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Data Sufficiency Butler: June 2026
June 3DS 1DS 2

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DS 1

A coffee shop is running a customer raffle in which each slip in a raffle box is labeled either Coffee or Sandwich but not both, indicating the prize on that slip. For one promotional drawing, one slip is selected at random, returned to the box, and then a second slip is selected at random. What is the probability that one selected slip is labeled Coffee and the other is labeled Sandwich?

(1) The probability that the two selected slips have the same label is 5/8.
(2) The number of Sandwich slips in the box is 3 times the number of Coffee slips.

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DS 2

At the beginning of 2023, a fitness club had 800 active members. During 2023, some of the members who had been active at the beginning of the year canceled their memberships and did not rejoin, and every new member who joined during 2023 remained active through the end of the year. If x percent of the members active at the beginning of 2023 canceled their memberships and did not rejoin, and if the number of active members at the end of 2023 was y percent greater than the number active at the beginning of 2023, how many new members joined the club during 2023?

(1) x - y = 7
(2) x + y = 35
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Problem Solving Butler: June 2025
June 4PS 1PS 2

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PS 1

The nth term of a series is denoted by \((–1)^n 3^{(n-1)} + 2n\). What is the sum of the first six terms of this series?

A. 182
B. 194
C. 203
D. 224
E. 234
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PS 2

At a circular dinner table, five couples are seated randomly. What is the probability that everybody sits adjacent to his or her partner?

A. 24/10!
B. 24/9!
C. 48/9!
D. 48/10!
E. 768/9!
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Critical Reasoning Butler: June 2026
June 4CR 1CR 2

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CR 1

Landlord: Lead-based paint is the cause of many respiratory infections for our tenants, ranging from persistent coughs to even the onset of emphysema and lung cancer. In addition to the suffering of our residents, the lead-based paint in our buildings exposes us to the threat of lawsuits, which could jeopardize the building entirely and has already forced us to divert property-improvement funds to pay for a legal retainer. Because of the threat that lead paint poses to our community, we should therefore treat each affected wall with a coat of sealant.

Which of the following, if true, would most strengthen the landlord’s argument?

(A) Exposure to lead-based paint is particularly harmful to infants, children, and the elderly, groups that make up a disproportionate percentage of the building’s tenants.

(B) Removing lead-based paint is a less costly option than is paying to litigate or settle tenant claims of negligence.

(C) Sealing lead-based paint prevents lead chips and dust from being released into the air and vents, where it is most easily breathed by tenants.

(D) Every unit in the building has at least one wall that has been painted with lead-based paint.

(E) Other nearby buildings that have removed lead paint from their walls have reported massive decreases in the incidence of respiratory problems among tenants.
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CR 2

Hiker: With the storm approaching quickly we need to make a decision on how to proceed. We can try to descend the mountain to seek shelter, but then we will undo our progress and lose our chance to reach the summit – and we may not make it all the way down, anyway. We can continue on our current course and hope that the storm misses us or passes quickly, but doing so risks a true disaster. So we should bivouac here, using the time before the storm to set up a safe, secure camp to weather the elements.

The hiker’s argument does which of the following?

A. Claims that her decision is the only logical course of action.
B. Chooses a course of action based on experience in similar situations.
C. Makes a unilateral decision by dismissing the recommendations of others.
D. Recommends a choice based on the elimination of alternative options.
E. Outlines the risks and benefits of all available courses of action.
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Data Sufficiency Butler: June 2026
June 4DS 1DS 2

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DS 1

A concert venue’s ticket revenue increased by 15% from 2023 to 2024, while its operating costs increased by 10% over the same period. If net income is defined as ticket revenue minus operating costs, by what percent did the venue’s net income increase from 2023 to 2024?

(1) The venue’s net income in 2023 was $120,000.
(2) In 2023, the venue’s ticket revenue was twice its operating costs.
_____________________
DS 2

At a school fundraiser, a snack table sold four types of items: fruit cups, granola bars, sandwich boxes, and salad bowls. A fruit cup cost $3, a granola bar cost $5, a sandwich box cost $9, and a salad bowl cost $14. If the snack table sold 21 items in total, at least one of each type, and collected $180, how many granola bars were sold?

(1) The number of granola bars sold was equal to the number of sandwich boxes sold.

(2) The snack table sold 4 fruit cups and 5 sandwich boxes.
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Problem Solving Butler: June 2025
June 5PS 1PS 2

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A team of 4 has to be made from a group of 4 boys and 5 girls. How many teams are possible such that there are at least two boys in the team?

A. 126
B. 121
C. 120
D. 86
E. 81

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The value x is randomly chosen from the integers 0 to 15 (both inclusive). The selected value is then put in the expression: f(x) = x^2 – 7x – 18. What is the probability that the resultant value of f(x) is negative?
A. 9/16
B. 11/16
C. 3 /4
D. 5/8
E. 1/2
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Critical Reasoning Butler: April 2026
June 5CR 1CR 2

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CR 1

Many people argue that setting the minimum age for drinking alcohol at 21 in the United States actually promotes teenage drinking. Believers in this position say that teenagers derive excitement from breaking the rules and drinking before they are allowed to. However, in several European countries where there is no legal drinking age, the incidence of teenage drinking is similar to the rate in the United States.

