E-book

The Ivy 147 Problem Solving and Data Analysis

Number of people who voted19. October 2018 US The scatterplot above shows the number of registered voters,
150 x, and the number of people who voted in the last election, y,
140 for seven districts in a town. A line of best fit for the data is
130 also shown. Which of the following could be the equation of
120 the line of best fit?
110
100 a) y = –0.5x
90 b) y = 0.5x
80 c) y = –2x
70 d) y = 2x
60120 140 160 180 200 220 240 260 280 300
Number of registered votes

20. March 2019 US The scatterplot above shows eight data points in the xy-plane. A
line of best fit is also shown for the data. If each data point is
y shifted 3 units upward and a new line of best fit for the shifted
9 points is drawn, how will the value of the y-intercept of the new
8 line compare with that of the line shown?
7
6 a) It will increase.
5 b) It will decrease.
4 c) It will remain the same.
3 d) There will no longer be a y-intercept.
2
1
00 1 2 3 4 5 6 7 x

21. March 2019 US (milCliitornuss opfromdeutrctiicotnons) The scatterplot above shows the citrus production, in millions of
metric tons, in China from 2006 through 2014. Which of the
Citrus Production in China 2 6–2 14 following could be the slope of a line of best fit for these data?
4305
32210505 a) 2.12
150 b) 5.25
0 0 2 Years4since 20606 8 c) 7.80
d) 10.29

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Problem Solving and Data AnalysisTemperature (°Re) 148 The Ivy

22. December 2016Num rbeetruorfnesdurveys The graph above shows the function r that relates the
Relationship between temperature r(x), measured in degree Reaumur, and the
temperature x, measured in degrees Fahrenheit. If A(122, 40) is a
Degrees Reaumur and Degree Fahrenheit point on the graph of r, which of the following must be true?
80
60 a) A temperature of 122 degree Reaumur corresponds to a
40 A temperature of 40 degrees Fahrenheit.
20
0 0 50Tempe1r0a0ture (°1F5)0 200 b) A temperature of 40 degree Reaumur corresponds to a
temperature of 122 degrees Fahrenheit.
23. May 2017 US
ReturneSdurfvroemys1S7enNtetiogahnbdorhoods c) For every increase of 40 degrees Fahrenheit, there is an
increase of 122 degrees Reaumur.
876500000000
43210000000000 20N0um4b0e0r o6f0s0urv8ey00s s1e,n0t00 d) For every increase of 40 degrees Reaumur, there is an
increase of 122 degrees Fahrenheit.

The scatterplot above shows the number of surveys sent to and
returned from people in 17 different neighborhoods. A line of
best fit for the data is also shown. For the neighborhood that had
surveys sent to 800 people, which of the following is closest to
the positive difference between the actual number of surveys
returned and the number predicted by the line of best fit shown?

a) 150
b) 170
c) 200
d) 250

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The Ivy 149 Problem Solving and Data Analysis

Questions 24 and 25 refer to the following information.
A researcher is investigating the relationship between the price of a four-pack of AA batteries at a convenience store and the number
of competitors the store has. She defines a store’s competitor as another similar store within a 1-mile radius of the store she selects.
She selects 32 convenience stores across a state at random, and for each store, she records the number of its competitors and its price
for a four-pack of AA batteries. The results are shown, along with the line of best fit, in the scatterplot below.

y
6.0

Price (dollars) 5.0

4.0

3.0 0 1 2 3 4 5 6 x
Number of competitors

24. April 2017 25. April 2017
Another convenience store in the same state sells a four-pack The line of best fit passes through the point (18, –0.12).
of AA batteries for $4.89. If the store’s price is more than that Which of the following can be concluded from this?
predicted by the line of best fit, what is the least number of
competitors the store could have? a) The line of best fit will not model the price well for
a store with a large number of competitors.
a) 1
b) 2 b) A convenience store with 17 competitors can no
c) 3 longer sell four-packs of AA batteries.
d) 4
c) A convenience store with 17 competitors cannot
decrease its price any further.

d) A convenience store cannot have more than 17
competitors.

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Problem Solving and Data Analysis 150 The Ivy

Questions 26 and 27 refer to the following information.Percwietnhtacgaebloeftehloeuvsiseihoolnds
A cable company recorded the percentage of households in the United States that had cable television from 1987 to 1997. In the
scatterplot below, x represents the number of years since 1987 and y represents the percentage of households with cable television.
The line of best fit for the data is shown.

y CHaobulseehToelldesviwsiiothn
766666555506206408482 0Num2ber o4f yea6rs sin8ce 119087 x

26. April 2017 27. April 2017
Which of the following is the best interpretation of the slope
Which of the following is closest to the equation of the line of of the line of best fit shown for these data?

best fit shown? 7 a) The actual increase in the percentage of household
5 with cable television each year
a) y = 54x +
b) The predicted increase in the percentage of
b) y = 7 x – 54 households with cable television each year
5
7 c) The actual increase in the number of households
c) y = 5 x + 54 with cable television each year

d) y = 7 x d) The predicted increase in the number of household
5 with cable television each year

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The Ivy 151 Problem Solving and Data Analysis

Questions 28 and 29 refer to the following information.

1.0Annual population
growth rate (percent)
0.5

0.0 0 5 10 15 20
–0.5

Years after 1990

The graph above shows the percent of annual population growth rate, r, in a certain country for the number of years after 1990, t, at
5-year intervals. A linear model fitting the plotted points is also shown. The equation for the linear model is r = –0.07t + 1.06.

28. October 2016 29. October 2016
Based in the model, which of the following is closest to the The actual annual population growth rate in the country in
year in which the predicted annual population growth rate of 1995 was 0.9%. For the year 1995, what is the actual annual
the country is –1%? (Oct 2016) population growth rate minus the annual population growth
rate predicted by the model? (Oct 2016)
a) 2010
b) 2015 a) –0.07%
c) 2020 b) 0.19%
d) 2025 c) 0.67%
d) 1.06%

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The Ivy 152 Problem Solving and Data Analysis

Data Inference

1. 3. October 2016
A quality control researcher at an electronics company is Researcher in Australia carried out an experiment to
testing the life of the company’s batteries in a certain determine if the color of a coffee mug affects how people
camera. The researcher selects 100 batteries at random from rate the flavor intensity of the coffee. Volunteers were
the daily output of the batteries and finds that the average randomly assigned to taste coffee in mugs that differed only
life of the batteries has a 95% confidence interval of 324 to by color: some white and some clear. The same type of
360 camera pictures. Which of the following conclusions is coffee was used in both mugs. The researchers concluded
the most reasonable based on the confidence interval? that the mean flavor intensity rating was significantly higher
for those who drank coffee in a white mug than for those
a) 95% of all the batteries produced by the company who drank coffee in a clear mug. Based on this study, which
that day have a life between 324 and 360 pictures. of the following statements is correct?

b) 95% of all the batteries ever produced by the a) The color of the mug was the cause of the
company have a life between 324 and 360 pictures. difference in mean intensity rating for these
volunteers, and this conclusion can be generalized
c) It is plausible that the true average life of batteries to all coffee drinkers.
produced by the company that day is between 324
and 360 pictures. b) The color of the mug was the cause of the
difference in mean intensity rating for these
d) It is plausible that the true average life of all volunteers, but it is not reasonable to generalize
batteries ever produced by the company is between this conclusion to all coffee drinkers.
324 and 360 pictures.
c) It is not reasonable to conclude that the color of the
2. June 2016 mug was the cause of the difference in mean
A random sample of 100 Popcorn haven consumers was intensity rating for these volunteers.
selected to taste Popcorn Haven’s new caramel and
chocolate covered popcorn. Of these consumers, 95 disliked d) It is not possible to draw any conclusion from this
the taste of the new popcorn flavor. Which of the following experiment because volunteers were used.
is the best inference to make from the results?

a) It is likely that most people who have never eaten
Popcorn Haven popcorn would enjoy the new
popcorn flavor.

b) It is likely that most people who have eaten
Popcorn Haven popcorn would dislike the new
popcorn flavor.

c) On average, 95% of all popcorn eaters would
dislike the new population flavor.

d) Of all the Popcorn Haven popcorn eaters, exactly
95% of them would dislike the new popcorn
flavor.

