Extend the square pattern of 8 black and 17 white square tiles by attaching a border of black tiles around the square. What is the ratio of black tiles to white tiles in the extended pattern?
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Border around an n × n square has 4n + 4 tiles. White count is unchanged.
Show solution
Original is 5 × 5 = 25 tiles (8 black + 17 white). New 7 × 7 has 49 tiles.
Added border tiles: 49 − 25 = 24, all black. New black total: 8 + 24 = 32. White still 17.
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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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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Everyone owns at least one. So car-only count = total − motorcycle-owners.
Show solution
Each non-motorcycle-owner must own a car (since every adult has at least one).
Each of the following four large congruent squares is subdivided into combinations of congruent triangles or rectangles and is partially bolded. What percent of the total area is partially bolded?
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Compute each square's shaded fraction; add; divide by 4 (since there are 4 squares of equal area).
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Shaded fractions of the four squares: 1/4, 1/8, 3/8, 1/4.
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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All sums are odd + even = odd. List the 9 sums and count distinct values.
Carmen takes a long bike ride on a hilly highway. The graph indicates the miles traveled during the time of her ride. What is Carmen's average speed for her entire ride in miles per hour?
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Average speed = total distance / total time. The graph's endpoints give you both.
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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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.
Show solution
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.
The graph shows the number of minutes studied by both Asha (black bar) and Sasha (grey bar) in one week. On the average, how many more minutes per day did Sasha study than Asha?
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Add the daily differences (Sasha − Asha) and divide by 5.
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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Fix Angie's seat. Carlos lands in any of the 3 remaining seats with equal probability; only 1 is opposite.
Show solution
Fix Angie in any seat. Carlos has 3 equally likely seats among the remaining 3.
Two congruent squares, ABCD and PQRS, have side length 15. They overlap to form the 15 by 25 rectangle AQRD shown. What percent of the area of rectangle AQRD is shaded?
Show hint (soft nudge)
Together the two squares cover area 2 · 152, but the rectangle is only 25 · 15 — the overlap is counted twice.
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Overlap area = (sum of both squares) − (rectangle).
Show solution
Each square has area 225, total 450. Rectangle area = 25 × 15 = 375.
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?
Show hint
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?
Show hint (soft nudge)
Each triangle is isosceles. Drop the altitude to the unequal side and use the Pythagorean theorem.
Show hint (sharpest)
Look for 15-20-25 (3-4-5 scaled) in both pictures.
Show solution
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.
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?
Show hint (soft nudge)
"At least half too low" with 10 guesses means age > 5th-smallest guess = 36, so age ≥ 37.
Show hint (sharpest)
"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.
Show solution
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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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).
Show hint (sharpest)
Split into two cases by units digit.
Show solution
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.
A circle with radius 1 is inscribed in a square and circumscribed about another square as shown. Which fraction is closest to the ratio of the circle's shaded area to the area between the two squares?
Show hint (soft nudge)
Inner square has diagonal = diameter = 2, so its side is √2 and area is 2.
Show hint (sharpest)
Shaded area inside the circle = π · 12 − 2 = π − 2. Area between the two squares = outer area − inner area = 4 − 2 = 2.
Show solution
Outer square: side 2, area 4. Inner square: diagonal = 2 (diameter), so side √2 and area 2.