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Problem Solving Butler: January 2025
January 26PS 1PS 2

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PS 1

If \((\frac{1}{50})^a(\frac{1}{20})^4 = \frac{1}{50} (100)^{-6}\), a = ?

A. 3
B. 4
C. 5
D. 6
E. 7

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PS 2

What is the highest integer k for which 125^k perfectly divides 200! ?

A. 49
B. 48
C. 16
D. 13
E. 1
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Critical Reasoning Butler: January 2025
January 26CR 1CR 2

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CR 1

A new town ordinance that requires all home owners to install burglar alarms and other anti-theft measures on their property has received stiff opposition from those who believe that citizens of a free society must be allowed to freely take risks, as long as those risks do not harm others. Resultantly, they conclude that it should be each home owner’s decision to install the anti-theft measures.

Which of the following, if true, most seriously weakens the conclusion drawn above?

A. Many new homes come with pre-installed anti-theft measures.
B. Property owners who do not install anti-theft measures, on average, experience greater monetary loss due to burglary than those who do install such measures.
C. Banks are required to install extremely stringent anti-theft measures.
D. The rate of burglary in states that do not have laws mandating the installation of anti-theft measures is greater than that in states that do have such laws.
E. Theft insurance rates are higher for everyone because of the need to pay for the increased losses sustained by those without anti-theft measures.
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CR 2

Lunacorp and Salem Industries are two sanitation firms that employ an equal number of workers who work the same number of hours per week. According to each company’s end of year report, the employees of Lunacorp took fewer sick leaves last year than did the employees of Salem Industries. Therefore, employees of Lunacorp are less likely to fall ill than are employees of Salem Industries.

Which of the following, if true, would most strengthen the conclusion above?

A. Lunacorp’s expenditure on new job-related medical claims is greater than Salem Industries’.
B. Salem Industries holds more health inspections than does Lunacorp.
C. Salem Industries’ infirmary is more modern than Lunacorp’s.
D. Salem Industries provides sanitation services for environments that are more hazardous for its employees to work in than does Lunacorp.
E. Salem Industries provides more types of health-care benefits than does Lunacorp.
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Data Sufficiency Butler: January 2025
January 26DS 1DS 2

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DS 1

Lily has only red and blue balls in a jar. If she removes three balls, is the probability of getting all red balls greater than the probability of getting at least one blue ball?

(1) The number of red balls is more than three times the number of blue balls.
(2) Less than 1 / 4 th of the balls in the jar are blue.

___________________________
DS 2

How many people did Casax Inc. employ in 2008?

(1) In 2007, Casax Inc.’s workforce numbered 100,000 people.
(2) In 2008, Casax Inc.’s workforce increased by 2 percent, or 2,000 people, over 2007.
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Problem Solving Butler: January 2025
January 27PS 1PS 2

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PS 1

K is a factor of 30! + 40!. If K is a prime number less than 40, the highest possible value of K is?

A. 19
B. 23
C. 29
D. 31
E. 37
____________________
PS 2

If y = 80x + 36, where x is a positive integer, which of the following cannot be factor of y ?

A. 3
B. 7
C. 12
D. 14
E. 24
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Critical Reasoning Butler: January 2025
January 27CR 1CR 2

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Technician: It is true that the Nebula G10 smartphone is not as valuable as the Outlook M1 smartphone because, on average, insurance companies do not pay as much to replace a lost Nebula G10 as to replace a lost Outlook M1. Repair costs show a different picture though. It can’t be denied that the repairs to Nebula G10 phones are especially labor-intensive and labor is a significant factor in accident repair costs. As a result, the average amount that insurance companies pay to repair a phone under warranty that is damaged by an accident is typically higher for the Nebula G10 than for the Outlook M1.

Which of the following, if true, most strongly supports the technician’s explanation?

A. Reliable statistics show that Outlook M1 phones are more frequently damaged in accidents than Nebula G10 phones are.
B. Outlook M1 phones are considered more economically valuable largely for the ease of repair of the phones.
C. Availability of technicians specializing in the repairs of Nebula G10 is higher than that of technicians specializing in the repairs of Outlook M1.
D. No significant difference exists between the cost of routine upkeep of Outlook M1 and that of Nebula G10.
E. Consumer data indicates that the Nebula G10 phones are returned for defects at a rate greater than that for Outlook M1 phones.
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CR 2

Forensic scientists trained to authenticate antiques do so by analyzing not only the physical form of the antique but also such properties as the chemical composition of the materials it is composed of and microscopic signs of damage that indicate its age. Even the most adept forgers cannot duplicate all of the properties the scientists analyze.

