549
http://www.illustrativemathematics.org/illustrations/549
The grocery store sells beans in bulk. The grocer's sign above the
beans says,
5 pounds for $4.
At this store, you can buy any number of pounds of beans at this
same rate, and all prices include tax.
Alberto said, “The ratio of the number of dollars to the number of
pounds is 4:5. That's $0.80 per pound.”
Beth said, "The sign says the ratio of the number of pounds to the
number of dollars is 5:4. That's 1.25 pounds per dollar."
1. Are Alberto and Beth both correct? Explain.
2. Claude needs two pounds of beans to make soup. Show
Claude how much money he will need.
3. Dora has $10 and wants to stock up on beans. Show Dora
how many pounds of beans she can buy.
4. Do you prefer to answer parts (b) and (c) using Alberto's rate
of $0.80 per pound, using Beth's rate of 1.25 pounds per
dollar, or using another strategy? Explain.
114
http://www.illustrativemathematics.org/illustrations/114
Four different stores are having a sale. The signs below show the
discounts available at each of the four stores.
 Buy one and get 25%
 Two for the price of one
off the second
 Buy two and get 50% off the
 Three for the price of
second one
two
1. Which of these four different offers gives the biggest
percentage price reduction? Explain your reasoning clearly.
2. Which of these four different offers gives the smallest
percentage price reduction? Explain your reasoning clearly.
135
http://www.illustrativemathematics.org/illustrations/135
Alexis needs to paint the four exterior walls of a large rectangular
barn. The length of the barn is 80 feet, the width is 50 feet, and the
height is 30 feet. The paint costs $28 per gallon, and each gallon
covers 420 square feet. How much will it cost Alexis to paint the
barn? Explain your work.
107
http://www.illustrativemathematics.org/illustrations/107
Mariko has an 80:1 scale-drawing of the floor plan of her house.
On the floor plan, the dimensions of her rectangular living room
are 1 7/8 inches by 2 1/2 inches.
What is the area of her real living room in square feet?
470
http://www.illustrativemathematics.org/illustrations/470
Travis was attempting to make muffins to take to a neighbor that
had just moved in down the street. The recipe that he was working
with required 3/4 cup of sugar and 1/8 cup of butter.
Travis accidentally put a whole cup of butter in the mix.
a. What is the ratio of sugar to butter in the original recipe?
What amount of sugar does Travis need to put into the mix to
have the same ratio of sugar to butter that the original recipe
calls for?
b. If Travis wants to keep the ratios the same as they are in the
original recipe, how will the amounts of all the other
ingredients for this new mixture compare to the amounts for
a single batch of muffins?
c. The original recipe called for 3/8 cup of blueberries. What is
the ratio of blueberries to butter in the recipe? How many
cups of blueberries are needed in the new enlarged mixture?
86
http://www.illustrativemathematics.org/illustrations/86
Nia and Trey both had a sore throat so their mom told them to
gargle with warm salt water.
 Nia mixed 1 teaspoon salt with 3 cups water.
 Trey mixed 1/2 teaspoon salt with 1 1/2 cups of water.
Nia tasted Trey’s salt water. She said,
“I added more salt so I expected that mine would be more salty,
but they taste the same.”
1. Explain why the salt-water mixtures taste the same.
2. Find an equation that relates s, the number of teaspoons of
salt, with w, the number of cups of water, for both of these
mixtures.
3. Draw the graph of your equation from part b.
4. Your graph in part c should be a line. Interpret the slope as a
unit rate.
633
http://www.illustrativemathematics.org/illustrations/633
Antonio and Juan are in a 4-mile bike race. The graph below
shows the distance of each racer (in miles) as a function of time (in
minutes).
1. Who wins the race? How do you know?
2. Imagine you were watching the race and had to announce it
over the radio, write a little story describing the race.
578
http://www.illustrativemathematics.org/illustrations/578
In order to gain popularity among students, a new pizza place near
school plans to offer a special promotion. The cost of a large pizza
(in dollars) at the pizza place as a function of time (measured in
days since February 10th) may be described as
1.
2.
3.
4.
5.
(Assume t only takes whole number values.)
If you want to give their pizza a try, on what date(s) should you
buy a large pizza in order to get the best price?
How much will a large pizza cost on Feb. 18th?
