Name Class Date 1-8
Rates, Ratios, and Proportions
Extension: Dimensional Analysis
Essential question: How can you use units to help solve real-world problems?
Unit Analysis When evaluating expressions that represent real-world
situations, you should pay attention to the units of measurement
attached to the parts of the expression. For instance, if p people go
to a restaurant and agree to split the $50 cost of the meal equally,
50  ​are dollars, the
then the units in the numerator of the expression ​ __
p
units in the denominator are people, and the units for the value of the
expression are dollars per person.
Video Tutor
CC.9–12.A.SSE.1
1
EXAMPLE
Evaluating Real-World Expressions
A Sheila is participating in a multi-day bike trip. On the first day, she rode
100 miles in 8 hours. Use the expression __
​ dt  ​where d is the distance traveled
and t is the travel time to find her average rate of travel. Include units when
evaluating the expression.
d  ​= _____________
​ __
​ 
  
   ​=
t
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B If Sheila continues riding at her average rate for the first day, then the
expression 100 + 12.5t gives the total distance that she has traveled after
riding for t hours on the second day. Evaluate this expression when t = 7,
and include units.
100 + 12.5t = 100
+ 12.5
·
=
REFLECT
1a. What are the terms in the expression 100 + 12.5t? What does each term represent
in the context of Sheila’s bike trip?
1b. What is the coefficient of the term 12.5t? What does it represent in the context
of Sheila’s bike trip?
Chapter 1
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Lesson 8
1c. If you write only the units for the expression 100 + 12.5t, you get
mi + ___
​ mi   ​· h where “mi” is the abbreviation for miles and “h” is the
h
abbreviation for hours. Explain what the following unit analysis shows:
mi   ​· h = mi + mi = mi
mi + ​ ___
h
1d. How can you modify the expression 100 + 12.5t so that the units are feet when the
expression is evaluated?
CC.9–12.N.Q.1
2
EXAMPLE
Using Unit Analysis to Guide Modeling
Lizzie has volunteered 20 hours at her town library. From now on, she plans to
volunteer 5 hours per week at the library. Write an algebraic expression to represent
the total number of hours she will volunteer.
A
Use unit analysis to help you get the correct units for the expression.
hours
hours
+ _________
​ 
  
 
​·
week
Write a verbal model.
C
Choose a variable for the unknown quantity.
D
Let
represent the
hours
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B
=
.
Write an algebraic expression to represent the situation.
REFLECT
2a. Explain why you chose the units you chose in Part A.
Chapter 1
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Lesson 8
2b. How many hours will Lizzie have volunteered at the library by the end of 10 weeks?
2c. Lizzie has also volunteered 10 hours at an animal shelter and she plans
to volunteer there for 3 hours a week beginning in 1 week. Rewrite the
algebraic expression you wrote in Part D above based on this new
information. (Assume that the number of weeks is at least 1.) Simplify,
if possible.
2d. How many hours will Lizzie have volunteered at the library and the animal shelter
combined by the end of 20 weeks?
practice
1. Henry drives in town at a rate of 25 miles per hour. It takes him 15 minutes
to go to the library from his house. The algebraic expression rt represents
distance traveled, where r is the average rate (in miles per hour) and t is the
travel time (in hours).
© Houghton Mifflin Harcourt Publishing Company
a.Can you multiply 25 and 15 to find the distance Henry traveled to
the library? Explain.
b.Show how to find the distance from Henry’s house to the library.
Include units in your calculation.
2. Sarah works 4 hours her first week of a part-time job and earns $60.
Her total pay after the second week can be represented by the
p
expression 60 + __
​ t  ​· s where p represents her pay for t hours of work
and s represents the hours she works in the second week.
a.What are the units of the fraction?
b.Rewrite the expression substituting the given values for p and t.
What are the units of each term of your new expression? Explain.
c.Evaluate your expression for s = 5. Include units.
Chapter 1
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Lesson 8
3. To convert dog years to human years, you count 10.5 dog years per human year
for the first two human years and then 4 dog years per human year for each
human year thereafter.
a.Show how to use unit analysis to get the correct units when you convert dog
years to human years.
b.Write and simplify an algebraic expression for converting dog years to
human years when the number of human years is 2 or more. Define what
the variable represents.
