Here is a list of the numbers of fish that Tyler caught in nine outings last summer: 2, 0, 1, 3, 0, 3, 3, 1, 2. Which statement about the mean, median, and mode is true?
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Answer: C — mean < median < mode.
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Hint 1
Sort the list. Mode = most common value, median = middle (5th of 9), mean = sum/9.
In a town of 351 adults, every adult owns a car, motorcycle, or both. If 331 adults own cars and 45 adults own motorcycles, how many of the car owners do not own a motorcycle?
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Answer: D — 306.
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Hint 1
Everyone owns at least one. So car-only count = total − motorcycle-owners.
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Approach: everyone has at least one ⇒ car-only = total − motorcycle-owners
Each non-motorcycle-owner must own a car (since every adult has at least one).
Bag A has three chips labeled 1, 3, and 5. Bag B has three chips labeled 2, 4, and 6. If one chip is drawn from each bag, how many different values are possible for the sum of the two numbers on the chips?
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Answer: B — 5 different values.
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Hint 1
All sums are odd + even = odd. List the 9 sums and count distinct values.
The taxi fare in Gotham City is $2.40 for the first 12 mile and additional mileage charged at the rate $0.20 for each additional 0.1 mile. You plan to give the driver a $2 tip. How many miles can you ride for $10?
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Answer: C — 3.3 miles.
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Hint 1
Subtract the $2 tip from $10. Then subtract the $2.40 flag-drop for the first half-mile. Convert the rest at $0.20 per 0.1 mile = $2 per mile.
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Approach: peel off fixed costs, then divide the remainder
Available for fare: $10 − $2 = $8.
After the first 1/2 mile costing $2.40: $8 − $2.40 = $5.60 left.
Additional rate: $0.20 / 0.1 mile = $2 per mile. So $5.60 buys 2.80 miles.
Angie, Bridget, Carlos, and Diego are seated at random around a square table, one person to a side. What is the probability that Angie and Carlos are seated opposite each other?
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Answer: B — 1/3.
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Hint 1
Fix Angie's seat. Carlos lands in any of the 3 remaining seats with equal probability; only 1 is opposite.
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Approach: fix one seat, count Carlos's options
Fix Angie in any seat. Carlos has 3 equally likely seats among the remaining 3.
There are 270 students at Colfax Middle School, where the ratio of boys to girls is 5 : 4. There are 180 students at Winthrop Middle School, where the ratio of boys to girls is 4 : 5. The two schools hold a dance and all students from both schools attend. What fraction of the students at the dance are girls?
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Answer: C — 22/45.
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Hint 1
Each ratio uses 9 parts. Compute girls at each school, then total girls / total students.
Let A be the area of the triangle with sides of length 25, 25, and 30. Let B be the area of the triangle with sides of length 25, 25, and 40. What is the relationship between A and B?
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Answer: C — A = B.
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Hint 1 of 2
Each triangle is isosceles. Drop the altitude to the unequal side and use the Pythagorean theorem.
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Hint 2 of 2
Look for 15-20-25 (3-4-5 scaled) in both pictures.
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Approach: drop altitudes, both reveal a 15-20-25 triangle
Triangle with base 30: half-base 15, hypotenuse 25 ⇒ height = √(252 − 152) = 20. Area A = (1/2)(30)(20) = 300.
Triangle with base 40: half-base 20, hypotenuse 25 ⇒ height = √(252 − 202) = 15. Area B = (1/2)(40)(15) = 300.
The three big rectangles drawn count, plus every rectangle assembled from the small pieces created where they overlap.
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Hint 2 of 2
Three originals; then look for every small piece (alone or glued to a neighbour) that is itself a rectangle.
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Approach: count originals, atoms, and rectangular unions
The three drawn rectangles themselves contribute 3.
The overlapping lines cut the figure into smaller atomic rectangles. The center is where all three overlap. Each individual atom is a rectangle (there are several), and pairs of adjacent atoms that share a full side form another rectangle.
Adding up all distinct rectangles — the three originals plus every smaller rectangle formed by the cuts — gives 11.
Students guess that Norb's age is 24, 28, 30, 32, 36, 38, 41, 44, 47, and 49. Norb says, "At least half of you guessed too low, two of you are off by one, and my age is a prime number." How old is Norb?
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Answer: C — 37.
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Hint 1 of 2
"At least half too low" with 10 guesses means age > 5th-smallest guess = 36, so age ≥ 37.
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Hint 2 of 2
"Two off by one" means age is squeezed between two guesses that differ by 2. The only such pair above 36 is 36 and 38, or 47 and 49.
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Approach: apply each clue in turn
Sorted guesses: 24, 28, 30, 32, 36, 38, 41, 44, 47, 49. "At least half too low" ⇒ age > 36.
"Two are off by one" ⇒ age sits between two guesses 2 apart. Candidates: 37 (between 36 and 38) or 48 (between 47 and 49).
How many 4-digit positive integers have four different digits, where the leading digit is not zero, the integer is a multiple of 5, and 5 is the largest digit?
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Answer: D — 84.
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Hint 1 of 2
All digits come from {0, 1, 2, 3, 4, 5}, with 5 present (it's the largest). Units digit is 0 or 5 (divisible by 5).
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Hint 2 of 2
Split into two cases by units digit.
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Approach: casework on the units digit
Case A: units = 0. The remaining three slots contain 5 and two distinct digits chosen from {1, 2, 3, 4}: C(4, 2) = 6 ways to pick the other two; 3! = 6 ways to arrange them. Subtotal: 6 × 6 = 36.
Case B: units = 5. The remaining three slots use three distinct digits from {0, 1, 2, 3, 4}. Choose and arrange: 5 · 4 · 3 = 60. Subtract leading-zero arrangements: 4 · 3 = 12 with 0 first. Subtotal: 60 − 12 = 48.
