D 0DNLQJ3XQFK Ratios, Rates, and Mixture Problems

Transcription

0DNLQJ3XQFK
Ratios, Rates, and
Mixture Problems
Learning Goals
In this lesson, you will:
f Use ratios to make comparisons.
f Use rates and proportions to solve mixture problems.
D
o you like smoothies? Perhaps one of the best things about smoothies is that
you can make one with just about any ingredients. Just throw them in the blender
and turn it on!
Smoothies can be very healthy too. Try this healthy smoothie recipe sometime.
●
1 banana
●
1 cup of vanilla yogurt
●
1 cup of grapes
●
1
__
●
2 cups of spinach leaves
2
of an apple
If this recipe serves 3 people, how much of each ingredient would you need to
© 2011 Carnegie Learning
make smoothies your whole class?
H
1.2
Ratios, Rates, and Mixture Problems

15
Problem 1
Four recipes for grapefruit
punch are given. Students write
ratios that compare the number
of parts of grapefruit juice to
the total number of parts in
each of the four given recipes.
Next, they will write ratios that
compare the number of parts
of lemon-lime soda to the total
number of parts in each recipe.
Problem 1
Each year, your class presents its mathematics portfolio to parents and community
members. This year, your homeroom is in charge of the refreshments for the reception that
follows the presentations. Four students in the class give their recipes for punch. The
class wants to analyze the recipes to determine which will make the punch with the
strongest grapefruit flavor, and which will make the strongest lemon-lime soda flavor.
The recipes are shown.
Bobbi’s Recipe
Adam’s Recipe
Grouping
t Ask a student to read the
4 parts lemon-lime soda
8 parts grapefruit juice
Zeb’s Recipe
Carlos’s Recipe
information and recipes aloud.
Complete Question 1 as
a class.
t
May the Best Recipe Win
1 part lemon-lime soda
1. How many total parts are in each person’s recipe?
Have students complete
Questions 2 through 5 with
a partner. Then share the
responses as a class.
Adam’s recipe: 12 total parts; Bobbi’s recipe: 8 total parts; Carlos’s recipe: 5 total
parts; Zeb’s recipe: 5 total parts
2. For each recipe, write a ratio that compares the number of parts of grapefruit juice to
the total number of parts in each recipe. If possible, simplify each rate.
8 parts grapefruit juice 2 parts grapefruit juice
Adam’s recipe: _____________________ 5 _____________________
12 total parts
3 total parts
Bobbi’s recipe: _____________________
8 total parts
Carlos’s recipe: _____________________
5 total parts
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Chapter 1
Ratios and Rates
Zeb’s recipe: _____________________
5 total parts
Share Phase,
Questions 2 through 5
t Who’s recipe contains the
3. Which recipe will make the punch with the strongest grapefruit taste? Explain how
most number of parts?
you determined your answer.
t
Who’s recipe contains the
least number of parts?
t
Can any of the ratios be
reduced? Which ones?
96
80 ; __
5 5 ____
75 ; __
3 5 ____
4 5 ____
72 ; __
2 5 ____
Sample Answer: __
3 120 8 120 5 120 5 120
5, and __
3 , Zeb’s recipe will have the strongest taste
4 is greater than __
2, __
Because __
5
5
3 8
of grapefruit flavor.
t
Are these ratios considered
part-to-part or part-to-whole
ratios? Explain.
t
Which recipe will have the
weakest taste of grapefruit?
Explain.
t
Which recipe will have the
weakest taste of lemon-lime
soda? Explain.
First, I had to scale up all the rates to rates with a common denominator. Then,
I was able to determine which recipe would have the strongest grapefruit flavor.
4. For each recipe, write a rate that compares the number of parts of lemon-lime soda to
the total number of parts in each recipe. If possible, simplify each rate.
