25 random problems — one per position, pulled from random authored years. Hints and solutions are locked until you submit. Retake as often as you want — every attempt is saved to your test history (if you're logged in).
The average cost of a long-distance call in the USA in 1985 was 41 cents per minute, and the average cost of a long-distance call in the USA in 2005 was 7 cents per minute. Find the approximate percent decrease in the cost per minute of a long-distance call.
In the small country of Mathland, all automobile license plates have four symbols. The first must be a vowel (A, E, I, O, or U), the second and third must be two different letters among the 21 non-vowels, and the fourth must be a digit (0 through 9). If the symbols are chosen at random subject to these conditions, what is the probability that the plate will read "AMC8"?
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Answer: B — 1/21,000.
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Hint 1 of 2
Count the total number of allowed plates by multiplying choices for each slot. The probability of AMC8 is 1 over that total.
Of the 500 balls in a large bag, 80% are red and the rest are blue. How many of the red balls must be removed from the bag so that 75% of the remaining balls are red?
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Answer: D — 100 red balls.
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Hint 1
Blue balls don't change. 75% red ⇒ 25% blue, so the 100 blue balls represent 25% of the new total.
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Approach: blue stays constant, so use blue to find new total
Initial: 400 red, 100 blue.
After removal, 25% blue means total = 100 / 0.25 = 400 balls.
Cassie leaves Escanaba at 8:30 AM heading for Marquette on her bike. She bikes at a uniform rate of 12 miles per hour. Brian leaves Marquette at 9:00 AM heading for Escanaba on his bike. He bikes at a uniform rate of 16 miles per hour. They both bike on the same 62-mile route between Escanaba and Marquette. At what time in the morning do they meet?
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Answer: D — 11:00 AM.
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Hint 1
By 9:00, Cassie has covered 6 miles. Then 56 miles remain at combined speed 28 mph ⇒ 2 hours.
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Approach: head-start, then combined speed
By 9:00 AM Cassie has biked (1/2)(12) = 6 miles. Gap: 62 − 6 = 56 miles.
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.
For each weight, find the lowest price dot in that column. Then compute price ÷ weight.
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Hint 2 of 2
Lowest-price-per-ounce will favor a weight where the cheapest available pepper drops well below the dollar-per-ounce line.
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Approach: lowest dot in each column, then divide
Lowest price at each weight (reading off the scatter): 1 oz ≈ $1.25 (rate ≈ 1.25), 2 oz ≈ $2 (1.00), 3 oz ≈ $2.5 (≈ 0.83), 4 oz ≈ $3.9 (≈ 0.97), 5 oz ≈ $4.5 (≈ 0.90).
The 3-ounce option has the lowest rate (~$0.83/oz). Answer: 3 ounces.
A five-legged Martian has a drawer full of socks, each of which is red, white or blue, and there are at least five socks of each color. The Martian pulls out one sock at a time without looking. How many socks must the Martian remove from the drawer to be certain there will be 5 socks of the same color?
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Answer: D — 13.
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Hint 1
Worst case: 4 of each color (12 socks) without yet getting 5 of one color.
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Approach: pigeonhole on the worst case
After 12 socks, possible to have 4 red, 4 white, 4 blue (no color has 5).
Pauline can shovel snow at the rate of 20 cubic yards for the first hour, 19 cubic yards for the second, 18 for the third, and so on, always shoveling one cubic yard less per hour than the previous hour. If her driveway is 4 yards wide, 10 yards long, and covered with snow 3 yards deep, then the number of hours it will take her to shovel it clean is closest to
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Answer: D — 7.
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Hint 1 of 2
First find the total volume of snow: 4 × 10 × 3.
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Hint 2 of 2
Add 20 + 19 + 18 + … until you reach that volume.
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Approach: accumulate the decreasing hourly amounts
The snow is 4 × 10 × 3 = 120 cubic yards.
Running totals: 20, 39, 57, 74, 90, 105, 119 — after 7 hours she's at 119, just shy of 120, so the time is closest to 7 hours.
Answer: E — Same area, but quadrilateral I has the smaller perimeter.
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Hint 1 of 2
Work out both areas first — they come out equal.
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Hint 2 of 2
Then compare the slanted sides to see which perimeter is larger.
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Approach: compare area, then perimeter
Both shapes have area 1 (I is a 1×1 parallelogram; II is two triangles of area ½).
They share two equal slant sides, but I's other two sides are unit length while II has a longer slant, so II's perimeter is bigger — meaning I's is less (choice E).
The hundreds digit of a three-digit number is 2 more than the units digit. The digits of the three-digit number are reversed, and the result is subtracted from the original three-digit number. What is the units digit of the result?
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Answer: E — 8.
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Hint 1 of 2
Let units = u, tens = t, hundreds = u + 2. Original − reversed simplifies to a constant.
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Hint 2 of 2
Reversing flips the hundreds and units digits. Difference = 99 · (hundreds − units).
There is a list of seven numbers. The average of the first four numbers is 5, and the average of the last four numbers is 8. If the average of all seven numbers is 647, then the number common to both sets of four numbers is
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Answer: B — 6.
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Hint 1 of 2
Add the two group-sums together — every number is counted once, except the shared one, counted twice.
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Hint 2 of 2
So (sum of the two fours) minus (sum of all seven) leaves exactly the common number.
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Approach: the overlap gets counted twice
The first four total 4 · 5 = 20 and the last four total 4 · 8 = 32. Adding gives 52, which counts every number once except the shared middle one, counted twice.
All seven total 7 · 6⁴⁄₇ = 46, counting each number once.
Subtracting strips one copy of everything, leaving the doubled number: 52 − 46 = 6.
The digits 1, 2, 3, 4, and 5 are each used once to write a five-digit number PQRST. The three-digit number PQR is divisible by 4, the three-digit number QRS is divisible by 5, and the three-digit number RST is divisible by 3. What is P?
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Answer: A — P = 1.
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Hint 1 of 2
QRS div by 5 means S ends in 0 or 5; the only digit available is 5. So S = 5.
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Hint 2 of 2
PQR div by 4 means QR is divisible by 4. With S = 5 used, QR is a 2-digit from {1, 2, 3, 4}. Options: 12, 24, 32.
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Approach: narrow each divisibility constraint in order
S = 5 (the only digit from {1,2,3,4,5} that makes QRS end in 0 or 5).
QR must be a 2-digit number from {1, 2, 3, 4} divisible by 4: {12, 24, 32}.
Test each. QR = 12: leftover {3, 4} for P, T. RST = 25T, digit sum 7 + T; T ∈ {3, 4} gives 10 or 11, neither div by 3.
QR = 24: leftover {1, 3}. RST = 45T, digit sum 9 + T; T = 3 gives 12 (div 3) ✓. So Q = 2, R = 4, T = 3, P = 1. Number: 12435.
QR = 32: leftover {1, 4}. RST = 25T sum 7 + T; T ∈ {1, 4} gives 8 or 11, neither div by 3.
A point is chosen at random from within a circular region. What is the probability that the point is closer to the center of the region than it is to the boundary of the region?
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Answer: A — 1/4.
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Hint 1 of 2
A point at distance r from the center is r from the center and R − r from the boundary.
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Hint 2 of 2
Closer to the center means r < R − r, i.e. r < R/2.
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Approach: compare distances, then take an area ratio
Being closer to the center than the boundary means r < R − r, so r < R/2 — the point lies in the inner circle of radius R/2.
Its area is a fraction (R/2)² / R² = 1/4 of the whole.