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Problem Solving and Data Analysis 153 The Ivy

4. April 2017 6. May 2017 US
To determine whether residents of a community would vote To determine if cooking with olive oil reduces the risk of
in favor of a ballot proposal to use $100,000 of local taxes heartburn for men, researchers interviewed a random sample
for additional playground equipment at a community park, of 5,500 men who had no history of heartburn. Study
Jennifer surveyed 60 adults visiting the park with their participants were identified as either regular or occasional
children during one week in June. She found that 45 of those olive users. Five years later, researchers interviewed the men
surveyed reported that they would vote in favor of the again. They found that the population of men who
proposal. Which of the following statements must be true? experienced frequent heartburn was significantly lower for
men identified as regular olive oil users. Which of the
a) When the actual vote is taken, 75 percent of the following is the most appropriate conclusion of the study?
votes will be in favor of the proposal.
a) Olive oil use causes a reduction in the risk of
b) No prediction should be made about the vote on heartburn for men and women.
the proposal because the sample size is too small.
b) Olive oil use causes a reduction in the risk of
c) The sampling method is flawed and may produce heartburn for men but not necessarily for women.
biased results.
c) There is an association between olive oil use and
d) The sampling method is not flawed and is likely to the risk of heartburn for men and women, but it is
produce unbiased results. not necessarily a cause-and-effect relationship.

5. November 2017 d) There is an association between olive oil use and
To determine if gender and number of siblings are related to the risk of heartburn for men, but it is not
whether 9-year-old children like to read, Sarah selected a necessarily a cause-and-effect relationship, and the
random sample of 40 boys with fewer than 3 siblings and a association may not exist for women.
random sample of 30 girls with 3 or more siblings tram the
9-year-old children in her city. For each child, she recorded
the child’s age, gender, and whether the child reported liking
to read. Why is it inappropriate for Sarah to draw a
conclusion from her study?

a) The two samples are not of equal size.
b) The two samples should have been chosen from

different cities.
c) There is no upper bound on the number of siblings

of the 30 girls.
d) Sarah will not be able to tell whether a difference

in liking to read is related to the difference in
gender or to the difference in number of siblings.

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The Ivy 154 Problem Solving and Data Analysis

7. October 2017 US 9. December 2017
Near the end of a US cable news show, the host invited In a study of 24 adults and 8 adolescents, all participants
viewers to respond to a poll on the show's website that took an 8-week mindfulness seminar. After the seminar
asked, "Do you support the new federal policy discussed concluded, the participants reported improvement in stress-
during the show?" At the end of the show, the host reported related symptoms. Based on the study, which of the
that 28% responded ''Yes," and 70% responded "No." Which following is the most appropriate conclusion?
of the following best explains why the results are unlikely to
represent the sentiments of the population of the United a) Participation in a mindfulness seminar improves
States? stress-related symptoms but only in the participants
of this study.
a) The percentages do not add up to 100%, so any
possible conclusions from the poll are invalid. b) Participation in a mindfulness seminar improves
stress-related symptoms in all adults and
b) Those who responded to the poll were not a adolescents.
random sample of the population of the United
States. c) Participation in a mindfulness seminar improves
stress-related symptoms in adults but not in
c) There were not 50% "Yes" responses and 50% adolescents because the sample size for
"No" responses. adolescents is too small.

d) The show did not allow viewers enough time to d) No conclusion can be drawn from the study about
respond to the poll. the causal relationship between participation in a
mindfulness seminar and stress-related symptoms
because everyone in the study took the seminar.

8. October 2018
A paleontologist unearths the fossil of a dinosaur femur
bone that is 1.50 meters long. Previous research has found
that the femur bone of an adult Tyrannosaurus rex has a
mean length of 1.28 meters with a standard deviation of
0.30 meters. Which of the following statements is most
appropriate?

a) The length of the femur bone is consistent with the
expected length of an adult Tyrannosaurus rex
femur bone.

b) The length of the femur bone is much longer than
expected for an adult Tyrannosaurus rex.

c) The length of the femur bone is much shorter than
expected for an adult Tyrannosaurus rex.

d) It is not possible to compare the length of the
femur bone to the expected length of an adult
Tyrannosaurus rex femur bone.

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Problem Solving and Data Analysis 155 The Ivy

10. May 2018 12. October 2018 US
One hundred park-district members will be selected to A retail company has 50 large stores located in different
participate in a survey about selecting a new park-district areas throughout a state. A researcher for the company
coordinator. Which of the following methods of choosing believes that employee job satisfaction varies greatly from
the 100 members would result in a random sample of store to store. Which of the following sampling methods is
members of park-district? most appropriate to estimate the proportion of all employees
of the company who are satisfied with their job?
a) Obtain a numbered-list of all park-district
members. Use a random number generator to a) Selecting one of the 50 stores at random and then
select 100 members from the list. Give the survey surveying each employee at that store
to those 100 members.
b) Selecting 10 employees from each store at random
b) Obtain a list of all park-district members sorted and then surveying each employee selected
alphabetically. Given the survey to the first 100
members on the list. c) Surveying the 25 highest-paid employees and the
25 lowest-paid employees
c) Tell the park-district members that volunteers are
needed to take the survey. Give the survey to the d) Creating a website on which employees can
first 100 members who volunteer. express their opinions and then using the first 50
responses
d) Obtain a list of all park-district members who are
attending an upcoming event. Give the survey to
the first 100 members on the list.

11. October 2018 13.
One hundred students will be selected to participate in a A school district wants to conduct a sample survey to
survey about proposed changes to the school’s vending determine the average number of sports played by high
machines. Which of the following sampling methods will school seniors in the district. Which of the following survey
generate a data set that is appropriate to use to draw a methods is most likely to produce valid results?
conclusion about the entire school population?
a) The district surveys 500 randomly selected high
a) Give the survey to the first 100 students who arrive school seniors in the district who play baseball.
on campus on a Monday morning.
b) The district surveys every tenth student to enter the
b) Give the survey to the 100 students participating in football stadium at the district championship game.
the school musical.
c) The district surveys 500 randomly selected high
c) Give the survey to the first 100 students who use school seniors in the district.
the school’s vending machines on a given day.
d) The district selects one high school in the district
d) Use random numbers to choose 100 students from and surveys all of its students.
a numbered list of all students at the school. Give
the survey to these students.

