 lbs. Meat Farms Deli Italian Hero $39 for 3 ft. Buffalo Wings - $7.50 for a platter of 25 wings Pasta Salad -$2.97 for 3 lbs. Closure: In order to compare prices you must. Practice: 1. Mr. Cohen needs help solving this problem. A hot dog truck sells 9 hot dogs for $ a) Find the unit price? b) If he wants to buy 3 hot dogs for Mr. DeMeo, how much will it cost? 2. Which is the better price? 32 ounces for $3.84 or 40 ounces for $ If you spend $11.13 for 8 gallons of gasoline, how much would you spend on 14 gallons? 4. You can buy 4 apples at Stop and Shop for $0.96. You can buy 6 of the same apples at Pathmark for $1.50. Which store has the better buy? 10

11 Lesson 3 Homework: Computing Unit Rate Find the unit rate of each: 1) If a runner ran 102 meters in 12 seconds, how many meters did he/she run per second? 2) Ticketmaster sold 1200 tickets to the Mets-Yankees game in 3 hours. How many tickets were sold is one hour? Which is the better bargain?? Find the unit price for each and compare them. 3) Pens: $4.50 for 3 pens or $3.20 for 20 pens 4) Pencils: 16 for $8.32 or 35 for $ ) DVD s: 4 for $79.96 or 5 for $ ) Lucy went away on vacation for 10 days and when she came home she had 280 s. How many s did she get per day? 7) Derek just got a new I-Phone and downloaded 348 songs in 6 hours. How many songs did he download per hour? 8) Ryan and his brother are comparing the prices of two brands of cereal. Frosted Flakes costs $2.25 for a 15- ounce box. Lucky charms costs $3.90 for a 30-ounce box. Which brand is more expensive and by how much per-ounce? 9) The table shows the prices that Mrs. Kurka paid at 3 different gas stations. Complete the table to determine which gas station had the better price per gallon. Gas Station Gallons Price Price per Gallon (Show work here) Hess 15 $43.50 Coastal 10 $29.40 Amoco 12 $

12 Lesson 4: Unit Rate with Complex Fractions/Conversion Factor Find the Unit Rate. 1) The swim team had their end of the year pizza party and consumed unit rate? pizza s in of an hour. What is the ) Mrs. Aronow is baking cookies for her Math team and of a cup of sugar for 1 batches of cookies. How 4 2 much sugar calls for one recipe? 3) John is in a 10 mile walkathon for breast cancer. He looked at his watch when he started walking- it was 7:02. After a half mile, he saw that it was 7:17. So this means John walks 2 1 mile in 15 minutes, which is 4 1 hour, how many miles would he walk in one hour? Conversion factor: A conversion factor is a rate that equals 1. For example, since 60 min= 1 h, both and equal 1. You can use as a conversion factor to change hours into minutes. 7 h = = 420 min Divide the common unit, hours (h). The result is in minutes. The table shows some common conversion factors for converting between the metric system and the customary system. Example: Convert 15 inches to centimeters. 15 in = Use since conversions is to centimeters. = (15)(2.54)cm Simplify (the inches cancel) = 38.1 cm LENGTH 1 in = 2.54 cm 1 km = 0.62 mi CAPACITY 1 L = 1.06 qt WEIGHT AND MASS 1 oz = 28g 1 kg = 2.2 lb Use conversion factors to convert each measure: Round to the nearest tenth if necessary. 4) 22 in cm 5) 26.4 lb kg 6) 20.5 oz g 7) 500 g oz 8) 5km mi 9) 20 L qt 12

13 Lesson 4 Homework: Unit Rate with Complex Fractions/Conversion Factor 1) Jamie was painting a large room. It took her 2 cans of paint to paint of the room. How many cans of paint will she need to use to paint the entire room? 2) The soccer team had their end of the year pizza party and consumed unit rate? pizzas in of an hour. What is the 4 2 Convert the following measures using a conversion factor. Round each to the nearest tenth if necessary. 3) 16.5 oz g 4) 19 in cm 5) 28.4 mi km 6) 32 cm in LENGTH 1 in = 2.54 cm 1 km = 0.62 mi CAPACITY 1 L = 1.06 qt WEIGHT AND MASS 1 oz = 28g 1 kg = 2.2 lb 7) 3.7 L qt 8) 77 g oz 9) 62 kg lb 10) 9.4 qt L 11) 100 lb kg 12) 32 mi km 13

