CCM6+ Unit 9 Packet 2017-18 – Page 1
UNIT 9 Ratios, Rates, Proportions, and Measurement Conversions
CCM6+ 2017-2018 Name: ___________________________ Math Teacher: _____________________ Estimated Quiz Date: ________________ Estimated Test Date: ________________ Mini-Me Project Due Date: ____________ Topic Unit Overview Unit 9 Vocabulary Daily Warm-Ups Ratios Unit Rates Ratio and Rate Reasoning Proportions/Constant of Proportionality Converting Measurement Mini-Me Project Instructions (minor grade) Unit 9 Study Guide 1
Page # 2 3 4 5-8 9-12 13-31 32-43 44-48 49-50 51-53
CCM6+ Unit 9 Packet 2017-18 – Page 2
CCM6+ Unit 9 Overview Understandings
Essential Questions
Students will understand that…
the concept of a ratio and the different ways ratios can be written. the concept of a unit rate and that a unit rate is a comparison to one unit. you can solve real-world problems by reasoning about tables of equivalent ratios, tape diagrams, double number lines or equations. ratio reasoning can be used to convert measurement units two equivalent ratios form a proportional relationship. the constant of proportionality can be displayed in an equation, table, or graph. a proportional relationship can be represented using an equation, graph or table. a proportional relationship can be graphed and that the graph will be a straight line through the origin.
How can you use ratios to make predictions? What are the benefits of using models to represent and solve ratio problems? How is the constant of proportionality used to solve real world problems with graphs, equations, and tables? How can you determine proportional relationships from a graph, table or equation? What is the importance of a unit rate? How can finding the "best buy" make you a savvy shopper? How can ratio and rate reasoning help you to solve real world problems?
Knowledge
Skills
Students will know…
Students will be able to…
the three ways to write a ratio and what each represents. the difference between a part to part ratio and a part to whole ratio. the difference between a rate and a unit rate. how to use the unit rate to find the “best buy”. how to use ratio and rate reasoning to solve real-world problems. how to recognize and represent proportional relationships. that all fractions are ratios but that not all ratios are fractions. that measurements can be converted between units. how constant of proportionality is related to a unit rate. how to determine if ratios form a proportion. what a proportional relationship looks like in a table, graph, and equation.
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describe a relationship between two quantities. write a ratio three different ways. simplify ratios. find equivalent ratios and make tables of equivalent ratios. find missing values in tables. use tables to compare ratios. calculate and compare unit rates associated with ratios of fractions. convert units of measurement in the metric and customary systems. decide whether two quantities represent a proportional relationship. identify the constant of proportionality in tables, graphs, and equations. represent a proportional relationship with an equation, graph, table. explain what a point (x,y) on the graph of a proportional relationship means in terms of the situation. exhibit the eight Mathematical Practice Standards.
CCM6+ Unit 9 Packet 2017-18 – Page 3
CCM6+ Unit 9 Vocabulary Vocabulary Word Double Number Lines Equivalent ratios
Definition A number line with a scale on top and a different scale on the bottom so that you can organize and compare items that change regularly according to a rule or pattern.
Part-to-Part
Ratios that name the same comparison A relationship between one part of the whole and another part of the whole
Part-to-Whole
A relationship between one part of the whole and the whole
Ratio
Tape Diagram
A comparison of two quantities using division A drawing that looks like a segment of tape, used to illustrate number relationships; also known as a strip diagram, bar model, fraction strip, or length model
Unit Rate
A rate in which the second quantity in the comparison is one unit
MATH 6 PLUS ADDITIONAL VOCABULARY
Constant of Proportionality
A complex fraction is a fraction where the numerator, denominator, or both contain a fraction The constant value of the ratio of two proportional quantities x and y; usually written y = kx, where k is the factor of proportionality
Proportions
Two ratios (or fractions) are equal
Proportional Relationships
A relationship between two equal ratios
Complex Fractions
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Daily Warm-Ups 1. What is the value for the shaded part of the fraction bars? What are all the possible ways of expressing the model?
