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Problem Solving Butler: July 2026
July 27PS 1PS 2

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PS 1

Alan's monthly income is m dollars. Every month he spends m/5 dollars on food, m/4 dollars on clothes and saves the rest. At those rates, in how many months will it take Alan to save $110, in terms of m?

A. 20/m
B. 200/m
C. m/20
D. m/200
E. 200m

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PS 2

How many integers between 11 and 353 are divisible by 3?

A) 111
B) 112
C) 113
D) 114
E) 115
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Critical Reasoning Butler: July 2026
July 27CR 1CR 2

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CR 1

­Many websites that owe their financial success to Oodles, a popular search engine, now claim that Oodles has destroyed their business by making slight alterations to the very search algorithm that initially fostered the websites’ success. However, such claims can be confidently rejected as greedy and self-entitled: the websites’ own data shows that the changes in Oodles’ algorithm have only resulted in 8-12% less traffic for the websites than they have grown accustomed to.

Which of the following, if true, would best strengthen the websites’ argument?

A. In recent months internet sales have become so competitive and profit margins so thin that even a slight drop in traffic is catastrophic for a website.
B. Leaked internal Oodles memos show that the search engine was aware that a number of the websites would be affected by the changes.
C. Oodles is not the only search engine on the internet, so the websites could make up the missing traffic elsewhere.
D. The drop in traffic is only temporary while Oodles finetunes the new algorithm.
E. Before making the changes to its algorithm, Oodles asked and received permission from the affected businesses to make the changes.

________________________
CR 2

Santa Clara island is home to a large population of wild horses, but horses are not native to the island. It is known that Spanish settlers colonized Santa Clara island and lived there continuously from 1650 to 1740, and the bones of horses were recently found in an excavation site that can be dated to the early 1700s. Therefore it is likely that the Spanish settlers were the ones who introduced horses to Santa Clara island.

The answer to which of the following questions would be most useful in evaluating the validity of the conclusion drawn above?

A. Did the Spanish settlers introduce horses to any other islands near Santa Clara?
B. Have any excavations found evidence of human settlers who lived on Santa Clara prior to 1650?
C. Were any other nonnative species introduced to Santa Clara island by the Spanish settlers?
D. Have any human settlements remained on Santa Clara island since 1740?
E. Does Santa Clara island contain any horse bones that can be dated to before 1650?
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Data Sufficiency Butler: July 2026
July 27DS 1DS 2

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DS 1

­At SoHo Corp, what is the ratio of managers to non-managers?

(1) If there were 3 more managers and the number of salespeople remained the same, then the ratio of managers to non-managers would double.

(2) There are 4 times as many non-managers as managers at SoHo Corp.
________________________
DS 2

­Joyce has a recipe for a batch of brownies and a recipe for a coffee cake. How many cups of flour does her recipe for a batch of brownies require?

(1) When Joyce bakes a batch of brownies and a coffee cake following her recipes, she uses a total of 9/2 cups of flour.

(2) When Joyce bakes 2 batches of brownies and 3 coffee cakes following her recipes, she uses a total of 11 cups of flour.
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Problem Solving Butler: July 2026
July 28PS 1PS 2

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PS 1

Mrs. Sharma has a few chocolate candies which she wants to distribute among her five grand-daughters. If she distributes the candies among them in the ratio of 1 : 4 : 5 : 7 : 9, then 1 candy is left out. If she distributes the candies in the ratio of 2 : 3 : 3 : 4 : 5, then 3 candies are left out. However, if she distributes the candies among them in the ratio of 1 : 2 : 3 : 7 : 8, then no candies are left over. What is the minimum number of candies that any of the five grand-daughters can receive in any of the three cases?

A. 3
B. 4
C. 5
D. 6
E. 12
_____________________
PS 2

An article, if sold at a discount of 30%, results in a loss of 35 dollars. However, there is a profit of 5 dollars if sold at a discount of 10%. What would be the profit or loss if it was sold without any discount?