14 Lesson 5: Currency Exchange Scenario: A 13 year old girl is going on a trip around the world and decided to buy a pair of Ugg boots everywhere she went. Help her convert her U.S. dollars into the country s currency that she is visiting. The table shows the exchange rates for certain countries compared to the U.S. dollar: $165 United States Dollars Country Rate United Kingdom Egypt Australia China U S Foreign = U S Foreign United Kingdom Egypt Australia China 1) If Mary went to the casino in China and won 1242 Yen. How much is that worth in US dollars? 2) You won the lottery! The amount you receive is $30,000. a) How much is this worth in Egypt? b) How much is this worth in the United Kingdom? 3) Egyptian currency is called a pound. How many pounds is $1, equivalent to? 14

15 Lesson 5 : Currency Exchange HOMEWORK ITEM PRICE US Dollar 1) $329 USD United States Dollars 2) United Kingdom Pounds $200 USD United States Dollars 3) MXP Mexican Pesos 1, FRF France Francs 1 United States Dollar (USD) United Kingdom Pounds (GBP) 1 United States Dollar (USD) Mexican Pesos (MXN) 1 United States Dollar (USD) 4.35 France Francs (FRF) USD United States Dollars Draw a picture and set up a proportion to solve. 4) A 6 foot tall tree has a shadow of 8 feet. How long is the shadow of a house that is 48 feet tall? 1 On a map of the United States, the scale is inch 10mile. Set up proportions to complete the following: 2 5) The measure from Albany to Buffalo was 6.5 inches. How many miles is it from Albany to Buffalo? Review 6) Evaluate: ) Simplify. 6x + 9y 3x 11y 8) Solve for x. 2(x + 4) = 12 9) Factor completely. 4x 12 15

16 Lesson 6 - Problem Solving with Similar Figures You can use proportional relationships to find missing side lengths in similar figures. Fill in the blank with the appropriate word, phrase, or symbol to make a true statement. 1. Similar figures have the same but not necessarily the same. 2. The symbol means "is similar to". 3. A drawing is an enlarged or reduced drawing that is similar to an actual object or place. 4. In similar triangles, corresponding are congruent and corresponding are in proportion. Find the missing side lengths in each pair of similar figures. 1. ABC ~ XYZ 2. ABC ~ XYZ A X A X x 25 6 Z 8 10 Y 5 50 Z 20 x Y C B C 40 B 3. ABC ~ XYZ 4. ABC ~ XYZ A X A X 16 C Z y B 15 Y x 14.4 B 5 Y C 16

17 Use similar triangles to find the missing information. (Draw a picture to help ) 7. A giraffe is 18 feet tall and casts a shadow of 12 feet. Corry casts a shadow of 4 feet. How tall is Corry? 8. When a Ferris wheel casts a 20-meter shadow, a man 1.8 meters tall casts a 2.4-meter shadow. How tall is the Ferris wheel? 9. A flagpole casts a shadow 28 feet long. A person standing nearby casts a shadow eight feet long. If the person is six feet tall, how tall is the flagpole? 10. A photograph measuring four inches wide and five inches long is enlarged to make a wall mural. If the mural is 120 inches wide, how long is the mural? 11. A 9-foot ladder leans against a building six feet above the ground. At what height would a 15-foot ladder touch the building if both ladders form the same angle with the ground? 12. Chris wants to reduce a triangular pattern with sides 16, 16 and 20 centimeters. If the longest side of the new pattern is to be 15 cm, how long should the other two sides be? 17

18 Lesson 6 - Problem Solving with Similar Figures Homework Find the indicated length for each pair of similar figures. 1) 4 inches 7.5 inches x 3.2 inches 2) 12 cm 3 cm x 20 cm 3) Two rectangles are similar. The first is 4 in. wide and 15 in. long. The second is 9 in. wide. Find the length of the second rectangle. 4) Which of the following are similar to Figure X? (there may be more than one) a) b) c) Figure X d) ) Brandon want to reduce a figure that is 9 inches tall and 16 inches wide so that it will fit on a 9-inch by 12- inch piece of paper. If he reduces the figure proportionally, what is the maximum size the reduced figure could measure? A. 12 inches by 1 21 inches B. 9 inches by 12 inches 3 C. 1 5 inches by 9 inches D inches by 12 inches 4 Review 6) Factor: 2x + 6 7) Simplify: 8(x + 8) 2x 8) Simplify: (4x + 20)