2. Use the table to answer the following questions
a. What is the ratio of boys to girls in room 207? b. What is the ratio of girls to all students in room 162? c. What is the ratio of girls to boys in room 417? 3. Mary is making a painting in art class. She cannot find the right color purple so she decides to make her own. She knows that to make the color she is looking for she needs 2 ounces of red paint and 3 ounces of blue paint. If she needs 20 ounces total, how many ounces of red paint and how many ounces of blue paint will she need? Use a ratio table or double number line diagram to solve. 4.A sixth grade class is running a 5k race. The class will begin practicing to increase their endurance, starting with 1,500 meters and adding 500 meters each week. How many weeks will it take to be ready for the race? 5. A sixth grade class is running a three-mile race. The class will begin practicing to increase their endurance, starting with 880 yards and adding 440 yards per week. How many weeks will it take to be ready for the race? 6.Zuri, the baby elephant, was born August 10, 2009 at Hogle Zoo. The calf weighed 251 lbs. at birth. If the baby elephant gains 48 ounces a day, how much will she weigh at the end of 7 days? 7.Carey and her friends want to make matching outfits for twin day at school. Her mom has 23 yards of cloth. If each outfit requires 3 ¼ yards of material, how many outfits can her mom make? Use a double number line to solve.
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RATIOS NOTES Big Ideas Ratio
3 ways to write a ratio
Notes
Examples
A ratio is a comparison of two things in There are 2 peanut butter sandwiches the same unit. for every 3 turkey sandwiches.
1. as a _______________
1.
2. with a ________ (
2.
)
3. with the word “_____” part-to-part ratio
3.
Each number is a _______ of the entire a) In a gumball machine there are only red and purple gumballs. If there are 20 gumballs in set. the machine and 12 are red, what is the ratio of purple to red?
**Your final answer should b) What is the ratio of white to dark gumballs?
be _______________.
Why are these part-to-part?
part-towhole ratio
This is always a ________ compared to the entire set. **Your final answer should be _______________. What is the ratio of cats to all animals?
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RATES and UNIT RATES Key Word
Definition
Examples a) The car went 120 miles on 4 gallons.
A rate is a ___________ comparing two things that are different ____________.
Rates have to be _______________, too!
b) To make 24 cookies you need 2 eggs in the recipe.
Rate ____ per ____:
𝑓𝑖𝑟𝑠𝑡 𝑖𝑠 𝑜𝑛 𝑡𝑜𝑝 𝑠𝑒𝑐𝑜𝑛𝑑 𝑖𝑠 𝑜𝑛 𝑏𝑜𝑡𝑡𝑜𝑚
= Why are these examples RATES not RATIOS?
first # ÷ second #
A unit rate is a special kind of rate where
Unit Rate
a) Dear Aunt Sally drove 84 miles in 3 hours. What was her average speed in miles per hour?
you _____________ to get a number of this unit per _________ of that unit.
b) The grocery store is selling a 6-pack of Hershey bars for $3.94. What is the price per bar?
Unit Rates help you find the ________ ________ at the grocery store. c) Food Lion sells a dozen donuts for $3.60. Currently they are on sale for half-price. If your unit rate is a price, it is called a
What is the normal price per donut?
___________ ___________. This is the _______ per ____ of the items. Unit Price = _______ ÷ the # of items **Use a calculator to divide!**
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What is the sale price per donut?
CCM6+ Unit 9 Packet 2017-18 – Page 10
Practice with Unit Rates
MATHSCAPE BUYER BEWARE What’s the Best Buy? Hot Words: Unit Price: _______________________________________________________________________ Rate: ____________________________________________________________________________ Shoppers need to be able to calculate unit prices to find the best buy. In this lesson you will compare various-sized packages of cookies made by the same company to decide which size is the best buy. Then, you will compare two different brands of chocolate chip cookies to decide which one gives you more cookie for your money. The Buyer Beware Consumer Research Group has collected data on chocolate chip cookies. They want you to find out which package of Choco Chippies is the best buy. To find the best buy, you need to find the unit price, or the price per cookie, for each size package. You can find the price of one cookie in a package if you know the total amount of cookies in the package and the price of the package.
Snack Size
Regular Size
Family Size
Giant Size
Use the chart on the next page to answer the following questions.
1. Use a calculator to figure out the price per cookie for each package. Round your answers to the nearest cent. 2. Decide which package size is the best buy. Explain how you figured it out. 3. List the different Choco Chippies package sizes in order from best buy to worst buy. 10
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Choco Chippies Prices Number of Cookies
Package Price
Snack
4
$0.50
Regular
17
$1.39
Family
46
$3.99
Giant
72
$5.29
Package Size
Price Per Cookie
How did you figure out the order of best to worst prices of cookies?
Did this surprise you? Why or why not? .
