Twelve friends met for dinner at Oscar's Overstuffed Oyster House, and each ordered one meal. The portions were so large, there was enough food for 18 people. If they shared, how many meals should they have ordered to have just enough food for the 12 of them?
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12 meals = 18 people ⇒ 1 meal feeds 1.5. Solve m · 1.5 = 12.
Ms. Hamilton's eighth-grade class wants to participate in the annual three-person-team basketball tournament. Lance, Sally, Joy, and Fred are chosen for the team. In how many ways can the three starters be chosen?
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Choosing 3 starters from 4 = choosing the 1 non-starter.
Ms. Hamilton's eighth-grade class wants to participate in the annual three-person-team basketball tournament. The losing team of each game is eliminated from the tournament. If sixteen teams compete, how many games will be played to determine the winner?
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Each game eliminates exactly one team. 15 must be eliminated.
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Need 15 teams eliminated (everyone except the winner).
After Sally takes 20 shots, she has made 55% of her shots. After she takes 5 more shots, she raises her percentage to 56%. How many of the last 5 shots did she make?
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Made so far: 0.55 · 20 = 11. Target made: 0.56 · 25 = 14. Difference is how many she made in the last 5.
An athlete's target heart rate, in beats per minute, is 80% of the theoretical maximum heart rate. The maximum heart rate is found by subtracting the athlete's age, in years, from 220. To the nearest whole number, what is the target heart rate of an athlete who is 26 years old?
Handy Aaron helped a neighbor 114 hours on Monday, 50 minutes on Tuesday, from 8:20 to 10:45 on Wednesday morning, and a half-hour on Friday. He is paid $3 per hour. How much did he earn for the week?
The numbers −2, 4, 6, 9 and 12 are rearranged according to these rules: The largest isn't first, but it is in one of the first three places. The smallest isn't last, but it is in one of the last three places. The median isn't first or last. What is the average of the first and last numbers?
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Largest (12) and smallest (−2) and median (6) all cannot be the first or last. That leaves 4 and 9 for the endpoints.
Show solution
12 (largest) not first or last (must be in 2nd or 3rd). −2 (smallest) not last (4th or 3rd). 6 (median) not first or last.
First and last cannot be 12, −2, or 6, so they're 4 and 9.
Niki usually leaves her cell phone on. If her cell phone is on but she is not actually using it, the battery will last for 24 hours. If she is using it constantly, the battery will last for only 3 hours. Since the last recharge, her phone has been on 9 hours, and during that time she has used it for 60 minutes. If she doesn't talk any more but leaves the phone on, how many more hours will the battery last?
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Idle uses 1/24 per hour; using uses 1/3 per hour. So far: 8 hr idle + 1 hr in use.
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Used: 8 · (1/24) + 1 · (1/3) = 1/3 + 1/3 = 2/3.
Remaining: 1/3 of battery. At idle rate 1/24 per hour: time = (1/3) · 24 = 8 hr.
Amy, Bill and Celine are friends with different ages. Exactly one of the following statements is true. I. Bill is the oldest. II. Amy is not the oldest. III. Celine is not the youngest. Rank the friends from the oldest to the youngest.
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If I is true, II is also true (Bill oldest ⇒ Amy isn't); that's two trues, forbidden. So I is false.
Show hint (sharpest)
Then Bill is not oldest. If II is true (Amy not oldest), Celine must be oldest ⇒ III also true (since the youngest can't be the oldest); forbidden again.
Show solution
I true ⇒ II also true. Two trues forbidden ⇒ I false.
II true ⇒ Celine oldest, making III true. Forbidden ⇒ II false.
So III is the only true one and I, II are false. Bill not oldest (I false); Amy is oldest (II false). Then by III, Celine is not youngest, so Bill is youngest.
Thirteen black and six white hexagonal tiles were used to create the figure below. If a new figure is created by attaching a border of white tiles with the same size and shape as the others, what will be the difference between the total number of white tiles and the total number of black tiles in the new figure?
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Ring n contains 6n hexagons. The new border (3rd ring) has 18 tiles.
Two 600 mL pitchers contain orange juice. One pitcher is 1/3 full and the other pitcher is 2/5 full. Water is added to fill each pitcher completely, then both pitchers are poured into one large container. What fraction of the mixture in the large container is orange juice?
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Total OJ: 600(1/3) + 600(2/5) = 200 + 240 = 440. Total volume: 1200.
Five friends compete in a dart-throwing contest. Each one has two darts to throw at the same circular target, and each individual's score is the sum of the scores in the target regions that are hit. The scores for the target regions are the whole numbers 1 through 10. Each throw hits the target in a region with a different value. The scores are: Alice 16, Ben 4, Cindy 7, Dave 11, Ellen 17. Who hits the region worth 6 points?
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Each region is used exactly once. Start with the smallest score (Ben = 4): the only pair from 1–10 is {1, 3}.
Show solution
Ben (4): only 1 + 3.
Cindy (7): from remaining digits {2, 4, 5, 6, 7, 8, 9, 10}, the only pair is 2 + 5.
Dave (11): only 4 + 7.
Remaining for Alice and Ellen: {6, 8, 9, 10}. Alice 16 = 6 + 10. Ellen 17 = 8 + 9.
A whole number larger than 2 leaves a remainder of 2 when divided by each of the numbers 3, 4, 5, and 6. The smallest such number lies between which two numbers?
Two-thirds of the people in a room are seated in three-fourths of the chairs. The rest of the people are standing. If there are 6 empty chairs, how many people are in the room?
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1/4 of the chairs are empty = 6 ⇒ chairs total. Then 3/4 of chairs gives # seated people.
Show solution
Empty chairs (1/4 of total) = 6 ⇒ chairs = 24.
Seated people = (3/4)(24) = 18, which is 2/3 of all people.
Spinners A and B are spun. On each spinner, the arrow is equally likely to land on each number. What is the probability that the product of the two spinners' numbers is even?
At a party there are only single women and married men with their wives. The probability that a randomly selected woman is single is 2/5. What fraction of the people in the room are married men?
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Pick 5 women. 2 single, 3 married. Married women bring 3 husbands. Total people: 5 + 3 = 8.
Show solution
Let women = 5. Single: 2; married: 3.
Married men = 3 (one per married woman). Total people: 5 + 3 = 8.
In the figure, ABCD is a rectangle and EFGH is a parallelogram. Using the measurements given in the figure, what is the length d of the segment that is perpendicular to HE and FG?
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Area of parallelogram EFGH = rectangle area − 4 corner triangles.
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HE is a hypotenuse of the 3-4 right triangle ⇒ HE = 5. Use Area = base · height.
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Corner triangles: at A (3, 4): area 6. At C (3, 4): area 6. At B (5, 6): area 15. At D (5, 6): area 15. Total: 42.
Two 4 × 4 squares intersect at right angles, bisecting their intersecting sides, as shown. The circle's diameter is the segment between the two points of intersection. What is the area of the shaded region created by removing the circle from the squares?
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Union of the two squares: 16 + 16 − overlap. Overlap is a 2×2 square (area 4).
Show hint (sharpest)
Circle's diameter = diagonal of overlap square = 2√2 ⇒ radius √2 ⇒ area 2π.