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The Ivy 156 Problem Solving and Data Analysis

14. March 2018 US
A political scientist wants to predict how the residents of
New Jersey will react to a new bill proposed in the state
senate. Which of the following study designs is most likely
to provide reliable results for the political scientist?

a) Mailing a questionnaire to each of 200 randomly
selected residents of New Jersey

b) Surveying a group of 300 randomly selected New
Jersey residents

c) Interviewing a group of students randomly selected
from a large public university in New Jersey

d) Surveying a group of 1,500 randomly selected US
residents

Sample and Population 2. March 2019 US
A sample of 40 fourth-grade students was selected at
1. March 2018 random from a certain school. The 40 students completed a
A researcher randomly selected 36 second-grade boys from survey about the morning announcements, and 32 though the
Wilson Elementary School to participate in a study about the announcements were helpful. Which of the following is the
effects of exercise on memory. At the end of the study, the largest population to which the results of the survey can be
researcher concluded that regular participation in moderate- applied?
level exercise has a positive effect on memory. The results
of the study can be generalized to which of the following a) The 40 students who were surveyed
populations? b) All fourth-grade students at the school
c) All students at the school
a) All boys at Wilson Elementary School d) A fourth-grade students in the county in which the
b) All students at Wilson Elementary School
c) All second-grade boys at Wilson Elementary school is located

School
d) All second-grade students at Wilson Elementary

School

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Problem Solving and Data Analysis 157 The Ivy

3. December 2018 5. April 2019 US
A polling agency conducted a survey about public opinion A scientist conducted an experiment and selected a random
on car-sharing services. The agency surveyed a random sample of runners from a list of all high school track
sample of adult residents from three large cities in the participants from a certain city. The scientist randomly
United States and reported that 73% support car-sharing assigned each runner to one of two treatment groups, and the
services, with an associated margin of error of 4%. Based on results of the experiment were found to be statistically
the study design, what is the largest population to which significant. To which of the following populations can the
these survey results can be generalized? results of the experiment be safely generalized?

a) All adult residents of the United States a) All high school athletes
b) All adult residents of large cities in the United b) All high school track participants from the city
c) All high school track participants from the country
States d) All runners
c) All adult residents who support car-sharing

services
d) All adult residents of the three large cities in the

survey

4. March 2019 6. May 2019
A town newspaper polled 500 voters selected at random A survey was conducted using a sample of history
from a list of all registered voters in the town. Of those professors selected at random from the California State
polled, 41% approved of their state governor's performance. Universities. The professors surveyed were asked to name
Which of the following is the largest population to which the the publishers of their current texts. What is the largest
results of the poll can be applied? population to which the results of the survey can be
generalized?
a) All registered voters who receive the newspaper
b) All registered voters who participated in the poll a) All professors in the United States
c) All registered voters in the state b) All history professors in United States
d) All registered voters in the town c) All history professors at all California State

Universities
d) All professors at all California State Universities

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The Ivy 158 Problem Solving and Data Analysis

7. 8. June 2016
City Councilwoman Kelly wants to know whether the A marketing executive plans to survey 300 potential car
residents of her district support a proposed school buyers to determine which car feature customers value most.
redistricting plan. Which of the following survey methods Which of the following sample groups is most likely to
will allow Councilwoman Kelly to make a valid conclusion represent all potential car buyers?
about whether residents of her district support the proposed
plan? a) 300 first-time car buyers selected at random
b) 300 potential car buyers selected at random
a) Ask her neighbors. c) The first 300 first-time car buyers who enter a
b) Ask the residents of Whispering Pines Retirement
certain car dealership
Community. d) The first 300 potential car buyers who enter a
c) Ask 200 residents of her district whose names are
certain car dealership
chosen at random.
d) Ask a group of parents at the local playground.

Margin of Error

1. 2.
Based on random samples of river heights, oceanographers A random sample of 35 four-door passenger vehicles had a
estimate that 4,800 cubic kilometers (km3) of freshwater is mean gas mileage, in miles per gallon (mpg), of 25.9 mpg.
discharged into the Arctic Ocean annually. The estimate has The estimate had a margin of error of 2.6 mpg at a 98%
a margin of error of 240 km3 at the 90% confidence level. confidence level. Of the following, which is the most
Which of the following is the most reasonable claim about plausible value for the true mean gas mileage of all four-
the volume of freshwater discharged annually into the Arctic door passenger vehicles?
Ocean?
a) 24 mpg
a) It is between 4,800 and 5,040 km3. b) 29 mpg
b) It is between 4,560 and 5,040 km3. c) 32 mpg
c) It is between 240 and 4,800 km3. d) 35 mpg
d) It is between 240 and 4,320 km3.

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Problem Solving and Data Analysis 159 The Ivy

3. 5. March 2019 US
A random sample of international flights in 2013 showed A park ranger asked a random sample of visitors how far
that 79% of the flights arrived within 15 minutes of their they hiked during their visit. Based on the responses, the
scheduled arrival time, and this estimate had a margin of estimated mean was found to be 4.5 miles, with an
error of 3% at a 95% confidence level. Which of the associated margin of error of 0.5 miles. Which of the
following conclusions is best? following is the best conclusion from these data?

a) Exactly 79% arrived within 15 minutes of their a) It is likely that all visitors hiked between 4 and 5
scheduled times. miles.

b) Exactly 76% arrived within 15 minutes of their b) It is likely that most visitors hiked exactly 4.5
scheduled times. miles.

c) Between 76% and 82% arrived within 15 minutes c) It is not possible that any visitor hiked less than 3
of their scheduled times. miles.

d) Between 92% and 98% arrived within 15 minutes d) It is plausible that the mean distance hiked for all
of their scheduled times. visitors is between 4 and 5 miles.

4. May 2018 6. May 2019
An ecologist selected a random sample of 30 prairie dogs At a large high school, 300 students were selected at random
from a colony and found that the mean mass of the prairie and were asked in a survey about a menu change in the
dogs in the sample was 0.94 kilograms (kg) with and school cafeteria. All 300 students completed the survey. It
associated margin of error of 0.12 kg. Which of the was estimated that 38% of the students were in support of a
following is the best interpretation of the ecologist’s menu change, with a margin of error of 5.5%. Which of the
findings? following is the best interpretation of the survey results?

a) All prairie dogs in the sample have a mass between a) The percent of the students at the school who
0.82 kg and 1.06 kg. support a menu change is 38%.

b) Most prairie dogs in the colony have a mass b) The percent of the students at the school who
between 0.82 kg and 1.06 kg. support a menu change is greater than 38%.

c) Any mass between 0.82 kg and 1.06 kg is a c) Plausible values of the percent of the students at
plausible value for the mean mass of the prairie the school who support a menu change are
dogs in the sample. between 32.5% and 43.5%.

d) Any mass between 0.82 kg and 1.06 kg is a d) Plausible values of the number of the students at
plausible value for the mean of the prairie dogs in the school who support a menu change are
the colony. between 295 and 305.

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The Ivy 160 Problem Solving and Data Analysis

7. May 2018 US 9. December 2018
An analysis of a random sample of a type of laptop A sample of students was selected at random from all ninth-
computer battery estimated that the mean working time was grade students in a large school district. The proportion of
4.7 hours with a margin of error of 0.7 hours. Which of the the students in the sample who were 14 years old was 80%,
following is the most appropriate conclusion based on this with an associated margin of error of 5%. Which of the
analysis? following is NOT a plausible number of 14-year-olds in the
3,601 ninth-grade students in the school district?
a) This type of laptop computer battery has a mean
working time of at least 4.7 hours. a) 2,750
b) 2,900
b) This type of laptop computer battery has a mean c) 3,050
working time of at least 5.7 hours. d) 3,200

c) This type of laptop computer battery has a mean
working time of between 4.0 and 5.4 hours.

d) This type of laptop computer battery has a mean
working time of between 0.0 and 0.7 hours.

8. October 2018 US 10. April 2019 US
A research surveyed a random sample of students from a A city with 120,000 residents is voting on a proposal that
large university about how often they see movies. Using the would eliminate overnight parking of vehicles on the city’s
sample data, the researcher estimated that 23% of the streets. An independent company randomly surveys 1,200
students in the population saw a movie at least once per residents to see whether or not residents would support this
month. The margin of error for this estimation is 4%. Which proposal. The outcome of the survey shows that 60% of the
of the following is the most appropriate conclusion about all residents surveyed approve of the proposal with a margin of
students at the university, based on the given estimate and error of 2%. Which of the following statements is a plausible
margin of error? conclusion from the outcome of the survey?

a) It is unlikely that less than 23% of the students see a) Exactly 60% of city residents approve eliminating
a movie at least once per month. overnight parking.

b) At least 23%, but no more than 25%, of the b) There are 72,000 city residents who approve
students see a movie at least once per month. eliminating overnight parking.

c) The researcher is between 19% and 27% sure that c) About 2% of the city residents do not approve
most students see a movie at least once per month. eliminating overnight parking.

d) It is plausible that the percentage of students who d) Between 58% and 62% of the city residents
see a movie at least once per month is between approve eliminating overnight parking.
19% and 27%.