4 parts lemon-lime soda 1 part lemon-lime soda
Adam’s recipe: _______________________ 5 ______________________
12 total parts
3 total parts
Bobbi’s recipe: _______________________
8 total parts
Carlos’s recipe: _______________________
5 total parts
1 part lemon-lime soda
Zeb’s recipe: ______________________
5 total parts
5. Which recipe will make the punch with the strongest lemon-lime soda flavor?
Explain how you determined your answer.
48 ; __
40 ; __
3 5 ____
45 ; __
24
2 5 ____
1 5 ____
1 5 ____
Sample Answer: __
3 120 8 120 5 120 5 120
3, and __
1 , Carlos’s recipe will have the strongest taste
1, __
2 is greater than __
Because __
5
5
3 8
of lemon-lime soda flavor.
First, I had to scale up all the rates to rates with a common denominator. Then,
I was able to determine which recipe would have the strongest lemon-lime
soda flavor.
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17
Problem 2
Each glass of punch contains 6
fluid ounces. One hundred sixty
people will attend the reception.
Students use proportions
to determine unit rates and
numbers of fluid ounces of
punch are needed for each part
if they were to use each recipe
for punch in Problem 1. They
will then organize their data in
a table and answer questions
related to the data.
Problem 2
Making the Refreshments
1. You are borrowing glasses from the cafeteria to serve the punch. Each glass holds
6 fluid ounces of punch. Your class expects that 70 students and 90 parents and
community members will attend the reception. You decide to make enough punch so
that every person who attends can have one glass of punch. How many fluid ounces
of punch will you need for the reception?
Number of people attending: 70 1 90 5 160
Number of fluid ounces: 160(6) 5 960
I will need 960 fluid ounces of punch for the reception.
Previously, you wrote rates to compare parts of each ingredient to total parts of all the
ingredients. Recall that a rate is a ratio in which the units of the parts or the whole being
Grouping
compared are different.
Questions 1 through 7 with a
partner. Then have students
share their responses as a class.
2. Determine the unit rate for the fluid ounces of punch there would be in one part of the
recipe if your class uses Adam’s recipe.
80 fluid ounces
960 fluid ounces 5 _______________
________________
1 part
12 parts
There will be 80 fluid ounces of punch for each part of the recipe.
3. How many fluid ounces of lemon-lime soda and grapefruit juice are needed to make
enough punch if your class uses Adam’s recipe? Show all your work.
Number of fluid ounces of lemon-lime soda:
x fluid ounces
80 fluid ounces 5 _____________
_______________
1 part
4 parts
(80)(4) 5 320
x 5 320
Number of fluid ounces of grapefruit juice:
x fluid ounces
80 fluid ounces 5 _____________
_______________
1 part
8 parts
(80)(8) 5 640
x 5 640
I will need 320 ounces of lemon-lime soda and 640 ounces of grapefruit juice to
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Chapter 1
Ratios and Rates
make enough punch if my class uses Adam’s recipes.
Share Phase,
t How did you determine the
number of fluid ounces of
punch needed?
t
t
t
t
enough punch if your class uses Bobbi’s recipe? Show all your work.
Number of fluid ounces in one part of recipe:
120 fluid ounces
960 fluid ounces 5 ________________
________________
1 part
8 parts
How did you determine the
unit rate for the number of
fluid ounces of punch there
would be in one part of the
recipe?
x fluid ounces
120 fluid ounces 5 _____________
________________
1 part
3 parts
What is 960 divided by 12?
Are you scaling up or scaling
down to solve the proportions
in this problem scenario?
How did you determine the
amount of lemon-lime soda
needed in each recipe?
(120)(3) 5 360
x 5 360
x fluid ounces
120 fluid ounces 5 _____________
________________
1 part
5 parts
(120)(5) 5 600
x 5 600
make enough punch if my class uses Bobbi’s recipe.
enough punch if your class uses Carlos’s recipe? Show all your work.
