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Problem Solving Butler: June 2025
June 9PS 1PS 2

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PS 1

Letters of the word “POLITICS” are shuffled to form all possible distinct combinations. In how many cases do the letters P, O, and L come together?

A. 240
B. 720
C. 2160
D. 4320
E. 8640
________________________
PS 2

A group of five boys and six girls is to be seated in a row. How many such arrangements are possible wherein no two boys sit together?

A. 6! x 5!
B. 6! x 5! x 2
C. 6! x 6!
D. 6! x 5! x 21
E. 7! x 6!
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Critical Reasoning Butler: June 2026
June 9CR 1CR 2

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CR 1

­The average age of residents in Sunset City is well over 70 years of age, and the populace is composed almost entirely of retirees, with extremely few families having young children. However, Sunset City is also known for its many boutiques that specialize in selling clothing for infants and toddlers.

Which of the following, if true, best reconciles the seeming discrepancy described above?

A. The children's clothing boutiques in Sunset City purchase their clothing from wholesalers in Sunrise City, where there are many more families with small children.

B. The few children who do live in Sunset City do not have particularly large wardrobes.

C. The residents of Sunset City are known to spend more on health care than do residents of nearby cities.

D. The residents of Sunset City are generally wealthy and are known to spend large amounts of money buying gifts for their grandchildren and great-grandchildren who live in other cities.

E. Sunset City is not located on a major highway.

_____________________
CR 2

Tougher international restrictions on importing pandas to zoos in the United States are not to blame for the recent shortage of pandas on display in U.S. zoos. In fact, the real reason fewer zoos today have fewer pandas on display than 10 years ago is that the reproduction rates of pandas have decreased by 40% and the cost of upkeep of pandas has increased by 30% over the same time period.

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

A) Over the past 10 years, no pandas have been imported to the United States.
B) Since there are fewer pandas in the world today than 10 years ago, zookeepers have found it difficult to secure pandas for their zoos, especially since the cost to secure and support the panda is twice that of any other animal.
C) The recent drought in China and southeastern Asia has severely decreased the bamboo shoots available for pandas to eat throughout the world.
D) A recent study has concluded that the animal people look forward to seeing the most at zoos is the koala bear.
E) An increase in the number of zoos throughout the United States has increased the demand for pandas by medium-sized and small zoos.
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Data Sufficiency Butler: June 2026
June 9DS 1DS 2

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DS 1

At a bookstore promotion, each of five customers won a chance to receive a discount. For each customer, five sealed envelopes were prepared. The five envelopes prepared for any customer contained the discounts $10, $20, $30, $40, and $60, one discount per envelope. If each customer selected one of the five envelopes prepared for that customer, did any customer receive the $60 discount?

(1) The sum of the discounts received by the five customers was $150.
(2) No two customers received the same discount.


____________________________
DS 2

A warehouse uses Metro Courier to ship parcels. Metro Courier charges $12 per parcel for the first 12 parcels shipped in a day and $8 for each parcel after the first 12. If the warehouse used only Metro Courier on Tuesday, did the warehouse ship more than 40 parcels on Tuesday?

(1) If the warehouse had shipped 3 additional parcels on Tuesday, its shipping cost for that day would have been at least $400.
(2) The warehouse’s average (arithmetic mean) shipping cost per parcel on Tuesday was $9.
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Problem Solving Butler: June 2025
June 10PS 1PS 2

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PS 1

An event was attended by 12 members. Each member shook the hand of all others, except four siblings who didn’t shake each others’ hands as they came together to the event. What was the number of handshakes during the event?

A. 54
B. 60
C. 66
D. 120
E. 128
________________
PS 2

In how many ways can a group of 8 people be divided into three groups, with 2, 3, and 3 members each?

A. 140
B. 280
C. 420
D. 560
E. 840
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Critical Reasoning Butler: June 2026
June 10CR 1CR 2

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Recently proposed legislation would change the way that the federal government treats the donation of used cars to charity. Under current rules, people can donate used cars and take a deduction equal to the full value of the car on their taxes. Under new rules, only a small fraction of the value of the car could be deducted. Since many charities rely on the income produced from used car donations, the new rules will severely impair the financial capabilities of these charities, and possibly cause some to shut down.

The argument above assumes which of the following?

A. Without the incentive of a deduction of the full value of the car, at least some individuals will no longer donate their used cars to charity as they otherwise would have.

B. Money generated from the sale of donated used cars provides the only source of funding for many charities.

C. The only reason for adopting the proposed legislation is to prevent the abuse of the system by car brokers who make money by buying used cars donated to charities and then selling them for large profits.

D. People that donate cars to charities do not donate cash or other goods to charity.

E. Income tax laws should be changed so that people who wish to donate a car can sell the car themselves and then donate the proceeds for a deduction.

________________________
CR 2

A local hardware store is concerned that its sales will diminish when a competing hardware store opens in the same area next month. However, it is actually more likely that the existing store’s sales will increase. Over the next few months, due to high demand for mixeduse residential and commercial buildings, over three times as many construction projects will be started in the area as were started over the past five years.

The argument above depends on which of the following assumptions?

(A) The managers of the new construction projects will purchase an equal amount of material from both hardware stores.

(B) The construction projects will require the purchase of materials that are sold at the existing hardware store.

(C) The materials sold at the existing store are higher in quality than those to be sold at the new hardware store.

(D) The area revitalization spurred by new mixed-use developments will encourage local residents to perform improvements on their own homes.

(E) The existing hardware store will lower its prices below those of its competitors to attract new customers.
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Data Sufficiency Butler: March 2026
June 10DS 1DS 2

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DS 1

A city council distributed a $320,000 grant among exactly three programs: Youth Sports, Public Gardens, and Senior Services. Did any one of the three programs receive more than 37.5% of the total grant?