Which of the following statements can most reasonably be inferred from the information above?

A) Setting the minimum age for drinking alcohol at 21 cannot be the only factor that affects the incidence of teenage drinking.

B) Setting the minimum age for drinking alcohol at 21 does not play a role in causing teenagers to begin drinking.

C) Setting a drinking age does not reduce the amount of alcohol consumed by teenagers.

D) Drinking alcohol can cause teenagers to engage in other illegal and illicit activities, such as smoking tobacco.

E) A higher percentage of teenagers drink in the United States than do in France.

____________________
CR 2

A talent agent analyzed her company's records in an attempt to determine why they were placing so few actors in roles. She attributed the company's poor performance to the fact that often the actors sent to an audition were completely inappropriate for the role.

It can be concluded from the passage above that the agent believes that

A. certain actors are inappropriate for certain roles
B. the actors her company represents are not very good
C. it is difficult to predict how appropriate an actor will be for a role
D. her company does not send enough actors to audition for major roles
E. directors often cast actors who are inappropriate for the role
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Data Sufficiency Butler: June 2026
June 5DS 1DS 2

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DS 1

At a climbing gym, 4 junior climbers and 3 instructors were weighed for harness fitting. If the average (arithmetic mean) weight of all 7 people was 55 kilograms and the heaviest instructor weighed 80 kilograms, was the average (arithmetic mean) weight of the 4 junior climbers less than 53 kilograms?

(1) The average (arithmetic mean) weight of the other two instructors was 56 kilograms.

(2) The three lightest people in the group were junior climbers, each weighing 47 kilograms.


_________________________
DS 2


An electronics retailer ships Model X TV sets in shipping units. Each shipping unit consists of one TV set and its protective packaging. Before July 1, the protective packaging weighed 25% of the TV set’s weight. If, on July 1, the retailer begins shipping a newer version of the Model X TV set with revised protective packaging, by what percent will the shipping weight of each unit change on July 1?

(1) On July 1, the weight of the TV set will increase by 20%.

(2) On July 1, the revised protective packaging will weigh 50% of the newer version of the TV set’s weight.
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Problem Solving Butler: June 2025
June 8PS 1PS 2

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PS 1

The sequence \(x_1, x _ 2, ..., x _n\) is such that \(x_1 = 5\), \(x_2 = –5\), \(x_3 = 0\), \(x_4 = –2\), \(_x5 = 4\) and \(x_n = x_{n-5}\). If the sum of the first P terms as well as the sum of the first P–3 terms in the sequence is 18, P = ?

A. 39
B. 42
C. 45
D. 48
E. 51

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PS 2

Lily has a rubber ring. She makes a red mark on the ring and from that red mark, she puts a mark after every 72 degrees angle from the center of the ring. Then, she starts from the original red mark and puts a mark after every 45 degrees from the center of the ring. She now removes one piece each of every unique size as per the marks. Approximately, what fraction of the original ring does she remove?

A. 23%
B. 38%
C. 48%
D. 53%
E. 72%
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Critical Reasoning Butler: June 2026
June 8CR 1CR 2

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CR 1

­Dean: Anything that hinders the students hinders the University. Anything that hinders the University hinders the students. Therefore, the Senator's proposal, which includes a provision to restrict the employment of new professors, hinders the students.

The above argument depends logically on which of the following assumptions?

A) The Senator's proposal was designed to hinder the University.
B) Most professors at the University support the Senator's proposal.
C) The University will be unable to continue operation if the Senator's proposal is approved.
D) Educational institutions that are not universities also object to the Senator's proposal.
E) Restricting the employment of new professors would hinder the University.
___________________
CR 2

Every student at Overland High School receives guidance counseling from either Ms. Hanner or Mr. Land. A review of college admissions data for students graduating from Overland during the last five years shows that, on average, Ms. Hanner was the guidance counselor for fifty-five percent of the students who applied to four-year colleges and forty percent of the students who were accepted at four-year colleges. These figures were quite consistent from year to year over the last five years.

Which of the following can be logically inferred about the past five years from the statements above?

(A) The total number of students graduating from Overland each year has remained roughly constant.

(B) Mr. Land has been assigned the majority of students with poor study skills.

(C) The percentage of applicants who are accepted to four-year colleges is higher among students guided by Mr. Land than it is among students guided by Ms. Hanner.

(D) The career goals of the students under the guidance of Ms. Hanner and Mr. Land have remained largely unchanged.

(E) The overall grade point average of students under Mr. Land's guidance has been higher than the average of students under Ms. Hanner's guidance.
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