(7) Lesson 1.6 - Solve Proportional Relationships

1. Course 2, Lesson 1-6
1. The amount a cashier earns is shown in the table.
Determine whether the amount earned is proportional to the
number of hours worked by graphing on the coordinate
plane. Explain your reasoning.
2. An online music club charges a $10 enrollment
fee and $2 per month. Does the graph show a
proportional relationship. Explain your reasoning.
Number of Hours 0 1 2 3
Earnings ($) 0 8 16 24

2. Course 2, Lesson 1-6
ANSWERS
1. proportional; The graph is a
straight line that passes through
the origin.
2. The graph is a straight line but
does not pass through the origin. So, the relationship
is not proportional.

3. HOW can you show that
two objects are proportional?
Ratios and Proportional Relationships
Course 2, Lesson 1-6

4. Ratios and Proportional Relationships
Course 2, Lesson 1-6 Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices and Council
of Chief State School Officers. All rights reserved.
• 7.RP.2
Recognize and represent proportional relationships between quantities.
• 7.RP.2b
Identify the constant of proportionality (unit rate) in tables, graphs,
equations, diagrams, and verbal descriptions of proportional relationships.
• 7.RP.2c
Represent proportional relationships by equations.
• 7.RP.3
Use proportional relationships to solve multistep ratio and percent
problems. Examples: simple interest, tax, markups and markdowns,
gratuities and commissions, fees, percent increase and decrease, percent
error.

5. Ratios and Proportional Relationships
Course 2, Lesson 1-6 Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices and Council
of Chief State School Officers. All rights reserved.
Mathematical Practices
1 Make sense of problems and persevere in solving them.
2 Reason abstractly and quantitatively.
3 Construct viable arguments and critique the reasoning of others.
4 Model with mathematics.

6. • To write and solve proportions
• To use unit rates to write equations
Ratios and Proportional Relationships
Course 2, Lesson 1-6

7. Ratios and Proportional Relationships
Course 2, Lesson 1-6
• proportion
• cross products

8. Ratios and Proportional Relationships
Course 2, Lesson 1-6
Words A proportion is an equation stating that two ratios or rates
are equivalent.
Numbers Algebra
6 3
8 4
 , 0, 0
a c
b d
b d
  

9. 1
Need Another Example?
2
Find the cross products.3
4 Multiply
Divide each side by 7.5
6 Simplify.
Step-by-Step Example
1. After 2 hours, the air temperature had risen 7°F.
Write and solve a proportion to find the amount
of time it will take at this rate for the temperature
to rise an additional 13°F.
Write a proportion. Let t represent the time in hours.
temperature →
time →
=
← temperature
← time
7 • t = 2 • 13
7t = 26
t ≈ 3.7
7 It will take about 3.7 hours to rise an additional 13°F.

10. Answer
Need Another Example?
Solve = .
12

11. 1
Need Another Example?
2
3
4 Find the cross products.
Multiply5
6 Divide each side by 80.
Step-by-Step Example
2. If the ratio of Type O to non-Type O donors at a blood drive was
37:43, how many donors would be Type O, out of 300 donors?
Type O donors →
total donors →
Write a proportion. Let t represent the number of Type O donors.
Type O donors →
total donors →
← Type O donors
← total donors
37 • 300 = 80t
11,100 = 80t
138.75 = t Simplify.7
There would be about 139 Type O donors.

12. Answer
Need Another Example?
A recipe serves 8 people and calls for 3 cups
of flour. If you want to make the recipe for 14
people, about how many cups of flour should
you use?
; about 5 c

13. 1
Need Another Example?
2
3
4
Let c represent the cost. Let y represent the number of yogurts.
Replace y with 10.
5
6
Multiply.
Step-by-Step Example
3. Olivia bought 6 containers of yogurt for $7.68.
Write an equation relating the cost c to the number
of yogurts y. How much would Olivia pay for 10 yogurts
at this same rate?
Find the unit rate between cost and containers of yogurt.
= or $1.28 per container
The cost is $1.28 times the number of containers of yogurt.
c = 1.28y
= 1.28(10)
= 12.80
The cost for 10 containers of yogurt is $12.80.

14. Answer
Need Another Example?
Melvin purchased eight T-shirts for $64.
Write an equation relating the cost to the
number of T-shirts. How much would it cost
to purchase ten T-shirts?
c = 8t; $80

15. 1
Need Another Example?
2
3
4
Let c represent the cost. Let g represent the number of gallons.
Replace g with 11.
5
6
Multiply.
Step-by-Step Example
4. Jaycee bought 8 gallons of gas for $31.12. Write
an equation relating the cost c to the number of
gallons g of gas. How much would Jaycee pay for
11 gallons at this same rate?
Find the unit rate between cost and gallons.
= or $3.89 per gallon
The cost is $3.89 times the number of gallons.
c = 3.89g
= 3.89(11)
= 42.79
The cost for 11 gallons of gas is $42.79.

16. Answer
Need Another Example?
Haley bought 4 pounds of tomatoes for
$11.96. Write an equation relating the cost
to the number of pounds of tomatoes. How
much would Haley pay for 6 pounds at this
same rate? for 10 pounds?
c = 2.99p; $17.94; $29.90

17. How did what you learned
today help you answer the
HOW can you show that
two objects are proportional?
Course 2, Lesson 1-6
Ratios and Proportional Relationships

18. How did what you learned
today help you answer the
HOW can you show that
two objects are proportional?
Course 2, Lesson 1-6
Ratios and Proportional Relationships
Sample answers:
• The cross products of any proportion are equal.
• You can use unit rates to write an equation expressing
the relationship between two proportional quantities.

19. Explain how to
solve a proportion
to determine an
unknown value.
Ratios and Proportional RelationshipsRatios and Proportional Relationships
Course 2, Lesson 1-6