Which of the following can be logically concluded from the passage above?

A. The time it takes for the forensic scientists to examine an antique and determine its authenticity makes their services impractical for commercial use.
B. Nobody can secure authentication from a forensic scientist trained to authenticate antiques by relying solely on a skill for forging antiques.
C. Forensic scientists trained to authenticate antiques will soon be hired by most museums.
D. Forensic forgery-detection training has taken many years to formulate and perfect.
E. In many cases even genuine antiques are denied authentication by forensic scientists trained to authenticate antiques.
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Data Sufficiency Butler: January 2025
January 27DS 1DS 2

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Each of the 600 cars in a certain parking lot is either a hatchback or a sedan. How many of the hatchbacks are electric?

(1) 75% of all cars in the parking lot are electric.
(2) The number of electric sedans is 25% more than the number of electric hatchbacks.
_______________________
DS 2

Last week, Cosmo repaired multiple motorbikes and cars. How many hours did Cosmo spend repairing motorbikes last week?

(1) Last week, Cosmo spent a total of 40 hours repairing both cars and motorbikes.
(2) Last week, the average (arithmetic mean) number of hours that Cosmo spent repairing each motorbike was 20 percent greater than the average number of hours he spent repairing each car.
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Problem Solving Butler: January 2025
January 28PS 1PS 2

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PS 1

If \(\frac{9^{x-1}}{27^{3x+2}} = 81^{x+1}\), x = ?

A. 11/12
B. 12/11
C. 4/3
D. -11/12
E. -12/11
_________________________
PS 2

If P is a positive integer and Q = 5P + 7, which of the following CANNOT be a divisor of Q?

A. 2
B. 3
C. 5
D. 7
E. 11
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Critical Reasoning Butler: January 2025
January 28CR 1CR 2

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CR 1

The introduction of ladybugs into a garden ecosystem is an excellent method of protecting ornamental flowers from aphids. Unfortunately, the ladybugs will only prey on the aphids for a week before flying away, and no aphid infestation severe enough to warrant the deployment of ladybugs can be cleared in a single week. However, although synthetic pesticides are not much longer lasting, they can be easily reapplied whenever the aphids reappear. Clearly, therefore, for protecting ornamental flowers from aphids, introducing ladybugs can play no useful role.

The argument is most vulnerable to the objection that it fails to

A. consider that there might be less severe aphid infestations that can be cleared in a week
B. allow for the possibility of combining the two aphid-control methods it mentions
C. distinguish among the several varieties of synthetic pesticides
D. mention any alternative aphid control method that could be utilized in place of the rejected method
E. allow for the possibility that the introduction of insects other than ladybugs might have the same effect.
__________________________
CR 2

Environmentalist: Cod prices in our area need to be increased significantly. The cod population in the nearby ocean has been falling for years because cod is being fished at unsustainable levels. Due to the low price of cod, few consumers have been affected by the introduction of the hefty environmental tax on the sale of this endangered fish.

The two sections in boldface play which of the following roles in the environmentalist’s argument?

(A) The first is a conclusion that the argument provides support for, and which in turn provides support for the main conclusion; the second is the main conclusion.
(B) The first is the only conclusion; the second provides an explanation for the first.
(C) The first is an observation that the second provides an explanation for; the second is the main conclusion but not the only conclusion.
(D) The first is a premise that supports the main conclusion of the argument; so is the second.
(E) The first is the main conclusion; the second is a conclusion supported by the argument, and which in turn supports the first.
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Data Sufficiency Butler: January 2025
January 28DS 1DS 2

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DS 1

Is the median monthly salary of 15 randomly selected employees of a particular company greater than the average (arithmetic mean) monthly salary by at least 5 percent?

(1) The median monthly salary of the 15 employees is $250 greater than their average monthly salary.
(2) The sum of the monthly salaries of the 15 employees is less than $70,000.