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Problem Solving and Data Analysis 161 The Ivy

11. March 2019 13. October 2018
A study estimated that the mean number of times per year Ms. Ortiz collected responses from 250 women who were
each person in the town of Shelton shops at a grocery store selected at random from a list of all residents of Oregon
is 91, with an associated margin of error of 3. The study also aged 25-30. The women in the sample responded to
estimated that the mean number of times per year each questions about their coffee-drinking habits. After analyzing
person in the town of Whitsville shops at a grocery store is the data, Ms. Ortiz used the percentage of women in the
95, with an associated margin of error of 4. Based on the sample that consume coffee on a daily basis along with a
study, which of the following is an appropriate conclusion? margin of error to estimate that between 62% and 70% of
women aged 25-30 residing in Oregon consume coffee on a
a) The people of Whitsville shop at a grocery store daily basis. Which of the following statements must be true?
more times per person per year than the people of
Shelton. a) 66% of the women in the sample consume coffee
on a daily basis, and the associated margin of error
b) The people of Shelton shop at a grocery store more is 4%.
times per person per year than the people of
Whiteville. b) 66% of the women in the sample consume coffee
on a daily basis, and the associated margin of error
c) The people of both towns shop at a grocery store is 8%.
the same number of times per person per year.
c) Between 62% and 70% of all women residing in
d) There is insufficient information to determine Oregon consume coffee on a daily basis.
which town's people shop at a grocery store more
times per person per year. d) If a different random sample of 250 women aged
25-30 residing in Oregon were surveyed, 66% of
12. May 2017 US them would respond that they drink coffee on a
A polling agency wanted to test whether a ballot measure daily basis.
would pass with greater than 50% yes votes. The agency
sampled 1,000 registered voters selected at random, and
50.6% of the voters favored the ballot measure. The margin

of error associated with this poll was  3%. Based on the

poll’s results, which of the following statements must be
true?

a) The percentage of voters who will vote yes for the
ballot measure is 50.6%.

b) The ballot measure will pass with more yes votes
than no votes, but the percentage of votes it will
receive cannot be predicted.

c) The ballot measure will pass with at least 53.6% of
the vote.

d) The poll’s results do not provide sufficient
evidence to conclude that the ballot measure will
pass.

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The Ivy 162 Problem Solving and Data Analysis

14. December 2017 15. December 2017
Jasmine and Daniel are each conducting a survey among the Sample Percent in favor Margin of error
1,190 students at North High to determine the proportion of A 52% 4.2%
students who would be interested in a new after-school B 48% 1.6%
sports club. Jasmine randomly selected 100 students to
survey, and Daniel randomly selected 50 students to survey. The results of two random samples of votes for a proposition
In both surveys, 34% of students indicated they are are shown above. The samples were selected from the same
interested. If J represents the margin of error for Jasmine’s population, and the margins of error were calculated using
sample proportion, D represents the margin of error for the same method. Which of the following is the most
Daniel’s sample proportion, and both students have the same appropriate reason that the margin of error for sample A is
amount of confidence in their results, which of the following greater than the margin of error for sample B?
is most likely true?
a) Sample A had a smaller number of votes that could
a) J < D not be recorded.
b) J > D
c) J = D b) Sample A had a higher percent of favorable
d) The relationship between J and D cannot be responses.

determined. c) Sample A had a larger sample size.
d) Sample A had a smaller sample size.

Statistics

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Problem Solving and Data Analysis 163 The Ivy

Questions 1-2 refer to the following information. 3. March 2018
Average distance (AU) Absolute magnitude 16, 19, 26, 7, 5, 10, 19, 17,18
1.83 18.4
1.99 16.6 The list above shows the low temperatures, in degrees
2.24 16.8 Fahrenheit, for 9 days in December in a certain city.
2.35 16.2 What is the median low temperature, in degrees Fahrenheit,
2.37 17.9 for the 9 days?
2.40 17.2
2.51 16.5 4. November 2017
2.61 17.6 A data set consists of the values 250, 320, 430, 452, 600,
2.66 17.3 and 17,167. If the outlier is removed, what will happen to
2.75 15.8 the value of the mean of the data set?
2.82 17.1
2.91 15.6 a) The mean will remain the same.
2.98 17.0 b) The mean will decrease.
3.14 15.4 c) The mean will increase.
3.23 15.8 d) There is not enough information to determine how

The average distance from the Sun, in astronomical units the mean will change.
(AU), and the corresponding absolute magnitude of 15
asteroids are shown in the table. An astronomical unit (AU)
is approximately 150 million kilometers. Absolute
magnitude is a measure of brightness.

1. March 2019 5. December 2018
What is the median of the distances from the Sun, in AU, of The distribution of the number of days served in office by
the 15 asteroids listed in the table? the first 43 presidents of the United States has a mean of
approximately 1,900 days. The ninth president, William
2. March 2019 Henry Harrison, served only 31 days, the fewest number of
What is the range of the absolute magnitudes of the asteroids days as president. What would be the effect on the mean of
listed in the table? the distribution if the number of days Willian Henry
Harrison served is removed from the data?

a) The mean would decrease.
b) The mean would increase.
c) The mean would be unchanged.
d) The effect on the mean cannot be determined.

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The Ivy 164 Problem Solving and Data Analysis

Questions 6-8 refer to the following information. 8. March 2018
Class size Number of classes What effect would removing the outlier have on the mean
23 1 math class size and the median math class size?
28 2
29 6 a) The mean would increase, and the median would
30 6 stay the same.
31 5
b) The mean would increase, and the median would
The table above shows the distribution of class sizes for the increase.
20 math classes at a high school.
c) The mean would stay the same, and the median
6. March 2018 would stay the same.
What is the mean math class size?
d) The mean would stay the same, and the median
a) 28.2 would increase.
b) 28.5
c) 29.4
d) 30.0

7. March 2018 9. March 2018 US
The math department chairperson wants to show the A group of 10 students played a certain game. Every player
principal of the school the number of math classes by size as received a score equal to an integer from 1 to 10, inclusive.
a percentage of the total number of math classes. Which type For the 10 players, the mean score was 4. If more than half
of graph is most useful for this purpose? of the players received a score greater than 5, which of the
following is true about the mean score of the remaining
a) Scatterplot players?
b) Box plot
c) Line graph a) It must be less than 4.
d) Circle graph b) It must be equal to 4.
c) It must be between 4 and 5.
d) It must be greater than 5.

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Problem Solving and Data Analysis 165 The Ivy

10. April 2019 US

Monthly Enrollments in Art Classes

Community Jan Feb Mar Apr May Jun Jul
center

A 124 111 98 82 77 152 133

B 465 407 391 354 365 511 495

The table above shows monthly enrollments in art classes at two community centers for 7 consecutive months. Based on the table,

by how much does the median monthly enrollment in community center B exceed the median monthly enrollment in community

center A for the 7 months?