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ORDER Best to Worst
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Unit Rates and Better Buy PRACTICE Use your calculator to find the unit price for each of the following. Round answers to the nearest cent. 1. 7 oz of crackers for $1.19 $ per cracker 2.
14 oz of cottage cheese for $1.19
$
per ounce
3.
16 boxes of raisins for $5.60
$
per box
Find the better buy based on unit price. 4. A 35-oz. can of Best Brand Plum Tomatoes is on sale for $0.69. A 4-lb can of Sun Ripe Plum Tomatoes is $1.88.
5. A can of Favorite Dog Food holds 14 oz. Four cans are $1.00. The price of three cans of Delight Beef Dog Food, each containing 12 oz, is $0.58.
6. For each item in the chart below, calculate the unit price and circle the better buy. Item Jefferson Auto Stores Tom’s Auto Parts
a.
b.
c.
oil
anti-freeze
auto wax
12 qt for $10.99
6 qt for $5.99
$______________ per quart
$______________ per quart
12 oz for $3.79
6 oz for $1.79
$______________ per ounce
$______________ per ounce
6 cans for $14.29
5 cans for $12.98
$_______________ per can
$_______________ per can
7. Six cans of fruit drink are on sale for $1.95. Individually, the price of each can is $0.35. How much does Tanya save buying 6 cans on sale?
8. Tubes of oil paint can be bought in sets of 5 for $13.75 or bought separately for the unit price. What would be the price of two tubes of this oil paint?
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MATHSCAPE BUYER BEWARE The Best Snack Bar Bargain You can use a price graph to compare unit prices for different products. In this lesson you will use a price graph to determine the price at different quantities of a snack bar if you were paying by the ounce. Then you will construct a price graph to compare the prices of five different products. Use a Price Graph to Find Unit Price The graph below shows the prices for three different snack bars: Mercury bars are $1.00 for 2 oz. Jupiter bars are $2.98 for 3.5 oz. Saturn bars are $3.50 for 4.5 oz. Each of the three dots on the graph shows the price and the number of ounces for one of the snack bars. Each line shows the price of different quantities of the snack bar at the same price per ounce.
Price per ounce $4.00
Key:
$3.50
Mercury bars
$3.00
Jupiter bars
$2.50
Saturn bars
$2.00 $1.50 $1.00 $0.50 $0.00 0
1. What is the price of a 3-oz Mercury bar?
1
2
3
4
Ounces
2. What is the price of a 0.5-oz Saturn bar? 3. Which snack bar has the lowest unit price?________________ Highest unit price? _____________ How do you know this by just l
king at the graph?
4. What are the unit prices for each bar? Hint: Find the price per ONE OUNCE. Mercury: $____________/oz Jupiter: $____________/oz Saturn: $____________/oz 13
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RELATING TABLES AND GRAPHS Oatmeal
Tuna
Penne Pasta
Sourdough Pretzels
Whole Wheat Rolls
1 oz 3 oz 7.5 oz
$2.00
8 oz
$1.00
9 oz
$1.60
12 oz 14 oz
$2.60 $2.40
Graph the information onto your own price graph. Use different colors or styles of lines for each item. KEY: PRICE GRAPH _____ = Oatmeal $6.00 _____ = Tuna _____ = Penne Pasta _____ = Sourdough Pretzels $5.00 _____ = Whole Wheat Rolls $4.00
Price in $ $3.00
$2.00
$1.00
$0.00 1 2
3 4
5 6 7
14
8 9 10 11 12 13 14 15 16 17 18 Ounces
CCM6+ Unit 9 Packet 2017-18 – Page 15 1. Graph the prices per ounces shown in the chart above. For each item, use a different color or style and connect the point created to the origin (0 oz, is $0). 2. Using your graph, estimate the missing values above in the chart. 3. The most expensive product per ounce is _______________ because __________________________.
4. The least expensive product per ounce is ________________ because __________________________. 5. Besides looking at your price graph, what is another way you could have used the information in the table to fill in the missing values in the chart?
6. Using your price graph, about how much would you pay for 6 ounces of sourdough pretzels?
7. Using your price table, about how much would you pay for 5 ounces of whole wheat rolls? (Hint: look at the price per one ounce and then use that to calculate the price per five ounces.)
8. Crabtree Middle School is holding a fundraiser dinner for 2,000 people. If the students need to cook 4 ounces of pasta per person, how much will it cost to buy enough penne pasta for the fundraiser? Hint: First find the cost for 4 ounces….that’s for one person….you need it for 2,000 people!
