On the AMC 8 contest Billy answers 13 questions correctly, answers 7 questions incorrectly and doesn't answer the last 5. What is his score? (right = +1, wrong or N/A = +0)
Elisa swims laps in the pool. When she first started, she completed 10 laps in 25 minutes. Now she can finish 12 laps in 24 minutes. By how many minutes has she improved her lap time?
Initially, a spinner points west. Chenille moves it clockwise 214 revolutions and then counterclockwise 334 revolutions. In what direction does the spinner point after the two moves?
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Answer: B — East.
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Hint 1
Net rotation = clockwise − counterclockwise. Reduce mod 1 revolution.
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Approach: subtract revolutions, reduce mod 1
Net counterclockwise: 3.75 − 2.25 = 1.5 revolutions counterclockwise.
1.5 mod 1 = 0.5 ⇒ spinner turns half-revolution from west ⇒ east.
Sum each rectangle's perimeter, then subtract the buried edges. The two rectangles share a 2-inch segment internally; that hides 2 from each rectangle's perimeter.
The table shows some of the results of a survey by radio station KACL. What percentage of the males surveyed listen to the station? (Total surveyed: 200. Females: 96. Females who listen: 58. Males who don't listen: 26. Total listeners: 136. Total non-listeners: 64.)
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Answer: E — 75%.
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Hint 1
Total males = 200 − 96 = 104. Of those, 26 don't listen.
Antonette gets 70% on a 10-problem test, 80% on a 20-problem test and 90% on a 30-problem test. If the three tests are combined into one 60-problem test, which percent is closest to her overall score?
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.
Problems 14, 15 and 16 involve Mrs. Reed's English assignment. A Novel Assignment. The students in Mrs. Reed's English class are reading the same 760-page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 30 seconds. If Bob and Chandra both read the whole book, Bob will spend how many more seconds reading than Chandra?
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Answer: B — 11,400 seconds.
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Hint 1
Difference per page: 45 − 30 = 15 sec. Multiply by 760.
Chandra and Bob, who each have a copy of the book, decide that they can save time by "team reading" the novel. In this scheme, Chandra will read from page 1 to a certain page and Bob will read from the next page through page 760, finishing the book. When they are through they will tell each other about the part they read. What is the last page that Chandra should read so that she and Bob spend the same amount of time reading the novel?
Before Chandra and Bob start reading, Alice says she would like to team read with them. If they divide the book into three sections so that each reads for the same length of time, how many seconds will each have to read?
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Answer: E — 7200 seconds.
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Hint 1 of 2
Equal time means pages-per-reader are in inverse proportion to seconds-per-page (rates).
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Hint 2 of 2
Reading rates: Alice 1/20, Bob 1/45, Chandra 1/30. Ratio of pages = 1/20 : 1/45 : 1/30 = 9 : 4 : 6 (multiply by 180).
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Approach: page split, then time
Reading rates: 1/20, 1/45, 1/30 pages/sec for Alice, Bob, Chandra.
Pages each reads in equal time: ratio 9 : 4 : 6 (sum 19). Total 760 pages ⇒ Alice 360, Bob 160, Chandra 240.
A cube with 3-inch edges is made using 27 cubes with 1-inch edges. Nineteen of the smaller cubes are white and eight are black. If the eight black cubes are placed at the corners of the larger cube, what fraction of the surface area of the larger cube is white?
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Answer: D — 5/9.
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Hint 1
Symmetry: each face has the same pattern. Count white unit squares on one face.
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Approach: count one face
Each face is 3 × 3 = 9 unit squares. Corners (4 of them) show the black corner cubes ⇒ 4 black, 5 white.
A singles tournament had six players. Each player played every other player only once, with no ties. If Helen won 4 games, Ines won 3 games, Janet won 2 games, Kendra won 2 games and Lara won 2 games, how many games did Monica (the sixth player) win?
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Answer: C — 2 games.
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Hint 1
Total games = C(6, 2) = 15 ⇒ total wins = 15. Subtract the five known.
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Approach: total wins constraint
Total games: 15. Sum of known wins: 4 + 3 + 2 + 2 + 2 = 13.
An aquarium has a rectangular base that measures 100 cm by 40 cm and has a height of 50 cm. The aquarium is filled with water to a depth of 37 cm. A rock with volume 1000 cm3 is then placed in the aquarium and completely submerged. By how many centimeters does the water level rise?
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Answer: A — 0.25 cm.
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Hint 1
Rise = displaced volume / base area = 1000 / (100 · 40).
A box contains gold coins. If the coins are equally divided among six people, four coins are left over. If the coins are equally divided among five people, three coins are left over. If the box holds the smallest number of coins that meets these two conditions, how many coins are left when equally divided among seven people?
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Answer: A — 0.
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Hint 1
List numbers ≡ 4 (mod 6) and find the first one that's also ≡ 3 (mod 5).
Barry wrote 6 different numbers, one on each side of 3 cards, and laid the cards on a table, as shown. The sums of the two numbers on each of the three cards are equal. The three numbers on the hidden sides are prime numbers. What is the average of the hidden prime numbers? (Visible sides: 44, 59, 38.)
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Answer: B — 14.
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Hint 1 of 2
Even-visible cards (44, 38) need an odd prime to make an odd sum; the 59 card needs an even prime to make an odd sum. Equal sums ⇒ the 59 card pairs with the only even prime, 2.
Still stuck? Show hint 2 →
Hint 2 of 2
Common sum = 59 + 2 = 61.
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Approach: parity forces 2 behind 59
Sum behind 59 + 59 = 61 (odd) only if hidden is even ⇒ hidden = 2 (only even prime).