11. Center, Spread, and Shape of Distributions

Letter count Number of words listed On a message board, participants listed words whose letters are
all in alphabetical order, such as "almost." The table at left lists
4 letters 12 the number of words they listed with each letter count. What
was the median letter count of the words they listed?
5 letters 49

6 letters 12

7 letters 9

8 letters 1

12. The histogram to the left summarizes the measurements a
5
4 botanist made of the transpiration rates in millimoles of water
3 mmol
2 ( )per square meter per second m2s of several diamondleaf
1
Number of plants00.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 willow plants. In which interval does the median transpiration

Transpiration Rate rate occur? mmol
m2s
a) 0.5 – 0.7

b) 0.7 – 0.9 mmol
m2s
mmol
c) 1.1 – 1.3 m2s

d) 1.3 – 1.5 mmol
m2s

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The Ivy 166 Problem Solving and Data Analysis
13. Number of home runs by Giants player in 2014

0 5 10 15 20 25
The dot plot above depicts the number of home runs hit by members of the San Francisco Giants during the 2014 baseball season.
If the player who hit the most home runs was removed from the data set, how would the mean of the data set be affected?

a) It would increase.
b) It would decrease.
c) It would remain the same.
d) There is not enough information to determine the impact on the mean number of home runs.

14. 15. May 2018 US
Of 100 people who played a certain video game, 85 scored
30 35 40 45 50 more than 0 but less than 10,000 points, 14 scored between
DaiilnyAhingchhtoermagpee,rAatluarsekian, Adepgrriele1s–F7,ah2r0e1n4heit 10,000 and 100,000 points, and the remaining player scored
5,350,000 points. Which of the following statements about
The dot plot above depicts the daily high temperatures in the mean and median of the 100 scores is true?
degrees Fahrenheit during the first week of April, 2014 in
Anchorage, Alaska. What would happen to the median if the a) The mean is greater than the median.
lowest temperature, 31  Fahrenheit, was replaced with a b) The median is greater than the mean.
new lowest temperature of 40  Fahrenheit? c) The mean and the median are equal.
d) There is not enough information to determine
a) It would increase.
b) It would decrease. whether the mean or the median is greater.
c) It would remain the same.
d) There is not enough information to determine the

impact on the median temperature.

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Problem Solving and Data Analysis 167 The Ivy

16. October 2018 US 18. March 2019 US
List A: 1 2 3 4 5 6 If a is the mean and b is the median of nine consecutive
List B: 2 3 3 4 4 5 integers, what is the value of |a – b|?

The table above shows two lists of numbers. Which of the 19. October 2016
following is a true statement comparing list A and list B? Employee Absences

a) The means are the same, and the standard Number of days Number of employees
deviations are different. 08
14
b) The means are the same, and the standard 23
deviations are the same. 44
55
c) The means are different, and the standard 13 1
deviations are different.
The frequency table above shows the distribution of the
d) The means are different, and the standard number of days each of the 25 employees of a company was
deviations are the same. absent last month. What is the median number of days
absent for the 25 employees last month?
17. October 2018 US
The maximum value of a data set consisting of 25 positive a) 1
integers is 84. A new data set consisting of 26 positive b) 2
integers is created by including 96 in the original data set. c) 4
Which of the following measures must be 12 greater for the d) 5
new data set than for the original data set?

a) The mean
b) The median
c) The range
d) The standard deviation

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The Ivy 168 Problem Solving and Data Analysis

Questions 20-21 refer to the following information.

At regular checkups, a pediatrician measures Anne’s height and weight and reports each as a percentile for her age. For example,

when Anne was 24 months old, her weight percentile of 65 indicated that 65% of children that age were at or below Anne’s weight.

Anne’s results for 6 checkups are shown below.

Checkup age Height percentile Weight percentile

1 month 40 35

3 months 45 45

6 months 55 50

12 months 60 55

18 months 55 55

24 months 50 65

20. June 2016 21. June 2016
What is Anne’s median height percentile for these 6 Which of the following values is larger for the weight
checkups? percentile data than for the height percentile data?

a) 50.8 a) Mean
b) 52.5 b) Median
c) 55 c) Mode
d) 57.5 d) Range

Question 22 refer to the following information 22. April 2017
Lionel purchases 1-pound bags of each of the five seeds and
Nutritional Information for 1-Ounce nuts shown in the table. Of the following, which best
approximates the average (arithmetic mean) number of
Serving of Seeds and Nuts calories per bag? (1 pound = 16 ounces)

Seed or nut Calories Total fat Protein a) 150
(grams) (grams) b) 250
c) 1,500
Pecan 198 20.2 3.0 d) 2,500

Pistachio 80 6.5 3.0

Pumpkin 159 13.9 8.5

Sunflower 166 14.6 5.9

Walnut 185 18.5 4.3

The table above shows the calories, grams of fat, and grams

of protein in 1-ounce servings of selected seeds and nuts.

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Problem Solving and Data Analysis 169 The Ivy

23. October 2017 US 25. November 2017
The weights, in pounds, for 15 horses in a stable were
reported, and the mean, median, range, and standard Survey Results of Households
deviation for the data were found. The horse with the lowest
reported weight was found to actually weigh 10 pounds less Number of children frequency
than its reported weight. What value remains unchanged if per household
the four values are reported using the corrected weight?
0x
a) Mean
b) Median 13
c) Range
d) Standard deviation 22

35

42

In the table above, the median number of children per

household is 2. What is one possible value of x?

24. December 2016 26. April 2017
Age of Participants
Ticket Prices by Row Number
36 55 42
25 15 31 Row number Ticket price
25 38 49
44 37 69 1 – 2 $25
15 37 77
32 28 72 3 – 10 $20
A journal article reported the mean age of participants in a
study. The age of the 18 participants in the study are given 11 – 20 $15
in the table above. Which of the following statements best
describes why the mean age of participants alone may not be The price of a ticket to a play is based on the row the seat is
a useful measure of center for the distribution?
a) Outlier may cause the distribution to be skewed. in, as shown in the table above. A group wants to purchase
b) Repeated numbers may cause the distribution to be
skewed. 10 tickets for the play.
c) The number of participants is too small to calculate
and report the mean. They will purchase 3 tickets for seats in row 1.
d) Sets containing an even number of data points may
cause the distribution to be skewed. They will purchase 2 tickets for seats in row 3.

They will purchase 2 tickets for seats in row 4.

They will purchase 3 tickets for seats in row 12.

What is the average (arithmetic mean) ticket price, in

dollars, for the 10 tickets? (Disregard the $ sign when

gridding your answer.)

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The Ivy 170 Problem Solving and Data Analysis

Questions 27 and 28 refer to the following information. 29. October 2017 USNumber of games
Weights of Manhole Covers on
Two City Streets 109876543 aNSuomccbeerrToefaGmoianls2S9cGoraemdebsy
Street A Street B 210 0 1Num2ber 3of go4als s5cored6 7
261 271 Based on the graph above, in how many of the games played
290 253 did the soccer team score goals equal to the median number
267 272 of goals for the 29 games?
282 262
273 270 Questions 30 refer to the following information.
269 288
293 263 Census blocks are geographic areas defined by the US Census
288 265
279 259 Bureau. The residents in each block are counted each decade.

The table above shows the weights, in pounds, of the manhole The table below shows the total number of census blocks for
covers on two city streets. City regulations require that each
manhole cover weigh at least 250 pounds and no more than the 5 states that had the cost census blocks in 2000 and 2010.
300 pounds.
Number of Number of
27. November 2017
The median weight of the manhole covers on Street A is State census blocks census blocks
how many pounds greater than the median weight of the
manhole covers on Street B? in 2000 in 2010

28. November 2017 Alaska 21,874 45,292
The mean weight of the 9 manhole covers on Street B is 267
pounds. If an additional manhole and manhole cover are Hawaii 18,990 25,016
added to Street B, and if the mean weight, in pounds, for the
10 manhole covers on Street B is an integer greater than 267, Delaware 17,483 24,115
what is one possible weight, in pounds, of the additional
manhole cover? Rhode Island 21,023 25,181

Vermont 24,824 32,580

Total for these 5 104,194 152,184
states

30. December 2017

For these 5 states, what is the difference in the mean of the

number of census blocks in 2010 and the mean of the

number of census blocks in 2000?