9. While shopping for oatmeal, Daphne’s mom finds another brand of oatmeal on sale for $2.65 for 16 ounces. a. Which oatmeal is the better buy? b. How much will she save buying the better buy instead of the other brand (for 16 ounces)? SHOW YOUR WORK!
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Ratio Tables 1. My lemonade has a ratio of 2 lemons to 3 cups of water. If I want to have 20 cups of lemonade, how many lemons do I need? Lemons 2 ?
Total 5 20
Using the ratio table, I can see looking at the whole that 5 * 4 = 20, so to find how many lemons I need I would do 2 * 4 = __________. So I need __________ lemons for 20 cups of lemonade.
2.
3.
The SPCA must keep a 2:5 ratio of cats to dogs. If they have 12 cats, how many dogs should they have? Write the answer in the chart. Cats Dogs 5
12
?
The school band needs a 5:4 ratio of flutes to clarinets. If there are 27 students total who play the flute or clarinet, how many play the flute? Write your answer in the chart. Flute
4.
2
Clarinet
Total
Megan charges $30 for 3 hours of swimming lessons. How much does Megan charge for 2 hours of swimming lessons? Write your answer in the chart. Hours
Cost
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Ratio Task
a.
John, Marie, and Will all ran for 6th grade class president. Of the 36
students, 16 voted for John, 12 for Marie, and 8 for Will. What was the ratio of votes for John to votes for Will? What was the ratio of votes for Marie to votes for Will? What was the ratio of votes for Marie to votes for John?
b. Because no one got half the votes, they had to have a run-off election. Marie dropped out and convinced all her voters to vote for Will. What is the new ratio of Will's votes to John's?
c. John and Will also ran for Middle School Council President. There are 90 students voting in middle school. If the ratio of Will's votes to John's votes remains the same as it was in part (b), how many more votes will Will get than John?
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Ratio word problems 1) The perfect purple paint is made by mixing 3 cups of blue paint to every 2 cups of red paint. If you have 45 cups of blue paint, how much red paint will you need?
3)Lemonade is made by mixing sugar water and lemons in a ratio of 5 to 4. How much of each ingredient is needed to make 72 cups of lemonade?
5)An ad claims that hair salons that recommend Dove Shampoo out number the hair salons that don’t by a ratio of 5 to 3. If 264 hair salons were interviewed, how many recommended Dove Shampoo?
2)When making cookies, you use 4 cups of sugar and 6 eggs. If you had 12 cups of sugar, how many eggs would you need?
4)In Mr. Ronaldo’s class, there are 5 boys for every 2 girls. If there are 28 students total, how many more boys are in his class than girls?
6)Carl is selling bags of Boy Scout popcorn. For every 20 bags he sells, he raises $15 for charity. If Carl has a goal of raising $60, how many bags of popcorn does he need to sell?
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7)Foot Locker ordered new Nikes and Adidas shoes in a ratio of 5:2. The store has 14 Adidas shoes in stock. How many Nikes and Adidas shoes are in the store?
9)The ratio of baseballs to gloves at a sporting goods store is 5 to 3. If the store has 100 baseballs in stock, how many gloves does the store have in stock?
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8)In a video game for every 6 enemies defeated, you earn 3 points. If you defeated 48 enemies, how many points would you earn?
10)Tucker is helping his father mow lawns. He can mow 3 lawns in 25 minutes. How long will it take Tucker to mow 12 lawns?
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11)In the Media Center, 8 fiction novels are checked out for every 5 nonfiction novels. How many nonfiction novels are checked out when there are 32 fiction novels checked out?
12)Abdul can ride his bike 3 miles in 15 minutes. At this rate, how long will it take Abdul to ride 18 miles?
13)Jamal, Tony and Ashley score points in a basketball tournament in a ratio of 2:3:5. Tony scores 75 points. How many points do Jamal and Ashley score?
14)A plant grows at a constant rate of 3 feet every 48-months. At this rate, how much would a plant grow in 5 years?
15)Pranet and Katie ran for 6th grade class president. There were 36 students voting. Pranet got two votes for every vote Katie got. How many votes did each candidate receive?
16)The ratio of boys to girls at the basketball game was 9 to 6. If there were 45 boys, how many total students were at the game?
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Quilting Ratios A ratio is a comparison of one number to another by division. In this example you will use ratios to describe a design. Leticia is designing a quilt pattern that is made up of squares. One row of the quilt is shown below.