A. 10%
B. 12.1%
C. 13.5%
D. 14.3%
E. 15.2%
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Critical Reasoning Butler: July 2026
July 28CR 1CR 2

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CR 1

Film critic: The documentary film The Decline of English Literature presents the view that contemporary British writers lack the basic narrative skills common among nineteenth-century British authors. The film's comparison is persuasive because it analyzes 20 popular, critically respected British novels, 10 from the nineteenth century and 10 from the past two decades, and demonstrates that the contemporary novels are not nearly as well-written as the nineteenth-century novels.

Which of the following, if true, casts the most doubt on the argument above?

(A) A film viewer who was not familiar with literary criticism may not be convinced by the film’s analysis of 20 novels.

(B) The film critic may favor nineteenth-century fiction and may therefore be biased in favor of the filmmaker’s view.

(C) The film’s title may predispose viewers to accept the film’s conclusion even before viewing the comparisons in the film.

(D) The filmmaker could have chosen to analyze only books that support the position she was trying to establish.

(E) The literary methods used by contemporary British authors could require less narrative skill than methods used by writers in other parts of the world.
______________________
CR 2

It is inaccurate to say that a diet high in refined sugar cannot cause adult-onset diabetes, since a diet high in refined sugar can make a person overweight, and being overweight can predispose a person to adult-onset diabetes.

The argument above is most parallel, in its logical structure, to which of the following?

(A) It is inaccurate to say that being in cold air can cause a person to catch a cold, since colds are caused by viruses, and viruses flourish in warm, crowded places.

(B) It is accurate to say that no airline flies from Halifax to Washington. No single airline offers a direct flight, although some airlines have flights from Halifax to Boston and others have flights from Boston to Washington.

(C) It is correct to say that over-fertilization is the primary cause of lawn disease, since fertilizer causes lawn grass to grow rapidly and rapidly growing grass has little resistance to disease.

(D) It is incorrect to say that inferior motor oil cannot cause a car to get poorer gasoline mileage, since inferior motor oil can cause engine valve deterioration, and engine valve deterioration can lead to poorer gasoline mileage.

(E) It is inaccurate to say that Alexander the Great was a student of Plato; Alexander was a student of Aristotle, and Aristotle was a student of Plato.
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Data Sufficiency Butler: July 2026
July 28DS 1DS 2

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DS 1

­There are three times as many hot dogs as pizzas and five times as many brownies as hot dogs in "Junk Food 'R' Us" restaurant. The restaurant sells only whole products (you cannot buy half a pizza, for example). How many brownies are there in the restaurant?

(1) There are at least 10 pizzas in the restaurant

(2) There are 32 hot dogs at most in the restaurant
_____________________
DS 2

The number of cats in Libby’s house is twice the number of dogs and one third the number of fish. If cats, dogs and fish are the only pets in Libby’s house, what is the probability of randomly selecting two cats?

(1) The number of cats in Libby’s house is 12.
(2) The total number of pets in Libby’s house is 54.
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Problem Solving Butler: July 2026
July 29PS 1PS 2

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PS 1

Which of the following is a prime number?

I. 9! + 5
II. 11! + 15
III. 13! + 13

A) I only
B) II only
C) III only
D) II and III only
E) None

__________________
PS 2

The total amount of Mark’s cell phone bill for the month of April was $120.75. The bill consisted of a fixed charge of $82.50 plus a charge of $0.025 per minute for data usage. For how many hours of data usage was Mark charged for the month of April?

A. 22
B. 22.5
C. 23
D. 24
E. 25.5
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Critical Reasoning Butler: July 2026
July 29CR 1CR 2

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CR 1

­Even the smallest amounts of chlorofluorocarbons contribute to the needless and rapid destruction of the Earth's protective ozone layer. The destruction of the ozone layer will lead to catastrophic global warming and greatly increased incidence of skin cancer. The loss of the ozone layer is unacceptable, so chlorofluorocarbons should not be used, especially since _________.

Which of the following best completes the passage?

A. most people live in areas where the ozone layer is thickest

B. users of chlorofluorocarbons are generally unaware of the risks involved

C. most chlorofluorocarbons are used in small amounts

D. safe and equally cheap replacements are readily available

E. the ozone layer serves to protect human life in a variety of ways

______________________
CR 2

A large Southern state university has changed its teaching practices. Formerly, instructors without PhDs taught most introductory courses; now professors with PhDs will teach all introductory classes. That means the average class size will increase from 44 students per class to 600 per class, but overall the students’ learning experience should improve.

