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Problem Solving Butler: April 2025
April 21PS 1PS 2

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PS 1

If x^2 < x, what is the range of possible values of x ?

A. -1 < x < 1
B. -1 < x < 0
C. 0 < x < 1
D. x > 0
E. x < 0

_____________________
PS 2

If x = 4p + 10 and |p| < 7, which of the following represents the correct range of values of x ?

A. -7 < x < 7
B. x > -18
C. x < 38
D. -38 < x < 18
E. -18 < x < 38
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Critical Reasoning Butler: April 2025
April 21CR 1CR 2

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CR 1

Slack time, or float time, refers to the time between the completion of one professional task and the beginning of the next task. By scheduling projects to minimize slack time, organizations can improve their employees’ efficiency. For instance, a Chinese consultancy firm devised a streamlined workflow design and check-in system that allowed a Brazilian conglomerate to minimize slack time by 80 percent, almost doubling its staff’s efficiency.

In the argument given, the two portions in boldface play which of the following roles?

A. The first is the argument’s conclusion; the second is an opinion that supports that conclusion’s validity.
B. The two portions are both evidence that support the argument’s conclusion.
C. The first is a premise that supports the argument’s conclusion; the second is that conclusion.
D. The two portions are both premises supporting the argument’s conclusion.
E. The first is the argument’s conclusion; the second is an evidence that supports that conclusion.


______________________
CR 2

Luc Macron: It is not wise to uncritically assume that all traditions are valuable. Most traditions are maintained because they benefit one or more social classes, and people will adhere to a tradition if it benefits them in some way. Such people may not be necessarily trying to uphold valueless traditions, but they are also not interested in examining if the tradition they participate in is of any real value.

Mariah: People would not participate in a tradition if they did not think it held some value. So, it can be assumed that if enough people participate in a tradition, it has some real social value.

Mariah’s rebuttal is most susceptible to criticism on the grounds that it

A. does not address the assertion that people will not examine the real social value of a tradition if it benefits them
B. places undue emphasis on the benefits of traditions
C. fails to specify how many people must participate in a tradition before it is justifiable to consider it valuable
D. supports the assertion that those who adhere to traditions are not purposefully trying to uphold valueless traditions
E. fails to distinguish between people’s motives for adhering to tradition from their motives for obeying non-traditional practices
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Data Sufficiency Butler: April 2025
April 21DS 1DS 2

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DS 1

At a company, 124 employees completed at least one of the following three training modules: Safety, Data Privacy, and First Aid. Of these employees, 40 completed exactly one module. Is there at least one employee who completed both Data Privacy and First Aid, but not Safety?

(1) 16 employees completed all three training modules.

(2) The combined number of employees who completed Safety and Data Privacy only and those who completed Safety and First Aid only is 27.
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DS 2

A festival organizer is putting together the final set of food trucks for a weekend street-food event. Several food trucks applied, and exactly 3 of them will be chosen. How many different groups of 3 food trucks could be chosen?

(1) If 2 more food trucks had applied, the number of possible 3-truck groups would have been 120.
(2) If 2 fewer food trucks had applied, the number of possible 3-truck groups would have been 20.
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Problem Solving Butler: April 2025
April 22PS 1PS 2

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PS 1

If x ≠ 0 and -1 < x < 1, among x, 1/x , and x^3, which may be the smallest?

I. 1/x
II. x
III. x^3

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III
________________________
PS 2

Function x%y represents the remainder when a positive integer x is divided by another positive integer y. If a, b, c, d are positive integers and a%b = 6 and c%d = 9, what is the minimum possible value of b x d?

A. 54
B. 60
C. 63
D. 70
E. Greater than 70
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Critical Reasoning Butler: April 2025
April 22CR 1CR 2

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CR 1

Since tiger poaching was made a criminal offence in India, the decline in India’s tiger population has rapidly slowed down. Clearly, similar law should be implemented to counter cheetah poaching in South Africa.

Each of the following is an assumption made in the argument above EXCEPT:

A) The cheetah poaching problem in South Africa is similar in nature to the tiger poaching problem in India.
B) The reduction in economic benefits of animal poaching has not contributed to the reduction in tiger poaching in India.
C) Tiger populations were accurately reported before and after the law was implemented in India.
D) Poaching is the sole cause of the decline of the population of tigers as well as cheetahs.
E) Citizens of India and South Africa are similar in terms of abiding by the law.
_______________________
CR 2

Due to the nationalistic sentiment behind the ongoing efforts to realize commercial space travel undertaken by Austrich and Tipai, it was feared that the two nations would impose silence about the results of aeronautical research conducted by their respective space agencies. This constraint, in turn, would slow the development of space flight and aeronautical engineering.

Which of the following, if true, would most seriously tend to weaken the prediction regarding the imposition of silence, described above?