1. What is the ratio of black squares to white squares? ___________ Is this a part-to-part or a part-to-whole ratio? ________________ 2. What is the ratio of black squares to all the squares in the row? _____________ Is this a part-to-part or a part-to-whole ratio? ___________________ 3. Gino is designing a quilted placemat shown below. Find two different ratios within the design. Describe them using words and numbers, and classify them as part-to-part or part-to-whole.
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Double Number Lines and Ratios practice Write each ratio as a fraction in lowest terms. 1. 6 to 8
2. 8:44
3.
60
4. 20 to 30
32
Use double number lines to show equivalent ratios. Fill in the blanks then write the equivalent ratios. 5. 5:30
0
1
0 ____ 6. 12:15
0
0 7. 30:12
____
5
5
10
30
____
12
____
15
45
0
10
30
0
____
12
_____
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The data table below shows how some students spent their time from 4p.m. to 5p.m. yesterday. Decide if statements #9-12 are TRUE or FALSE. 9. One out of every three students did homework. How Students Spent Their Time # of Students Homework
10
Sports Practice
5
Music Practice
5
Chores or Job
6
Other
4
10. One out of every five students did chores. 11. The ratio of homework to music is 5 to 2. 12. The ratio of chores to music or sports is 2 to 3.
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CCM6+ Unit 9 Packet 2017-18 – Page 25 Ratio Practice Problems Using Double Number Lines Directions: Draw a double number line for each problem to help answer the question (extend the double number line if needed). Be sure to label what each bar represents. 1. There are 4 apples on each plate. If there are 6 plates, how many apples are there altogether?
2. There are 24 students in a classroom and 6 large round tables. How many children should be seated at each table if there must be the same number of children at each table?
3. 8 meters of wire length weigh 12 grams. How much will 1 meter of the same wire weigh?
Use Double Number Line Diagrams to work with Rates 4) A group of students is spending the day at the county carnival. Stores around the area are selling booklets of discounted carnival ride tickets. The stores’ ticket prices are as shown: Farm Fresh: 15 tickets for $12.50 Save-a-Lot: 20 tickets for $16.00 Mini-Mart: 25 tickets for $19.79 Which store offers the best deal? Justify your choice! 5) Karen and four friends had 175 tickets. After they each rode The Screamer five times, all of the tickets were gone. How many tickets does it cost for one person to ride The Screamer one time?
At this rate, how many tickets will it take for Karen to ride The Screamer five more times?
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CCM6+ Unit 9 Packet 2017-18 – Page 26 Practice Solving Ratios Using Tape Diagrams Directions: Draw a tape diagram for each problem to help answer the question. Be sure to label what each represents. Example: Emily created a new drink by mixing 2 parts melon juice with 3 parts orange juice. If Emily used 18 ounces of melon juice, how many ounces of orange juice would be needed? 18 Melon Juice Orange Juice ?
2 bars = 18 ounces, so 1 bar = 9 ounces. Orange juice has 3 bars so 3•9 = 27 ounces 1.
For every 5 calendars that Austin sold, Lara sold 4. Austin sold 45 calendars last month. How many calendars did Lara sell?
2. For every 3 cars that Kelly sells, Jane sells 4. Jane sold 32 cars last month. How many cars did Kelly sell?
3. The ratio of boys to girls who participated in the pie-eating contest was 7:2. There were 35 boys. How many girls participated?
4. The ratio of the length of Grace’s string to the length of Peter’s string is 5:6. Peter’s string measures 60 inches. How long is Grace’s string?
5. The ratio of blue marbles to yellow marbles is 5:6. If there are 42 yellow marbles how many blue marbles are there?
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6.
The ratio of trucks to minivans in the parking lot is 2:5. If there are 14 trucks, how many minivans are there?
7. Noah and Hunter shared some marshmallows in the ratio 3:2. If Hunter had 12 marshmallows, how many marshmallows did Noah have?
8. Megan created a new drink by mixing 2 parts guava juice with 5 parts mango juice. If Megan used 12 ounces of guava juice, how many ounces of mango juice would be needed?
9. Kara and Sam shared a cash prize in the ratio 3:7. If Sam received $77, how much money did Kara receive?
10. The ratio of the length of Mary’s wire to the length of Ayman’s wire is 5:6. Ayman’s wire measures 36 inches. How long is Mary’s wire?
11. The ratio of cats to dogs at the pet store is 5:6. If there are 121 total cats and dogs, how many cats and how many dogs are at the pet store?