(1) Senior Services received more funding than each of the other two programs.
(2) Youth Sports received 1.5 times as much funding as Public Gardens.

________________
DS 2

A bakery display case contains only croissants, muffins, and scones. If the ratio of croissants to muffins is 4 to 9 and the display case contains 120 pastries in total, how many more muffins than scones are in the display case?

(1) The ratio of croissants to scones is 4 to 7.
(2) The number of scones is 18 more than the number of croissants.
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Problem Solving Butler: June 2025
June 11PS 1PS 2

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PS 1

At a circular dinner table, five couples are seated randomly. Two arrangements are considered different only if the relative positions of people change. What is the total number of arrangements in which everybody sat adjacent to his or her partner?

A. 32
B. 48
C. 384
D. 576
E. 768

_________________
PS 2

From a group of 5 boys and 5 girls, 4 boys and 3 girls are to be selected and made to sit in a row. How many different arrangements are possible if boys cannot take the corner seats?

A. 9,000
B. 18,000
C. 27,000
D. 36,000
E. 72,000
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Critical Reasoning Butler: June 2026
June 11CR 1CR 2

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­Purchasers of laptop computers generally do not use them often enough to justify the cost. Presumably, laptop computers are more expensive than desktop computers due to the laptop computers’ portability. However, one survey showed that only 22% of laptop computer owners had used them outside the home in the past 30 days.

The conclusion that purchasers of laptop computers do not use them often enough to justify the cost is most called into question by which of the following?

(A) Some laptop computer owners do not like to take their laptops to school or work out of fear that the computers will be damaged or stolen.

(B) Laptop computer owners sometimes overestimate how often they use their laptops outside the home.

(C) Laptop computer owners often use their laptop computers in different locations around their own homes.

(D) Laptop computer owners usually prefer to purchase computers that have wireless Internet capability, so the computers are more mobile.

(E) Laptop computer owners who use their laptop computers outside the home tend to do so at least three times per week

____________________
CR 2

­­At Food World Supermarket, built in 1975, the number of successful thefts has risen dramatically in the last few months. Food world has a reliable electronic security system at all the customer exit doors, and this system is always operational. Therefore the thefts must have been committed by people who used exits other than the regular customer exit doors.

Which of the following is assumption that would make the conclusion above logically correct ?

(A) If a surveillance system is installed in a supermarket, it is always equipped at every possible exit.

(B) If an employee so wishes, he is allowed to leave thorugh an exit that is not monitored by surveillance system.

(C) If a store has reliable security system, it is impossible to pass through the system undetected.

(D) If a supermarket was built before 1980, it often has exits that cannot be equipped with electtronic surveillance.

(E) If a store has a reliable electronic security system but is still experiencing a rise in theft, it must be the case that the employees must are stealing.
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Data Sufficiency Butler: June 2026
June 11DS 1DS 2

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DS 1

A language-learning app charges customers on a weekly basis. For billing purposes, a partial week of a subscription is treated as a full week, and the customer is charged the full weekly fee for it. If Laura’s subscription length was converted into billable weeks under this rule, how many billable weeks was she charged for?

(1) Laura’s total subscription charge was $180.

(2) If Laura had ended her subscription 2 days earlier, she would have been charged $45 less.
____________________
DS 2

At a print shop, the manager recorded the number of flyers produced by each of 10 copy machines during one shift. What was the standard deviation of the numbers of flyers produced by the 10 copy machines?

(1) Each of the 10 copy machines produced either 40 or 60 flyers.

(2) Exactly 6 of the 10 copy machines produced the same number of flyers.
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Problem Solving Butler: June 2025
June 12PS 1PS 2

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PS 1

If a and b are two-digit integers such that a > 60 and b < 80, which of the following is closest to the maximum possible value of ab ?

(A) 490
(B) 4,800
(C) 5,600
(D) 8,000
(E) 10,000
___________________
PS 2

If x is the sum of the reciprocals of the five consecutive even integers starting with 40, which of the following is true?

(A) 0.00 <x<0.05
(B) 0.05 <x<0.10
(C) 0.10 <x<0.15
(D) 0.15 <x<0.20
(E) 0.20 <x <0.25
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Critical Reasoning Butler: April 2026
June 12CR 1CR 2

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CR 1

When parts of the skeleton of a Neanderthal male were assembled together recently, by scientists they noticed a fatal rib wound. They assumed that it was the result of a hunting accident or a conflict with a fellow Neanderthal. But modern research indicates that this Neanderthal may have been killed by a human being.

Which of the following, if true, would most strengthen the conclusion drawn by the researchers?

A. Neanderthals only hunted in the forests, using thrusting spears.

B. At that time, only humans had developed projectile weapons.

C. The Neanderthal's wound was, in all likelihood caused by a thrown spear.

D. At that time only human beings had the type of weapon that would inflict this kind of wound.

E. Neanderthals and humans often came into contact with each other.
__________________
CR 2

­Although it is commonly believed that a healthy diet consists largely of fresh produce, many nutritionists in fact recommend that certain fruits and vegetables be purchased frozen. Which of the following, if true, would most help explain the nutritionists' recommendations?

A. Fruits and vegetables are consumed on their own by some and are used in baking by others.

B. The cost of shipping frozen fruits and vegetables is generally lower than shipping them fresh.

C. Cold storage technology has increased the storage life of apples to almost a year.

D. Frozen produce, unlike fresh, is picked at its peak and thus contains a richer nutritional profile.

E. Nutritionists' advice tends to be distorted and reframed by the media.
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