19 Lesson 7 Scale Drawing Vocabulary Scale Drawing a proportional representation of an object. Scale The constant ratio of each actual length to its corresponding length in the drawing. This scale can be expressed as the scale factor. ( new over original ) Drawing Geometric Figures To draw geometric figures to scale, use your knowledge of similar figures and proportional relationships. The ratios of corresponding side lengths of similar figures are equal to the scale factor, so the scale factor indicates how much larger or smaller to make each side length in the scale drawing. Scaling of geometric figures preserves angle measures, so the corresponding angles are congruent. Determine if the following figure will get bigger or smaller if the scale factor is: 2 5 1) scale factor is 3 2) scale factor is 3) scale factor is 4) scale factor is Example First: Draw a rectangle that is 3 units with and 4 units high. 2 Next: Draw another rectangle using a scale factor of. 1 New width: New height: What is the area of the original rectangle? square units What is the area of the new rectangle? square units How does the ratio of their areas relate to the scale factor? ( ) 2 **If two figures are similar, the ratio of their areas is the square of the scale factor. The triangles shown below are similar figures. What is the ratio of the area of the scale triangle to the area of the original triangle? Original D New New Original = Scale factor is: Ratio of areas is: A 6cm C 9 cm 8 cm B E 12 cm F 19

20 Practice Directions: For questions 1 through 3, make a scale drawing of the figure using the given scale factor. Then write the ratio of the areas. 1. scale factor: 2 1 ratio of areas: 2. scale factor: scale factor: 3 5 ratio of areas: ratio of areas: ratio of areas: 20

21 Lesson 7 Scale Drawing Homework 1) Look at the triangles below a) What is the area of the smaller triangle? b) What is the area of the larger triangle? c) What is the ratio of area of the larger triangle to the smaller triangle? 2) Make a scale drawing of the figure using the given scale factor. Then write the ratio of the areas. scale factor: 2 Area of new figure Area of original Remember New over Original Ratio of areas 3) scale factor: 3 1 Area of new figure Area of original figure Ratio of areas 21

22 Lesson 8 Using Scales on Maps Maps are common examples of scale drawings. The distances on a map are proportional to the actual distances. Using the map distances, find the actual distance if the map uses the scale 4 in: 30 mi inches 2. 7 inches inches inches Example On the following map, what is the approximate distance from Martin to Greensburg? (The scale shows that 1 inch = 30 miles.) Measure the distance from Martin to Greensburg on the map inches 2 5) The distance from Martin to Greensburg is about miles. 6) The distance from Cleary to Clarkson is about miles. 22

23 The figure below is a scale drawing of a playhouse. In the drawing, the side of each square represents two and a half feet. Find the actual length of each segment. 7. The width of the house. 8. The height of the house. 9. The height of the first floor. 10. The height of the roof. 11. The width of the door. 12. The height of the door. 13. Alex made a scale drawing of his apartment so he could figure out how to rearrange the furniture. The scale in the drawing below is 1 cm: 3 ft. What is the actual length of Alex s bedroom? 23

24 Lesson 8 Using Scales on Maps Homework 1. Zainab drew an accurate map showing her house and her friend Cassidy s house. The scale on the map is centimeter = mile. If the actual distance from her house to Cassidy s house is 2 miles, what is the map 2 2 distance, in centimeters? Directions: Use the floor plan below to answer questions 9 through 14. Use a ruler to measure the dimensions. Scale: 1 inch = 8 feet 9. What is the actual length of the kitchen? 10. What is the actual width of the kitchen? 11. What is the actual area of the kitchen in square feet? 12. What is the actual length of the master bedroom, including the bath and closet? 13. What is the actual width of the master bedroom? Directions: Use the figure below to answer questions 7 and 8. Scale Figure (NEW) Original Figure 10 ft 6 ft A B. 9 9 ft What is the scale factor for the parallelograms? C. 5 3 D ft 16. What is the ratio of the area of the figure to the area of the original figure? 25 A. 9 4 B. 9 3 C. 5 2 D. 3 24