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Ratios Review…getting into Proportions Write each ratio as a fraction in lowest terms. 1. 6 to 8
2. 8:44
3.
60
4. 20 to 30
32
Find equivalent ratios. Find a math relationship and USE IT!
5
.
5 30
=
6.
12 15
=
7.
30 12
=
1
5
60
= =
=
10
=
60
24
120
The data table below shows how some students spent their time from 4p.m. to 5p.m. yesterday. Decide if statements #8-11 are TRUE or FALSE by seeing if they are equal ratios. How Students Spent Their Time
8. One out of every three students did homework.
# of Students 9. One out of every five students did chores. Homework
10
Sports Practice
5
Music Practice
5
Chores or Job
6
Other
4
10. The ratio of homework to music is 5 to 2.
11. The ratio of chores to music or sports is 2 to 3.
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Part to Part Ratios using tape Diagrams….how are the proportions different? Make a Proportion (2 equal ratios).
12.For every 5 calendars that Austin sold, Lara sold 4. Austin sold 45 calendars last month. How many calendars did Lara sell?
13.For every 3 cars that Kelly sells, Jane sells 4. Jane sold 32 cars last month. How many cars did Kelly sell?
14.The ratio of boys to girls who participated in the pie-eating contest was 7:2. There were 35 boys. How many girls participated?
15.The ratio of the length of Grace’s string to the length of Peter’s string is 5:6. Peter’s string measures 60 inches. How long is Grace’s string?
16.The ratio of blue marbles to yellow marbles is 5:6. If there are 88 marbles how many blue marbles are there?
17. The ratio of trucks to minivans in the parking lot is 2:5. If there are 14 trucks, how many minivans are there?
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18.Noah and Hunter shared some marshmallows in the ratio 3:2. If together they have 40 marshmallows, how many did Noah have?
19.Megan created a new drink by mixing 2 parts guava juice with 5 parts mango juice. If Megan used 12 ounces of guava juice, how many ounces of mango juice would be needed?
20.Kara and Sam shared a cash prize in the ratio 3:7. If the cash prize was $90, how much money did each person receive?
21.The ratio of the length of Mary’s wire to the length of Ayman’s wire is 5:6. Ayman’s wire measures 36 inches. How long is Mary’s wire?
22.The ratio of cats to dogs at the pet store is 5:6. If there are 121 total cats and dogs, how many cats and how many dogs are at the pet store?
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Set up two equal ratios to find the missing number. It might help to first find a unit rate. 1. An advertisement claims that dentists that recommended the new zigzag toothbrush outnumber the dentists that don’t by a ratio of 5 to 3. If 264 dentists were interviewed, how many recommended the toothbrush?
2. A quart (32 ounces) container of yogurt contains 920 calories. About how many calories would there be in a 5-ounce serving?
3. The scale on a map is 3 inches equals 4 miles. How far is the actual distance between two towns that are 12 inches apart on the map?
4. Aniya’s class is selling wrapping paper. For every 5 rolls they sell, they make a profit of $1.80. If her class has a goal of making a $180 profit, how many rolls of wrapping paper do they need to sell?
5. A recipe requires
¼
lb of onions to make 3 servings of soup. Mark has many servings can Mark make?
1½
lb of onions. How
6. The ratio of nitrogen to potassium in a sample of soil is 12:9. The sample has 36 units of nitrogen. How much potassium does the sample have? a. 21 units b. 27 units c. 33 units d. 48 units 3
7. To clean a tank, 4cup of disinfectant is needed for every 2 gallons of water. How many cups of disinfectant are needed for 20 gallons of water? 1
a. 72
b. 15
1
c. 222 31
d. 3
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Solving Proportion Notes Let’s review first:
Ex 1.
2 6 and 7 21
Ex 2.
8 6 and 24 20
Ex 3.
2 6 and 8 20
Are these proportional? ***Cross multiply to see if equal products
SOLVING PROPORTIONS
Solve for the unknown variable:
How can you solve for the variable?
Ex 1.
More examples: Solve for the unknown variable.
Ex 1.
You can use proportions to solve real-world problems, when we cannot figure the unit rate easily.
Example: A stack of 18 Jenga blocks is about 25.8 cm tall. What is the height, to the nearest tenth of a centimeter, of a stack of 11 Jenga blocks?
Use the information to set up a proportion and solve.