On what date, if any, will a large pizza cost 13 dollars?
Write an expression that describes the sentence "The cost of a
large pizza is at least A dollars B days into the promotion," using
function notation and mathematical symbols only.
Calculate C(9) − C(8) and interpret its meaning in the context of
the problem.
629
http://www.illustrativemathematics.org/illustrations/629
In (a)–(e), say whether the quantity is changing in a linear or
exponential fashion.
1. A savings account, which earns no interest, receives a deposit
of $723 per month.
2. The value of a machine depreciates by 17% per year.
3. Every week, 9/10 of a radioactive substance remains from the
beginning of the week.
4. A liter of water evaporates from a swimming pool every day.
5. Every 124 minutes, 1/2 of a drug dosage remains in the body.
645
http://www.illustrativemathematics.org/illustrations/645
The population of a country is initially 2 million people and is
increasing at 4% per year. The country's annual food supply is
initially adequate for 4 million people and is increasing at a
constant rate adequate for an additional 0.5 million people per
year.
1. Based on these assumptions, in approximately what year will
this country first experience shortages of food?
2. If the country doubled its initial food supply and maintained
a constant rate of increase in the supply adequate for an
additional 0.5 million people per year, would shortages still
occur? In approximately which year?
3. If the country doubled the rate at which its food supply
increases, in addition to doubling its initial food supply,
would shortages still occur?
231
http://www.illustrativemathematics.org/illustrations/231
According to the U.S. Energy Information Administration, a barrel
of crude oil produces approximately 20 gallons of gasoline. EPA
mileage estimates indicate a 2011 Ford Focus averages 28 miles
per gallon of gasoline.
1. Write an expression for G(x), the number of gallons of
gasoline produced by x barrels of crude oil.
2. Write an expression for M(g), the number of miles on
average that a 2011 Ford Focus can drive on g gallons of
gasoline.
3. Write an expression for M(G(x)). What does M(G(x))
represent in terms of the context?
4. One estimate (from www.oilvoice.com) claimed that the
2010 Deepwater Horizon disaster in the Gulf of Mexico
spilled 4.9 million barrels of crude oil. How many miles of
Ford Focus driving would this spilled oil fuel?
387
http://www.illustrativemathematics.org/illustrations/387
John makes DVDs of his friend’s shows. He has realized that,
because of his fixed costs, his average cost per DVD depends on
the number of DVDs he produces. The cost of producing x DVDs
is given by
C(x)=2500+1.25x.
1. Suppose John made 100 DVDs. What is the cost of
producing this many DVDs? How much is this per DVD?
2. Complete the table showing his costs at different levels of
production.
3. Explain why the average cost per DVD levels off.
4. Find an equation for the average cost per DVD of producing
x DVDs.
5. Find the domain of the average cost function.
6. Using the data points from your table above, sketch the graph
of the average cost function. How does the graph reflect that
the average cost levels off?
134
http://www.illustrativemathematics.org/illustrations/134
Joe was planning a business trip to Canada, so he went to the bank
to exchange $200 U.S. dollars for Canadian (CDN) dollars (at a
rate of $1.02 CDN per $1 US). On the way home from the bank,
Joe’s boss called to say that the destination of the trip had changed
to Mexico City. Joe went back to the bank to exchange his
Canadian dollars for Mexican pesos (at a rate of 10.8 pesos per $1
CDN). How many Mexican pesos did Joe get?
595
http://www.illustrativemathematics.org/illustrations/595
A wheel of radius 0.2 meters begins to move along a flat surface so
that the center of the wheel moves forward at a constant speed of
2.4 meters per second. At the moment the wheel begins to turn, a
marked point P on the wheel is touching the flat surface.
1. Write an algebraic expression for the function y that gives the
height (in meters) of the point P, measured from the flat
surface, as a function of t, the number of seconds after the
wheel begins moving.
2. Sketch a graph of the function y for t > 0. What do you notice
about the graph? Explain your observations in terms of the
real-world context given in this problem.
3. Write an algebraic expression for the function x that gives the
horizontal position (in meters) of the point P as a function of
t, the number of seconds after the wheel begins moving.
4. Sketch a graph of the function x for t > 0. Is there a time
when the point P is moving backwards? Use your graph to
justify your answer.