192 fluid ounces
960 fluid ounces 5 ________________
________________
5 parts
1 part
x fluid ounces
192 fluid ounces 5 _____________
________________
1 part
2 parts
(192)(2) 5 384
x 5 384
x fluid ounces
192 fluid ounces 5 _____________
________________
1 part
3 parts
(192)(3) 5 576
x 5 576
make enough punch if my class uses Carlos’s recipe.
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19
Share Phase,
Questions 6 and 7
t How did you determine the
enough punch for the reception if your class uses Zeb’s recipe? Show all your work.
amount of grapefruit juice
needed in each recipe?
192 fluid ounces
960 fluid ounces 5 ________________
________________
t Using the data in the table,
5 parts
1 part
which recipe has the weakest
taste of grapefruit?
x fluid ounces
192 fluid ounces 5 _____________
________________
1 part
t Using the data in the table,
1 part
192(1) 5 192
x 5 192
which recipe has the weakest
taste of lemon-lime?
x fluid ounces
192 fluid ounces 5 _____________
________________
1 part
4 parts
192(4) 5 768
x 5 768
I will need 192 ounces of lemon-lime soda and 768 ounces of grapefruit juice
to make enough punch if my class uses Zeb’s recipe.
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Chapter 1
Ratios and Rates
Amount of
Lemon-Lime
Soda
(fluid ounces)
Amount of
Grapefruit
Juice
(fluid ounces)
Total Amount
of Punch
(fluid ounces)
Adam’s recipe
320
640
960
Bobbi’s recipe
360
600
960
Carlos’s recipe
384
576
960
Zeb’s recipe
192
768
960
7. Complete the table with the calculations you determined for each person’s recipe.
Grouping
Questions 8 through 11 with
a partner. Then share the
responses as a class.
Share Phase,
t If the recipe has the weakest
taste of grapefruit, does that
mean it has the strongest
taste of lemon-lime soda?
t
t
t
What would the punch taste
like if there were equal parts
of grapefruit juice and lemonlime soda used in the recipe?
If you put more grapefruit juice
in the punch, does that result
in a stronger grapefruit taste?
If you put more lemon-lime
soda in the punch, does that
result in a stronger lemonlime soda taste?
grapefruit flavor? How does the table confirm your choice?
Zeb’s recipe will have the strongest grapefruit flavor. His recipe uses the most
grapefruit juice, 768 fluid ounces, out of the total 960 fluid ounces.
9. In Problem 1, Question 5 you determined which recipe would have the strongest
lemon-lime soda flavor? How does the table confirm your choice?
Carlos’s recipe will have the strongest lemon-lime soda flavor. His recipe has the
most lemon-lime soda, 384 fluid ounces, out of the total 960 fluid ounces.
10. If you would use 8-ounce glasses for the reception rather than 6-ounce glasses, how
would that affect the amount of punch you would need to make?
There are 160 people attending.
Number of total fluid ounces: (160)(8) 5 1280
1280 2 960 5 320
To use 8-ounce glasses rather than 6-ounce glasses would require 320 more fluid
1 more punch.
ounces of punch, or about __
3
11. Will the ratio of the parts for any of the recipes change by putting more punch in each
glass? Explain your reasoning.
The ratio of parts of any recipe will remain the same.
t
If the recipe has the weakest
taste of lemon-lime soda,
does that mean it has
the strongest taste of
grapefruit juice?
8. In Problem 1, Question 3 you determined which recipe would have the strongest
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21
Talk the Talk
Students explain how using
ratios and rates helped them
solve the problems in this
lesson.
Talk the Talk
1. Explain how ratios and rates helped you solve the problems in this lesson.
The ratios in Problem 1, Questions 1 and 2, indicated which punch tasted more like
grapefruit and which punch tasted more like lemon-lime soda. Rates were used to
Grouping
determine how many fluid ounces of lemon-lime soda and grapefruit juice were
needed to make each recipe.
Question 1 with a partner. Then
share the responses as a class.
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Chapter 1
Ratios and Rates
Be prepared to share your solutions and methods.