Set up the proportion:
You try this one:
1. Set up the proportion
Carmen bought 3 pounds of bananas for $1.08. At this rate, how many pounds of bananas can she buy with $1.80?
p 10 6 3
25 20
and
Ex 2.
45 x
Ex 2.
45 15 x 3
n 10
18
= 25.8
and
28 8
Ex 3.
11 𝑥
= 2. Solve the proportion.
*The trick is to always correspond! 32
4 5 5 w
Ex 3.
9 x
and
57 19
CCM6+ Unit 9 Packet 2017-18 – Page 33
PROPORTIONS
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Constant of Proportionality Notes What is a constant of proportionality?
The constant value of the ratio of two proportional quantities.
How to identify the constant of proportionality?
You can identify the constant of proportionality in tables, graphs, equations and other proportional relationships.
Also is classified as the unit rate.
*Recall how to compute the unit rate. Use those same strategies to find the constant of proportionality. Example 1: Tables
Analyze the table. number of pens (p) Cost (C)
3
5
8
10
15
$6
$10
$16
$20
$30
What is the cost of 1 pen?
*The cost of 1 pen is $2. 2 is the constant of proportionality because it is the constant value of the ratio between the number of pens and the cost.
*The equation can be written as C = 2p, which represents the total cost (C) equals 2 dollars times the number of pens (p) purchased. Example 2: Graphs
Using the graph, determine the constant of proportionality.
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To determine the constant of proportionality, find the unit rate. To find the unit rate, look where the Length is 1 unit. What is the Lateral Surface Area when the Length is 1?
*4 is the constant of proportionality. If you follow the ratio, the constant is 4 because 1:4, 2:8, 3:12, and etc.
*The equation for this would be A = 4L meaning the area (A) equations 4 times the length (L). Example 3: Equations
Since we know that proportional equations contain only multiplication or division, use the coefficient to identify the constant of proportionality.
1. The amount of sales tax paid on an item is proportional to the cost of the item. If the sales tax rate is 7%, then the amount of the sales tax (t) is .07 times the cost (c) of the item. The equation is t = .07c can be used to determine the amount of sales tax. What is the constant of proportionality? *The constant is .07 or 7% since that is the coefficient of the equation. Example 4: Verbal Descriptions
1
In probability, the chance to roll a 1 when rolling a number cube is6. In the long run, the number of times you get a 1 is proportional to the number of times you roll. If you roll 30 times, you would expect to roll a 1 five times. The constant is
1 6
because it is the constant value of the ratio
when comparing the number how many 1s are on a number cube (1:6).
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CCM6+ Unit 9 Packet 2017-18 – Page 38 Constant of Proportionality Worksheet 1. Find the constant of proportionality from the table below. Show your work! X
7.5
10
17.5
20
Y
4.5
6
10.5
12
2. Find the constant of proportionality from the table below. Show your work! X
1.5
2
3.5
5
Y
10.5
14
24.5
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3. Find the constant of proportionality from the table below. Show your work! X
2
4
5
7
Y
1
2
2.5
3.5
4. Find the constant of proportionality from the table below. Show your work! X
2
3
5
6
Y
6
9
15
18
5. Find the constant of proportionality from the table below. Show your work! X
2
4
7
9
Y
0.4
0.8
1.4
1.8
6. Find the constant of proportionality from the table below. Show your work! X
1.5
3
4.5
12
Y
1
2
3
8
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The graph above represents one of the contestants’ data at a hot dog eating contest. Answer the following questions based on your knowledge of ratios and proportional relationships. 1. Does the graph represent a proportional relationship? How do you know? 2. What is the constant of proportionality? 3. What ordered pair on the graph makes the constant of proportionality easy to determine? 4. What does the ordered pair (0,0) represent in this graph? 5. What is an equation that would represent the relationship shown in the graph?
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MEASURES Within the CUSTOMARY SYSTEM
Customary Units of Length
Customary Units of Weight
1 foot (ft) = ________ inches (in)
1 pound (lb) = ________ ounces (oz)
1 yard (yd) = ________ ft = _______ yd
1 ton = ______ lbs = __________ oz
1 mile (mi) = ________ ft = _______ yd What tricks have you learned in the past to help you remember these?