Which one of the following is an assumption required by this argument?

(A) Requiring professors with PhDs to teach all introductory classes will mean that the university must hire more faculty with doctorates.

(B) Students tend to participate in smaller classes more than they do in large lectures, even when the lectures are supplemented by weekly discussion sections.

(C) Major private universities already have implemented a format in which professors with PhDs teach all introductory classes as large lectures.

(D) A class taught by a PhD, even in a lecture format with hundreds of students, is a better learning environment than a smaller class taught by an instructor without a PhD.

(E) Services that rank colleges and universities usually consider the percentage of classes taught by PhDs when computing rank.
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Data Sufficiency Butler: July 2026
July 29DS 1DS 2

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DS 1

Paula has a collection of used books worth $104. How many are histories?

(1) Paula has only mysteries and histories; each mystery is worth $1, and each history is worth $25.
(2) Paula has exactly 4 mysteries in her collection that together are worth a total of $4.
___________________
DS 2

­If company Y’s profits decreased $1.5 million from last year to this year, what was the percent decrease in profits?

(1) If the profits had decreased by $2 million, there would have been a 40 percent decrease.

(2) This year’s profits were $3.5 million
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Problem Solving Butler: June 2026
July 30PS 1PS 2

_____________________
PS 1

Samantha purchased a five-year municipal bond that pays simple interest for $600. After the bond matured five years later, she had received a total of $720 in principal plus interest payments. What is the interest rate of Samantha's bond?

A. 4%
B. 5%
C. 10%
D. 12%
E. 20%
_____________________
PS 2

Mrs. Anderson divided $420 evenly among her children. Later, when Mrs. Anderson was away, Daniel, the eldest, took $5 from each of his brothers and sisters. Daniel ended up with $90. If Mrs. Anderson has more than 7 children, how many children does Mrs. Anderson have?

A. 12
B. 14
C. 16
D. 18
E. 20
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Critical Reasoning Butler: July 2026
July 30CR 1CR 2

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CR 1

­Susan read in her local paper that the average (arithmetic mean) income in her community last year was $38,000. She worked very hard to make $36,000 last year and was upset that she earned less than the majority of the earners in her community.

Which of the following, if true, most calls into question Susan’s belief that she earned less than the majority of the earners in her community?

(A) The standard deviation of earnings in her community was greater than $2,000.

(B) The mean earnings in the community were $3,000 greater than the median earnings.

(C) The mean earnings in the community were $1,000 greater than the median earnings.

(D) Doctors in Susan's community had an average income over $200,000.

(E) The average income of $38,000 is calculated from the incomes of both full- and part-time workers.
_____________________
CR 2

­Gas-fired and oil-fired power plants emit some pollutants into the air. However, coal-fired plants, as presently engineered, emit far more noxious air pollutants, causing acid rain to fall hundreds of miles away from the plants themselves. Therefore, in order to reduce air pollution, we should close down the coal-fired plants and replace them with gas and oil-fired plants.

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

A. In addition to the air emissions from coal-fired power plants, mining coal can pollute water, while gas and oil-fired plants pollute only air.

B. Coal plants can easily be redesigned to eliminate most emissions into the atmosphere.

C. Gas-fired power plants are more cost efficient than coal-fired power plants.

D. As supplies of oil decline, the demand for natural gas and coal will increase.

E. Engineers have developed a model to reduce emissions from oil and gas-fired plants to an imperceptible amount.
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Data Sufficiency Butler: July 2026
July 30DS 1DS 2

________________________
DS 1

­In a certain telephone poll, 600 people were asked whether they were in favor of, against, or undecided on a certain bill being debated in the legislature. How many of the people polled were in favor of the bill?

(1) The number of people who were in favor of the bill was 200 greater than the number of people who were against it.

(2) Two hundred people were undecided, which was twice as many as the number who were against the bill.

________________________
DS 2

­If Joe and Sam can complete a job in 2 hours working together, what fraction of the job was done by Joe?

(1) If Joe had worked alone, he would have completed the job in 3 hours.

(2) Sam did one-third of the job.
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