FILL BLANKThe graph shows the results of a word game in which two players were given identical sets of 12 different letters. Using only the 12 letters, each player listed as many words as the player could think of in three minutes, without repeating any letters within a word. Neither player listed a word with fewer than 3 or more than 7 letters.In order to determine the scores for the players, each word listed with 3 or 4 letters was given 1 point, each word listed with 5 or 6 letters was given 2 points, and each word listed with 7 letters was given 3 points. The score for each player was the sum of the points given for the words that the player listed.What was the absolute value of the difference between the score for Player A and the score for Player B?
Answer(s): A
From the bar graph:Player A· 3-letter words: 8· 4-letter words: 3· 5-letter words: 7· 6-letter words: 8· 7-letter words: 4Score for A· 34 letters (8 + 3) × 1 = 11· 56 letters (7 + 8) × 2 = 30· 7 letters 4 × 3 = 12Total for A = 11 + 30 + 12 = 53Player B· 3-letter words: 7· 4-letter words: 3· 5-letter words: 2· 6-letter words: 8· 7-letter words: 8Score for B· 34 letters (7 + 3) × 1 = 10· 56 letters (2 + 8) × 2 = 20· 7 letters 8 × 3 = 24Total for B = 10 + 20 + 24 = 54Absolute difference = |53 - 54| = 1
The graph shows the results of a word game in which two players were given identical sets of 12 different letters. Using only the 12 letters, each player listed as many words as the player could think of in three minutes, without repeating any letters within a word. Neither player listed a word with fewer than 3 or more than 7 letters.Of the 3-letter words listed by Player A or Player B, 4 words were listed by both players.How many of the 3-letter words listed were listed by only one of the two players?
From the graph, Player A listed 8 three-letter words and Player B listed 7 three-letter words. A total of 4 of these words were listed by both players.Number of three-letter words listed by only one player = 8 + 7 - 2 × 4 = 15 - 8 = 7.
The graph shows the results of a word game in which two players were given identical sets of 12 different letters. Using only the 12 letters, each player listed as many words as the player could think of in three minutes, without repeating any letters within a word. Neither player listed a word with fewer than 3 or more than 7 letters.What is the average (arithmetic mean) number of letters per word in the words listed by Player B?
Answer(s): C
Player B's word counts from the graph:· 3-letter: 7· 4-letter: 3· 5-letter: 2· 6-letter: 8· 7-letter: 8Total words = 7 + 3 + 2 + 8 + 8 = 28Total letters =3(7) + 4(3) + 5(2) + 6(8) + 7(8) = 21 + 12 + 10 + 48 + 56 = 147 Mean number of letters = 147 ÷ 28 = 5.25
When the integer n is divided by 12, the remainder is 9.Which of the following must be a divisor of n?
Answer(s): E
If leaves a remainder of 9 when divided by 12, then n = 12k + 9Every such number is divisible by gcd(12, 9) = 3, so 3 must divide n.
FILL BLANKSquare ABCD has sides of length 12. The midpoints of the edges of ABCD are the vertices of a second square, and the midpoints of the edges of the second square are the vertices of a square region that is shaded in the figure.What is the area of the shaded region?
Midpoints of the 12-by-12 square form a second square whose side is (distance between (6,0) and (12,6)).The midpoints of that second square's sides are (9, 3), (9, 9), (3, 9), (3, 3), giving a third, axis-aligned square with side 6.Area = 62 = 36.
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