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Problem Solving and Data Analysis 171 The Ivy

Data Distribution

1. May 2019 2. May 2019
Age of 25 Babies When They Began Walking
Value Frequency
8 9 10 11 12 13 14 15 16 1a
Age (months) 2 2a
3 3a
The dot plot above gives the ages, in months, at which 25 4 2a
babies began walking. Which of the following is true about 5a
the mean and the median of the data?
The frequency distribution above summarizes a set of data,
a) The mean is greater than the median. where a is a positive integer. How much greater is the mean
b) The mean is less than the median. of the set of data than the median?
c) The mean is equal to the median.
d) The relationship between the mean and the median a) 0
b) 1
cannot be determined form the dot plot. c) 2
d) 3

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The Ivy 172 Problem Solving and Data Analysis

3. 4. April 2019 US
Scores of Two Classes in a Quiz
Two independent surveys asked random samples of 500
01 234 5
Class A people about the distances they commute to work each day.

01 234 5 The results of the surveys are detailed in the table below.
Class B
Daily Commuting Distance
The dot plots above summarize the scores that two classes,
each with 24 students, at Central High School achieved on a Survey Mean (miles) Standard deviation (miles)
current events quiz. Which of the following correctly
compares the standard deviation of the scores in each of the A 13.9 1.5
classes?
B 15.1 1.5
a) The standard deviation of the scores in Class A is
smaller. Which statement is true based on the results of these

b) The standard deviation of the scores in Class B is surveys?
smaller.
a) There is a greater variation in the distribution of
c) The standard deviation of the scores in Class A
and Class B is the same. the distances people commute to work in Survey

d) The relationship cannot be determined form the A.
information given.
b) There is a greater variation in the distribution of

the distances people commute to work in Survey

B.

c) The variation in the distribution of the distances

people commute is the same in both surveys.

d) It is impossible to determine the variation in the

distribution of the distances people commute

because the means are different.

5. May 2018

A 25550 40430 49150 62590 73670 118780 126040
B 22860 55020 173730 300580 358920 456170 603300
Which of the following is true about the standard deviations of the two data sets in the table above?
a) The standard deviation of data set B is larger than the standard deviation of data set A.
b) The standard deviation of data set A is larger than the standard deviation of data set B.
c) The standard deviation of data set A is equal to the standard deviation of data set B.
d) There is not enough information available to compare the standard deviations of the two data sets.

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Problem Solving and Data Analysis 173 The Ivy

6. September 2020 7. May 2018 US
Daily Electric Fan Sales Of 100 people who played a certain video game, 85 scored
more than 0 but less than 10,000 points, 14 scored between
012 345678 10,000 and 100,000 points, and the remaining player scored
Number sold in a day 5,350,000 points. Which of the following statements about
April the mean and median of the 100 scores is true?

012 345678 a) The mean is greater than the median.
Number sold in a day b) The median is greater than the mean.
June c) The mean and the median are equal.
d) There is not enough information to determine
The dot plot shown summarize the daily sales of electric
fans at a store for the 25 days the store was open in April whether the mean or the median is greater.
and June, respectively. Which of the following was(were)
the same for daily sales in April and in June?

I. Mean
II. Standard deviation

a) I only
b) II only
c) Both I and II
d) Neither I nor II

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The Ivy 174 Problem Solving and Data Analysis

Boxplot

1. December 2018 2. March 2019 US
2010
1990 Boat A

0 2 4 6 8 10 12 14 16 18 Boat B
The box plots above show the distribution of the average
amount of energy used per person, in tons of oil equivalent 0 10 20 30 40 50 60 70
(toe), for 25 countries, in 1990 and in 2010. Number of fish caught
Which of the following correctly compares the median and
range of these distributions? The box plot above summarize the distribution of the
number of fish caught each day on two commercial fishing
a) The median in 2010 is greater than the median in boats for a season. By how many fish does the median
1990, and the range in 2010 is less than the range number of fish caught each day on Boat B exceed the
in 1990. median number of Boat A?

b) The median in 2010 is greater than the median in
1990, and the range in 2010 is greater than the
range in 1990.

c) The median in 2010 is less than the median in
1990, and the range in 2010 is less than the range
in 1990.

d) The median in 2010 is less than the median in
1990, and the range in 2010 is greater than the
range in 1990.

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Problem Solving and Data Analysis 175 The Ivy

3. October 2018Copper concentration (mg/kg) 5. May 2019
The box plots below show the distribution of copper
concentration, in milligrams (mg) per kilogram (kg) of soil, Acres of useful timberland
in soil samples collected from two locations in a particular The number of acres of useful timberland in 13 counties in
city. California is summarized in the box plot above. Which of
the following is closest to the median number of acres?
100.0
a) 4,399
80.0 b) 7,067
c) 8,831
60.0 d) 10,595

40.0 6. March 2019
box plot A
20.0 box plot B

0 20 30 40 50 60
Location A Location B Based on the two box plots above, which of the following
statements is true?
What is the approximate difference, in mg/kg, between
the range of copper concentration in the Location B a) The range of box plot A is greater than the range
samples and the range of copper concentration in the of box plot B.
Location A samples?
a) 10 b) The range of box plot A is less than the range of
b) 12 box plot B.
c) 29
d) 52 c) The range of box plot A is equal to the range of
box plot B.
4. May 2019
d) It cannot be determined if the range of box plot A
37 51 61 84 87 is different from the range of box plot B.

Which of the following statements about the data

represented in the box plot above must be true?

a) There are more data between 61 and 84 than

between 51 and 61.

b) There are no data between 37 and 51.

c) The mean of the data is 61.

d) The range of the data is 50.

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The Ivy 176 Problem Solving and Data Analysis

Questions 7 and 8 refer to the following information. 8. March 2018 US
For a particular office building with 1,420 employees, Tia Which of the following box plots could represent Amir’s
and Amir each conducted a survey about the average one- survey data?
way commute times, in minutes, between the employees’
home and office. Both Tia and Amir selected employees at a)
random, mailed out surveys, and collected data from the 10 20 30 40 50
returned surveys. For both surveys, respondents were asked Average one(m-winayutceos)mmute time
to report their average commute times to the nearest 5
minutes. Tia collected data from 150 employees, and Amir b)
collected data from 85 employees. The results from Tia’s 10 20 30 40 50
and Amir’s returned surveys are summarized below. Average one-way commute time
(minutes)
Tia s Survey Results
c)
5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 10 20 30 40 50
Average one-way commute time Average one-way commute time
(minutes) (minutes)

Number of employees 322222111110820860644286420 Amir s Survey Results 5 d)
27 45 10 20 30 40 50
5 10 50 Average one(m-winayutceos)mmute time
10 10 11 12 10
9.
15 20 25 30 35 40 Resting heart rates (beats per minute)
Average one-way commute time (minutes)
35 40 45 50 55 60 65 70 75 80 85 90 95 100 105
7. March 2018 US The box plot above summarizes the resting heart rates, in
beats per minute, of the members at a gym. What is the
If T is the median commute time of the employees who range of resting heart rates?

responded to Tia’s survey and A is the median commute a) 16 beats per minute
b) 31 beats per minute
time of the employees who responded to Amir’s survey, c) 62 beats per minute
d) 70 beats per minute
what is the value of T – A?

a) 10

b) 8

c) 5

d) 0

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The Ivy 177 Additional Topics

Circle Equations

1. June 2016 3. October 2016
(x – 6)2 + y2 = 64 (x – 3)2 + (y – 5)2 = 18
One equation in a system of equations is shown above. The The graph of the equation above is a circle in the xy-plane.
graph of the second equation (not shown) is a vertical line in What is the area of the circle?
the xy-plane that crosses the x-axis at (3, 0). For both y- a) 6
coordinates of the solutions to the system of equations, what b) 9
is the value of y2? c) 12
a) 32 d) 18
b) 35
c) 55 4. December 2016
d) 58 (x – 1)2 + (y – 3)2 = 16
The equation above forms a circle when graphed in the
2. June 2016 xy-plane. What is the radius of the circle?
x2 + y2 + 2x – 8y = 8
The equation of a circle in the xy-plane is shown above.
What is the radius of the circle?