4. Tracie buys tickets to a concert for herself and two friends. There is an 8% tax
on the cost of the tickets and an additional $10 booking fee. Write an algebraic
expression to represent the cost per person. Simplify the expression, if possible.
Define what the variable represents and identify the units for the expression.
5. Write two different algebraic expressions that could represent the phrase
“a number plus 2 times the number.” Then rewrite the phrase so that only
one of the algebraic expressions could be correct.
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Chapter 1
50
Lesson 8
1-8
Name
Name ________________________________________
Class Date __________________
Date Class __________________
Practice
Additional
Practice
Rates, Ratios,
and Proportions
1. Julia drove 135 miles in 4.5 hours. Find her average
rate in miles per hour.
___________________________
Find the average rate.
2. Four pounds of apples cost $1.96.
3. Sal washed 5 cars in 50 minutes.
_________________________________________
________________________________________
4. A giraffe can run 32 miles per hour. What is this
speed in feet per second? Round your answer
to the nearest tenth.
___________________________
Use unit analysis to write an algebraic expression to represent the
situation.
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5. Billie and Toni are driving from Chicago to Phoenix. The first
day they drove 420 miles. They want to drive at 65 miles per hour
for the rest of their trip.
a. Write an algebraic expression to represent the situation.
__________________
b. If they drive 8 hours a day for the next two days, how far
have they traveled by the end of the third day?
__________________
6. Frank ordered playoff tickets for himself and three friends.
There is a $7 service fee per ticket and an additional shipping
cost of $12 for the entire ticket order.
a. Write an algebraic expression to represent the situation.
__________________
b. Write an algebraic expression to represent the ticket
cost per person.
__________________
7. Sam is building a model of an antique car. The scale of his
1
model to the actual car is 1:10. His model is 18 inches long.
2
How long is the actual car in feet? Round your answer to the
nearest tenth of a foot.
______________________
8. The scale on a map of Virginia shows that 1 centimeter
represents 30 miles. The actual distance from Richmond,
VA to Washington, DC is 110 miles. On the map, how many
centimeters are between the two cities? Round your answer
to the nearest tenth.
______________________
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content
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Chapter
1 Copyright © by Holt McDougal. Additions and changes to51
Lesson 8
Holt McDougal Algebra 1
Name ________________________________________ Date __________________ Class __________________
Problem
Solving
Problem
Solving
Rates, Ratios, and Proportions
Write the correct answer.
2. Sally volunteers at the youth center
14 hours per week. If Sally volunteers
4 days a week, what is her average rate
in hours per day?
1. A donut shop bakes 4 dozen donuts
every 18 minutes. Find the average rate
to the nearest hundredth.
_________________________________________
3. The birth rate in Namibia is 35 babies to
every 1000 people. In 2001, the country
had a population of about 1,800,000
people. How many babies were there?
_________________________________________
4. A boat travels 160 miles in 5 hours.
What is its speed in miles per minute
to the nearest hundredth?
_________________________________________
_________________________________________
The Hendersons are driving cross country from Boston to Los
Angeles. The first day they drove 300 miles. They plan to drive at an
average speed of 60 miles per hour for the rest of the trip so that the
trip will be completed in less than one week.
A 300t + 60
C 60t + 300
B 300 − 60t
D 60t − 300
6. What are the units in the expression from
Problem 5?
F hours
H days
G miles
J weeks
F 4 days
G 5 days
H 6 days
J 7 days
7. If the Hendersons drive for 9 hours per
day, how many miles will they travel in a
day?
A 300 miles
C 60 miles
B 9t miles
D 540 miles
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
Chapter 1
52
Lesson 8
Holt McDougal Algebra 1
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8. The distance from Boston to Los Angeles
is approximately 3000 miles. How many
days should the Henderson’s trip take if
they drive 9 hours per day after the first
day?
5. Write an algebraic expression to
represent the length of the total trip.