Customary Units of Capacity 1 cup (c) = _____ fluid ounces (fl oz) 1 pint (pt) = _____c = _____fl oz 1 quart (qt) = _____ pt = ____ c = _____ fl oz 1 gallon (gal) = _____ qt = ____ pt = ____ c =_______fl oz
12 in = 1 ft 3 ft = 1 yd
16 oz = 1 lb 2000 lb = 1 t
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2 c = 1 pt 2 pt = 1 qt 4 qt = 1 gal
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Set up two equal ratios and find the missing piece. Use the conversion charts below. METRIC to CUSTOMARY:
CUSTOMARY to METRIC:
Now, let’s try some: a) 5 inches = _________cm *We are converting from inches to cm so use the chart on the _____. 1 𝑖𝑛𝑐ℎ 2.54 𝑐𝑚
b) 8 km = ________mi
=
𝑥
*We are converting from km to mi so use the chart on the _________. 1 𝑘𝑚
= 0.621 𝑚𝑖
c) 18 g = ________ oz
5 𝑖𝑛𝑐ℎ𝑒𝑠
8 𝑘𝑚 𝑥
*Use the chart to the ______________. _______ = _______
d) 3.5 qt = ______L
*Use the chart to the ______________. _______=_______
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More conversion practice
1) 16 in = ____________cm
2) 345 lb = __________kg
3) 56 g = ___________oz
4) 450 km = ____________mi
5) 1200 mL =__________fl oz
6) 40 m = ____________ft
7) Penny has a pencil that is 19 cm long. How long is this pencil in inches?
8) A cookie recipe (1 batch) calls for 1 lb of butter. How many grams of butter would be in 3 batches?
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CCM6+ Unit 9 STUDY GUIDE Ratios—remember, they MUST BE SIMPLIFIED!
1.
Using the shapes above, create a part-to-part ratio. Write it with numbers and words.
2.
Using the shapes above, create a part-to-whole ratio. Write it with numbers and words.
3.
If the ratio of boys to girls is 3:2 and there are 25 students in a class, make equal ratios to show how many students in the class are boys and how many are girls.
4.
If 4 out of every 9 jelly beans is orange and the package contains 108 jelly beans, how many are orange? Make equal ratios to solve this.
Rates/Unit Rates/Proportional Reasoning 5.
What is the unit rate if it takes 4 minutes to eat 8 slugs? (# slugs per minute)
6.
Which is the better buy—3 videos for $40.00 or 5 videos for $68?
7.
In a waffles recipe, it takes 2 eggs for every 24 waffles. If you need to make 60 waffles for a big party, how many eggs will it take? Make equal ratios to solve this problem. It might help to first find a unit rate.
8. It takes Munchy Marvin 3 days to eat a box of cereal. How many boxes of cereal will he eat in 12 days?
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Proportions and Constant of Proportionality In questions 9 and 10 use the tables to pull ratios and determine if the sets are proportional to each other. 9.
10.
x
2
3
5
8
12
x
0
3
4
8
12
y
1
1.5
2.5
4
6
y
2
5
6
10
14
Proportional? ___________
Proportional? ___________
Proportional? Explain and show proof: If proportional, what is the constant of proportionality? ______
and show proof: IfProportional? proportional, what isExplain the constant of proportionality? _____
Explain:
Explain:
In the next set of ratios, determine if they are proportional using what you know about equivalent ratios 11.
10 20 and 12 32
12.
6 9 and 10 15
13.
110 22 and 120 24
The following set of ratios are proportional. Find the value of the variable. 16.
7 x = 20 10
17.
1 2 = 9 1 x 22
6 19.
20.
6 18 = 18 x 30 d = 25 32
18.
3.2 : 8 and x : 32
21.
75 m = 25 60
2
22.Fill in all missing values in the ratio table to the right.
x 2 y
23. What is the constant of proportionality in the graph below? k = ________
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6 12 20
30
65 80
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Measurement 24. In the space to the right, draw gallon guy/gallon land. 25. Use gallon guy to answer these questions: a) 8 qt = _________ pts b) 4 gal = _________ qts c) 8 cups = _________pts 26. 1 yard = _________ feet, and 1 foot = _____________ inches, so 1 yard = ____________ inches 27. 1 pound = _____________ ounces, so in 80 ounces there are ___________ pounds. 28. In the space below draw the memory tool to order the metric prefixes (Hint: King Henry….)
29. Convert these metric measures using the tool above: a)
7 mm = ___________cm
b) 8 kg = ____________g
c) 4.5 cm = _______m
For the problems below, use the charts to the right of each problem. (It doesn’t matter which chart you use!) If necessary, round decimals to the nearest hundredth. 30. 18 in = ___________cm
31. 20 mL = __________fl oz
32. 25 kg = ___________lbs
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