2005 AMC amc10b 真题 答案 详解

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英文真题

2005 AMC 10B Problems
Problem 1
A scout troop buys 1000 candy bars at a price of five for $2. They sell all the candy bars at a price of two for $1. What was the profit, in dollars?
(A)$100 (B)$200 (C)$300 (D)$400 (E)$500

Problem 2
A positive number ( x ) has the property that ( x \% ) of ( x ) is 4. What is ( x )?
(A)2 (B)4 (C)10 (D)20 (E)40

Problem 3
A gallon of paint is used to paint a room. One third of the paint is used on the first day. On the second day, one third of the remaining paint is used. What fraction of the original amount of paint is available to use on the third day?
(A)(\frac{1}{10}) (B)(\frac{1}{9}) (C)(\frac{1}{3}) (D)(\frac{4}{9}) (E)(\frac{5}{9})

Problem 4
For real numbers ( a ) and ( b ), define ( a \diamond b = \sqrt{a^2 + b^2} ). What is the value of
( (5 \diamond 12) \diamond ((-12) \diamond (-5))? )
(A)0 (B)(\frac{17}{2}) (C)13 (D)13(\sqrt{2}) (E)26

Problem 5
Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs?
(A)(\frac{1}{5}) (B)(\frac{1}{3}) (C)(\frac{2}{5}) (D)(\frac{2}{3}) (E)(\frac{4}{5})

Problem 6
At the beginning of the school year, Lisa's goal was to earn an A on at least 80% of her 50 quizzes for the year. She earned an A on 22 of the first 30 quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she earn a grade lower than an A?
(A)1 (B)2 (C)3 (D)4 (E)5

Problem 7
A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. What is the ratio of the area of the smaller circle to the area of the larger square?
(A)(\frac{\pi}{16}) (B)(\frac{\pi}{8}) (C)(\frac{3\pi}{16}) (D)(\frac{\pi}{4}) (E)(\frac{\pi}{2})

Problem 8
An 8-foot by 10-foot floor is tiled with square tiles of size 1 foot by 1 foot. Each tile has a pattern consisting of four white quarter circles of radius ( \frac{1}{2} ) foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?

真题文字详解(英文)

Problem 1 The following problem is from both the 2005 AMC 12B #1 and 2005 AMC 10B #1, so both problems redirect to this page. A scout troop buys 1000 candy bars at a price of five for 2 dollars. They sell all the candy bars at the price of two for 1 dollar. What was their profit, in dollars? (A) 100 (B) 200 (C) 300 (D) 400 (E) 500 Solution Expenses = 1000 · 2/5 = 400 Revenue = 1000 · 1/2 = 500 Profit = Revenue - Expenses = 500 - 400 = (A) 100 Note: Revenue is a gain. Solution 2 (Faster) Note that the troop buys 10 candy bars at a price of 4 dollars and sells 10 bars at a price of 5 dollars. So the troop gains 1 dollar for every 10 bars. So therefore we divide 1000 ÷ 10 = 100. So our answer is (A) 100 ~HyperVoid See also Problem 2 The following problem is from both the 2005 AMC 12B #2 and 2005 AMC 10B #2, so both problems redirect to this page. A positive number x has the property that x% of x is 4. What is x? (A) 2 (B) 4 (C) 10 (D) 20 (E) 40 Solution Solution 1 Since x% means 0.01x, the statement "x% of x is 4" can be rewritten as "0.01x · x = 4": 0.01x · x = 4 => x^2 = 400 => x = (D) 20 Solution 2 Try the answer choices one by one. Upon examination, it is quite obvious that the answer is (D) 20 . Very fast. Solution by franzlisztt See also Problem 3 A gallon of paint is used to paint a room. One third of the paint is used on the first day. On the second day, one third of the remaining paint is used. What fraction of the original amount of paint is available to use on the third day?

真题文字详解(中英双语)

Problem 1 The following problem is from both the 2005 AMC 12B #1 and 2005 AMC 10B #1, so both problems redirect to this page. 以下题目来自2005年AMC 12B #1和2005年AMC 10B #1，因此这两个问题都重定向到这个页面。

A scout troop buys 1000 candy bars at a price of five for 2 dollars. They sell all the candy bars at the price of two for 1 dollar. What was their profit, in dollars?
(A) 100 (B) 200 (C) 300 (D) 400 (E) 500

Solution
Expenses = 1000 · 2/5 = 400
Revenue = 1000 · 1/2 = 500
Profit = Revenue - Expenses = 500 - 400 = (A) 100

Note: Revenue is a gain.

Solution 2 (Faster)
Note that the troop buys 10 candy bars at a price of 4 dollars and sells 10 bars at a price of 5 dollars. So the troop gains 1 dollar for every 10 bars. So therefore we divide 1000 ÷ 10 = 100. So our answer is (A) 100. ~HyperVoid

See also

Problem 2 The following problem is from both the 2005 AMC 12B #2 and 2005 AMC 10B #2, so both problems redirect to this page. 以下题目来自2005年AMC 12B #2和2005年AMC 10B #2，因此这两个问题都重定向到这个页面。

A positive number ( x ) has the property that ( x\% ) of ( x ) is 4. What is ( x )?
(A) 2 (B) 4 (C) 10 (D) 20 (E) 40

Solution
Solution 1
Since ( x\% ) means 0.01( x ), the statement "x% of ( x ) is 4" can be rewritten as "0.01( x \cdot x = 4 )":
0.01( x \cdot x = 4 \Rightarrow x^2 = 400 \Rightarrow x = (D) 20 ).

Solution 2
Try the answer choices one by one. Upon examination, it is quite obvious that the answer is (D) 20. Very fast.

Solution by franzlisztt

See also

Problem 3