A. Aeronautical research funded by Austrich and Tipai has reached a few conclusions that are of considerable importance to the scientific community.
B. To improve their nations' reputation, the governments of Austrich and Tipai encourage their respective space agencies to publish results, particularly the important ones.
C. As the research priorities of nation states are unlike those of institutes of learning, the financial support provided to research endeavors by nations tends to distort the agenda of research.
D. Independent researchers cannot build upon the results of scientific research when those results are kept secret.
E. Some of the problems that Austrich and Tipai devoted research resources to are of critical scientific importance but not expected to lead to any practical applications in the near future.
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Data Sufficiency Butler: April 2025
April 22DS 1DS 2

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DS 1

Maya and Leo are digitizing a set of archival files. Working together at their respective constant rates, they can finish the job in 20 fewer hours than Leo would need to finish the job alone. How many hours would Maya need to finish the job alone, working at her own constant rate?

(1) Working alone, Maya would need 15 fewer hours than Leo would need to finish the job alone.
(2) The two of them can finish the job together in 10 hours.
______________________
DS 2

A courier company assigned each of its drivers to exactly one of three delivery zones: Harbor, Central, or Ridge. If the numbers of drivers in Harbor, Central, and Ridge were in the ratio 2:3:5, respectively, was the average number of packages delivered per driver across all three zones combined greater than 30?

(1) The average numbers of packages delivered per driver in Harbor, Central, and Ridge were 38, 27, and 25, respectively.
(2) The total number of packages delivered by all the drivers was less than 300.
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Problem Solving Butler: April 2025
April 23PS 1PS 2

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PS 1

For which of the following is f(x + y) = f(x) + f(y) ?

A. f(x) = 1 / x
B. f(x) = 2x2
C. f(x) = x + 2
D. f(x) = –5x
E. f(x) = √x
______________________
PS 2

f(n + 1) = –n + f(n)/2 . If f(4) = 12, what is the value of f(0)?

A. 80
B. 114
C. 228
D. 252
E. 260
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Critical Reasoning Butler: April 2025
April 23CR 1CR 2

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CR 1

Many skin-whitening creams and similar cosmetics utilize heavy metal compounds to block the production of the skin pigment melanin. The metal compounds bleach the skin by destroying melanocytes, the cells that produce melanin, lightening the skin’s tone. Critics fear that exposure to these compounds could have numerous negative health effects, including a heightened risk of skin cancer, but a defender responds, “The average dose of metal compounds absorbed via skin-whitening creams is less than what one might absorb from a few minutes of exposure to gasoline smoke. Using these creams will not hurt anyone.”

Which of the following issues would it be most important to resolve in evaluating the dispute concerning the health effects of the skin-whitening creams?

A. Whether the creams affect all areas that they are applied to equally
B. Whether treatments for hyperpigmentation disorders use metal compounds to reduce melanin production
C. Whether the typical amount described by the defender is the amount absorbed via one use or a cumulative amount, absorbed over a lifetime
D. The cultural impact of skin-whitening creams
E. Whether the skin-whitening creams were tested for safety on a broad cross-section of the population
____________________
CR 2

Prosecutor: The current Mayor of Tivo was elected in 2015. Each year since then, at least seven companies have been prosecuted for violating anti-pollution laws. In 2015, under the previous mayor, only two companies were prosecuted for violating anti-pollution laws. So, the previous mayor was more tolerant of polluters than the new one is.

Journalist: But in 2015, far fewer companies actually violated the anti-pollution laws, and now many more companies regularly do so.

The journalist challenges the prosecutor’s argument by doing which of the following?

A) Including data from further into the past than the prosecutor’s data
B) Highlighting that the argument illegitimately forms a general conclusion with a sample of only a few cases as a basis
C) Challenging the veracity of the evidence presented in support of the conclusion reached by the prosecutor
D) Introducing evidence that undercuts an assumption in the argument made by the prosecutor
E) Stating that the argument assumes that information about some members of a group applies to all members of that group
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Data Sufficiency Butler: April 2025
April 23DS 1DS 2

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DS 1

A community theater sold tickets for the same show in January and in February. In January, every ticket was sold at the same fixed price, whereas in February, the ticket price increased, and every ticket was sold at the same higher fixed price. If fewer tickets were sold in February than in January, was the revenue from ticket sales in February greater than the revenue from ticket sales in January?

(1) The percent increase in the ticket price from January to February was equal to the percent decrease in the number of tickets sold from January to February.

(2) If the percent increase in the ticket price from January to February had been 8 percentage points greater than it actually was, while the percent decrease in the number of tickets sold remained the same, then the revenue from ticket sales in February would have been equal to the revenue from ticket sales in January.

______________________
DS 2

During a wedding season, two event venues, Cedar Hall and Willow Hall, rented out their banquet spaces for private receptions. Was the range of rental fees charged at Cedar Hall greater than the range of rental fees charged at Willow Hall?

(1) The highest rental fee charged at Cedar Hall was greater than the highest rental fee charged at Willow Hall.
(2) The lowest rental fee charged at Cedar Hall was greater than the lowest rental fee charged at Willow Hall.
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