________________________
DS 2

Jack paid $30 at a salon; this included the salon fee, taxes, and tip. How much tax did Jack pay in the transaction?

(1) Tax and Tip together contributed to 25% of the total expense.
(2) Salon fee and the tip together contributed to 90% of the total expense.
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Data Sufficiency Butler: January 2025
January 29DS 1DS 2

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DS 1

The average selling price per sofa-set for a showroom in 2016 was $40 more than that in 2015. What was the revenue for the showroom, from the sale of sofa-sets, in 2015?

(1) The showroom sold 20% more number of sofa-sets in 2015 than in 2016.
(2) The revenue of the showroom, from the sale of sofa-sets, was $30,000 more in 2015 than in 2016.

________________________
DS 2

A university offers 3 courses in mathematics: a BS program, an MS program, and a dual degree program. The annual tuition fee for the dual degree program is 20% less than the annual tuition fees for the other 2 programs taken together. If 60% of mathematics students in the upcoming class will attend the BS program and the remaining are equally divided between the other 2 programs, which of the 3 programs will generate the most tuition fees for the university in the coming year?

(1) The annual tuition fee for the MS program is 30% higher than that of the BS program.
(2) The annual tuition fee for the BS program is $30,000.
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Critical Reasoning Butler: January 2025
January 29CR 1CR 2

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CR 1

Candidate: Today’s voters expect their government to be accountable. Virtually, every improvement in government’s accountability inspires a new round of demands for greater accountability. If a politician is satisfied with the government’s current level of accountability, that politician will soon find that the constituents are not. As your representative in parliament, meeting your demands for accountability will be my goal!

Which of the following must be true on the basis of the statements above?

A. The candidate’s competitors will succeed in attracting voters only if those competitors adopt the candidate’s goals as their own.
B. The candidate’s goal is possible to meet, only if it is possible to continue the current advances in government accountability.
C. A politician who fails to accurately anticipate the demands of the constituents is certain to fail in advancing government accountability.
D. If a politician becomes satisfied with the accountability that the government displays, the government accountability will decline.
E. The candidate’s constituents are currently satisfied with the accountability displayed by the government.
___________________________
CR 2

The World Energy Council maintains ranking of nations, detailing which country’s citizens are the most and least energy conscious by measuring the amount of electricity consumed per every 100 citizens.

Of the following, each statement, if true, would serve as a logical objection to using the council’s ranking as a representation of the energy consciousness of citizens in each nation EXCEPT:

A. The refrigeration of food consumes significantly less electricity in countries with a temperate or arctic climate than it does in countries with a tropical climate.
B. Some countries have hundreds of millions of citizens and some barely have a few millions.
C. The average citizen in developed nations can afford to purchase more energy-efficient technology than can citizens of less developed nations.
D. Due to low electricity production, many developing countries regularly shut down residential electrical supply for as long as a few hours every day for several months in a year.
E. The power grids of some countries are in poor shape and thus waste more electricity than do power grids of other countries.
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Problem Solving Butler: January 2025
January 29PS 1PS 2

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PS 1

If a, b, c, d, e, and f represent different prime numbers each, which of the following may represent the value of 16! ?

A. a^15 x b^6 x c^3 x d^2 x e^2 x f
B. a^15 x b^5 x c^3 x d^2 x e x f^2
C. a^15 x b^6 x c^3 x d^2 x e x f
D. a^8 x b^5 x c^3 x d^2 x e x f
E. a^8 x b^5 x c^2 x d^2 x e x f
_________________________
PS 2

p, m, and n are positive integers greater than 1 such that p = m x n and m and n do not have any common prime factors. If p has 25 factors, m has r factors, and n has s factors, what is the value of (r + s)?

A. 4
B. 8
C. 10
D. 16
E. 25
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Problem Solving Butler: January 2025
January 30PS 1PS 2

_______________________________
PS 1

If 3p^5 = 96 and 7q = 32, what is the value of p/q?

A. 16/7
B. 7/16
C. ±16/7
D. ±7/16
E. Cannot be determined from the given information
_______________________________
PS 2

p, q, r, s are positive integers. When p is divided by q, the remainder is 7; when r is divided by s, the remainder is 12. Which of the following may be a value of q + s?

A. 15
B. 17
C. 19
D. 20
E. 23
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