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Additional Topics 178 The Ivy

5. April 2017 8. March 2018
The graph of x2 – 4x + y2 + 6y – 24 = 0 in the xy-plane is a A circle in the xy-plane has a center of (–6, –8), and the
circle. What is the radius of the circle? origin lies on the circle. Which of the following is an
equation for the circle?
a) 2 6
b) 11 a) (x + 6)2 + (y + 8)2 = 100
c) 37 b) (x + 6)2 + (y + 8)2 = 10
d) 76 c) (x – 6)2 + (y – 8)2 = 100
d) (x – 6)2 + (y – 8)2 = 10

6. April 2017 9. May 2018 US
Which of the following equations describes a circle with In the xy-plane, the points (2, 4) and (–2, –4) are the
radius 10 that passes through the origin when graphed in the endpoints of a diameter of a circle. Which of the following is
xy-plane? an equation of the circle?

a) (x – 5)2 + (y + 5)2 = 10 a) (x – 2)2 + (y + 4)2 = 80
b) (x – 5)2 + (y + 5)2 = 100 b) (x – 2)2 + (y + 4)2 = 20
c) (x – 10)2 + (y – 10)2 = 100 c) x2 + y2 = 80
d) (x – 5 2 )2 + (y + 5 2 )2 = 100 d) x2 + y2 = 20

7. October 2017 US 10. October 2018
y A circle in the xy-plane has the equation
49 = (x – 3)2 + (y + 5)2. Which of the following is the
(h, k) y-coordinate of a point that lies on the circle?

O (4, 0) (20, 0) x a) –13
b) 2
In the xy-plane above, the circle has center (h, k) and radius c) 3
10. What is the value of k? d) 5

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The Ivy 179 Additional Topics

11. October 2018 13.
A circle in the xy-plane has the equation
y x2 + y2 – 14y – 51 = 0. What is the center of the circle?

(− 1 , y) a) (51, 14)
4 b) (7, 10)
c) (0, 0)
x d) (0, 7)

Note: Figure not drawn to scale 14.
A circle in the xy-plane has its center at (11, 12). If the point
The center of the circle shown above is (0, 0), and the circle (13, 14) lies on the circle, which of the following is an
equation of the circle?
has a radius of 1. What is the y-coordinate of the point
a) x2 + y2 – 22x + 24y = –257
shown? b) x2 + y2 + 22x – 24y = –257
c) x2 + y2 + 22x + 24y = –257
a) 1 d) x2 + y2 – 22x – 24y = –257
16
3
b) 4

c) 15
4
3
d) 4

12. April 2019 US
(x – 6)2 + (y – 3)2 = 25

The graph in the xy-plane of the equation above is a circle. If
the circle is translated downward a units such that the circle
is tangent to the x-axis, the equation becomes
(x – 6)2 + (y – 3 + a)2 = 25. What is the value of a?

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Additional Topics 180 The Ivy

15.

A circle with center M in graphed in the xyy-plane below. Which of the following is an equation of the circle?

12 a) (x + 8)2 + (y + 8)2 = 100
10 b) (x + 8)2 + y2 = 100
8 c) (x + 8)2 + (y – 6)2 = 100
6 d) (x + 8)2 + (y + 6)2 = 100
4
M2
-18 -16 -14 -12 -10 -8 -6 -4 -2 -2 2 4x

-4
-6
-8
-10

16.

(x – 4)2 + (y + 6)2 = 9

Which of the following circle graphs is best represented by the equation shown above?

a) y b) y

4 2468 x 8
2 6
4 2 4x
– –– 2
–
– –8 –6 –4 –2 –2
– –4

c) y d) y

15 5 10 15 x 15
10 10
5 5
– – – – 5 10 15 x
– – –– –
– –
–

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The Ivy 181 Additional Topics

Complex Numbers

1. June 2016 3. December 2017

(3.63 + 3.01i) + (5.44 – 5.22i) The product of the complex numbers (2 + 3i) and (3 – i) can
Which of the following complex numbers equals the sum be written as a + bi, where a and b are real numbers and
above? i = −1 . What is the value of 100a + b?

a) 8.45 – 2.21i
b) 9.07 + 8.23i
c) 9.07 – 2.21i
d) 35.46 + 35.32i

2. December 2016 4. March 2018

In the complex number system, what is the quotient when (5 – 2i)(1 – i)
Which of the following is equivalent to the complex number
i4 −16 is divided by i 2 − 4 ? (Note: i = −1 ) expression above? (Note: i = −1 )
i4 − 25 i 2 − 9
a) 3 – 7i
b) 5 – 5i
c) 5 + 9i
d) 7 + 3i

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5. March 2018 US 8. April 2019 US

(3 + 4i) – (2 + 3i) What is the sum of the complex numbers 6 + 5i and 8 + 3i2?
In the complex number system, which of the following is (Note: i = −1 )
equivalent to the expression above? (Note: i = −1 )
a) 11 + 5i
a) 0 b) 14 – 2i
b) 1 + i c) 14 + 8i3
c) –1 – i d) 17 + 5i
d) –5 – 7i

6. May 2018 9. May 2019 US
In the complex number system, what is the value of the
expression 16i4 – 8i2 + 4? (Note: i = −1 ) i2 + (–i)2
In the complex number system, what is the value of the
7. May 2018 US given expression? (Note: i = −1 )
(15 – 4i)(6 – 3i) = a + bi
In the equation above, a and b are real numbers and a) –2
i = −1 . What is the value of a? b) 0
c) 2
d) 2i

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The Ivy 183 Additional Topics

Circle Theorems

1. March 2018 3.
S
The measure, in degrees, of an angle is 135º. The measure,
in radians, of the angle is aπ. What is the value of a? 3
8

The angle in the figure is a central angle in radians.
What is the length of arc S in the circle shown above?

2. March 2018 US
An angle measure of 540 degrees was written in radians as
xπ. What is the value of x?

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Additional Topics 184 The Ivy
4.
6. October 2016 B
4 A

 6 OC

In the circle above, radius OB has length 1. The measure of

In the diagram, is a central angle. AOC is equal to the measure of BOC, and the length of
What is the measure of θ in radians in the circle shown 8
arc ABC is 9 . What is the measure, in degrees,
above?
ofAOB? (Disregard the degree symbol when gridding

your answer.)

7.

5. May 2018 B
A
x°
x° O A
O
B

Segment OA and OB are radii of the semicircle above.

Arc AB has length 3π and OA = 5. What is the value of x? Note: Figure not drawn to scale.

In the figure above, the circle has center O and has radius
10. If the length of arc AB (shown in bold) is between 5 and
6, what is one possible integer value of x?

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The Ivy 185 Additional Topics
8. October 2016 9. January 2017 US

100° B C
C A x°

Point C is the center of the circle above. What fraction of the
area of the circle is the area of the shaded region?

Note: Figure not drawn to scale.

In the circle above, point A is the center and the length of arc

BC is 2 of the circumference of the circle. What is the
5

value of x?

Similar Triangles

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Additional Topics 186 The Ivy

1. December 2016 3. May 2017 US
B
D B

A EC E
In the figure above, AD = 4, AE = 6, and segment DE is
parallel to segment BC. What is the ratio of the length of AD C
segment BD to the length of segment CE?
If the figure above, BC = 5, and the length of line segment
a) 2:5
b) 3:5 AD is half the length of line segment CD. What is the length
c) 2:3
d) 1:2 of line segment DE?

a) 2
5
3
b) 5

c) 5
3
5
d) 2

2. December 2016 4. November 2017
P
Y
T 15 Q 5 R
12 B
S 6 10

What is the length of PQ in the figure above? A CX Z

a) 36 Right triangles ABC and XYZ above are similar. What is the
b) 39
c) 250 tangent of angle Z?
d) 369
a) 3
5
3
b) 4

c) 4
3
5
d) 3

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The Ivy 187 Additional Topics

5. November 2017 7. December 2017
C
B
53 16 D BD
AC

AE

E In  ACE above, BD|| AE . If BD = 19 and AE = 31, what
CD
Note: Figure not drawn to scale. is the value of DE ?

In the figure above,  B is congruent to  E. What is the a) 19
31
value of sin D? 12
3 19
a) 5 b)

b) 4 c) 19
5 12
31
c) 16 d) 19
5
16
d) 25

6. December 2017

C 8. March 2018 US

In triangle ABC, angle A measures 48˚, angle B measures

B ˚, and angle C measures ˚. Triangle ABC is similar to
LM MN LN
triangle LMN, such that AB = BC = AC = 3 . What is

A ED the measure, in degrees, of angle L?

In the figure above, CD = 5 and AD = 8. What is the value of

the cosine of  AEB?
5
a) 89

b) 8
89
5
c) 8

d) 8
5

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Additional Topics 188 The Ivy
9. May 2018
11. October 2018
B C

29 21 B 3.5 D
E A2
E 3
A 20 C D 5 F
Triangle ABC and DEF above are similar. How much longer In  ACD above, BE ||CD . What is the length of CD ?
than segment EF is segment DE?

a) 1
b) 2
c) 4
d) 8

10. May 2018 US 12. October 2018

B B
E

6

A FiguDre not drawn to C A 8D C
Note: scale.
In the figure above,  ABC and  BCD are right triangles,
In the figure above, ABC and DEC are right triangles.
AD = 8, and BD = 6. What is the length of DC ?
If CD = 20 and the tangent of angle ABC is 2.5, what is the

length of segment ED?

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The Ivy 189 Additional Topics

13. October 2018 US 15. March 2019 US
E
C

B BD

12 13 AE

A5 C D F Note: Figure not drawn to scale.
In the figure above, segments AE and BD are parallel. If
In the figure above, triangle ABC is similar to triangle DEF. angle BDC measures ˚ and angle ACE measure ˚, what
is the measure of angle CAE?
What is the value of cos E?
12 a) ˚
a) 5 b) ˚
c) ˚
b) 12 d) ˚
13
5 16. April 2019 US
c) 12 A
5
d) 5 B3E
13
C 9D
14. March 2019 US In the figure above,  ACD is a right triangle and BE is
parallel to CD . What is the perimeter of  ACD to the
Triangle ABC and triangle DEF are similar triangles, where nearest tenth of a unit?
AB and DE are corresponding sides. If DE = 2AB and the
perimeter of triangle ABC is 20, what is the perimeter of a) 29.7
triangle DEF? b) 36.0
c) 41.5
a) 10 d) 50.9
b) 40
c) 80
d) 120

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The Ivy 190 Additional Topics

Speed, Distance, Time

1. 3.

A car moving at a constant rate travels 96 miles in 2 hours. If an object travels at 5 feet per minute, how many feet does
If the car maintains this rate, how many miles will the car
travel in 5 hours? it travel in 2 seconds?

a) 480 a) 120
b) 240
c) 210 b) 10
d) 192 2
e) 144 c) 5

d) 1
6
1
e) 12

2. 4.

If an object travels at a speed of 3 feet per second, how If a car is traveling at a constant rate of 45 miles per hour,
many feet does it travel in half an hour? how many miles does it travel from 10:40 A.M. to 1:00 P.M.
of the same day?

a) 165
b) 150
c) 120
d) 105
e) 90

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Additional Topics 191 The Ivy

5. 7.

Carmen went on a trip of 120 miles, traveling at an average A motor boat traveling at 18 miles per hour traveled the
length of a lake in one-quarter of an hour less time than it
of x miles per hour. Several days later she returned over the took when traveling at 12 miles per hour. What was the
same route at a rate that was 5 miles per hour faster than her length in miles of the lake?

previous rate. If the time for the return trip was one-third of a) 6
an hour less than the time for the outgoing trip, which b) 9
c) 12
equation can be used to find the value of x? d) 15
120 1 e) 21
a) x+5 = 3

b) x = x+5 − 1
120 120 3

c) 120 5) = 1
x+ ( x+ 3

d) 120 = 120 + 1
x x+5 3

e) 120(x + 5) – 120x = 1
3

6. 8.

Lucy drove from her house to a friend’s house at an average A man drove a car at an average rate of speed of 45 miles
speed of 40 miles per hour. She returned home along the per hour for the first 3 hours of a 7-hour car trip. If the
same route at an average speed of 60 miles per hour. If her average rate of speed for the entire trip was 53 miles per
total driving time was 1 hour, how many total miles did hour, what was the average rate of speed in miles per hour
Lucy drive? for the remaining part of the trip?

a) 50
b) 55
c) 57
d) 59
e) 62

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The Ivy 192 Additional Topics

9. 11. March 2020 US

Angelica drove 100 miles to Baxterville at an average speed Minato drove 390 miles. Part of the drive was along local
of 50 miles per hour. She then increased her average speed roads, where his average speed was 20 miles per hour, and
by 20 percent and drove 30 miles to Center City. What was the rest was along a highway, where his average speed was
her average speed for the entire journey, in miles per hour? 60 miles per hour. The drive took 8 hours. What distance, in
miles, did Minato drive along local roads?

a) 30
b) 45
c) 90
d) 120

10. 12. May 2018

Phil is making a 40-kilometer canoe trip. If he travels at 30 George took a nonstop flight from Dallas to Los Angeles, a
kilometers per hour for the first 10 kilometers, and then at 15 total flight distance of 1233 miles. The plane flew at a speed
kilometers per hour for the rest of the trip, how many of 460 miles per hour for the first 75 minutes of the flight
minutes longer will it take him than if he makes the entire and at a speed of 439 miles per hour for the remainder of the
trip at 20 kilometers per hour? flight. To the nearest minute, for how many minutes did the
plane fly at a speed of 439 miles per hour?

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Additional Topics 193 The Ivy

Questions 13–15 refer to the following information 15. April 2019 US

For gym class, Shayla completed a 4–mile walking and What was the total distance that Shayla spent walking and
running, in kilometers? (Use 1 mile = 1.61 kilometers)
running exercise. She ran for 7t miles and she walked for
13 a) 0.40
3( 15 – t) miles, where t is the total amount of time, in hours, b) 4.00
c) 6.44
Shayla spent running. The equation 7t + 3( 13 – t) = 4 d) 10.53
15
models this situation.

13. April 2019 US

Which of the following is the best interpretation of the value

of 7 in the equation that models this situation?

a) Shayla walked at a speed of 7 miles per hour.

b) Shayla ran at a speed of 7 miles per hour.

c) Shayla walked for 7 minutes.

d) Shayla ran for 7 minutes.

14. April 2019 US 16. October 2019 US

What is the value of t in the equation that models this On a 210-mile trip, Cameron drove at an average speed of
60 miles per hour for the first x hours. He then completed
the trip, driving at an average speed of 50 miles per hour for
the remaining y hours. If x = 1, what is the value of y?

situation? 7
50
a)

b) 7
20
31
c) 60

d) 13
15

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