2012 AMC amc10a 真题 答案 详解

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中英双语真题

2012 AMC 10A
Problem 1
Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30 seconds. Working together, how many cupcakes can they frost in 5 minutes?
Cagney 每 20 秒可以给一个蛋糕上霜，Lacey 每 30 秒可以给一个蛋糕上霜，他俩合作 5 分钟内可以给多少个蛋糕上霜？
(A) 10 (B) 15 (C) 20 (D) 25 (E) 30

Problem 2
A square with side length 8 is cut in half, creating two congruent rectangles. What are the dimensions of one of these rectangles?
一个边长为 8 的正方形被分成两半，得到两个全等的矩形。问其中一个矩形的尺寸是多少？
(A) 2 by 4 (B) 2 by 6 (C) 2 by 8 (D) 4 by 4 (E) 4 by 8

Problem 3
A bug crawls along a number line, starting at -2. It crawls to -6, then turns around and crawls to 5. How many units does the bug crawl altogether?
一个虫子从-2这个点开始沿着数轴爬行。它先爬到-6，然后掉头再爬到5。那么这只虫子总共爬了多长？
(A) 9 (B) 11 (C) 13 (D) 14 (E) 15

Problem 4
Let ∠ABC = 24° and ∠ABD = 20°. What is the smallest possible degree measure for angle CBD?
令∠ABC=24°，∠ABD=20°。问角CBD最少是多少度？
(A) 0 (B) 2 (C) 4 (D) 6 (E) 12

英文真题

2012 AMC 10A Problems Problem 1 Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30 seconds. Working together, how many cupcakes can they frost in 5 minutes? (A) 10 (B) 15 (C) 20 (D) 25 (E) 30 Problem 2 A square with side length 8 is cut in half, creating two congruent rectangles. What are the dimensions of one of these rectangles? (A) 2 by 4 (B) 2 by 6 (C) 2 by 8 (D) 4 by 4 (E) 4 by 8 Problem 3 A bug crawls along a number line, starting at -2. It crawls to -6, then turns around and crawls to 5. How many units does the bug crawl altogether? (A) 9 (B) 11 (C) 13 (D) 14 (E) 15 Problem 4 Let ∠ABC = 24° and ∠ABD = 20°. What is the smallest possible degree measure for angle CBD? (A) 0 (B) 2 (C) 4 (D) 6 (E) 12 Problem 5 Last year 100 adult cats, half of whom were female, were brought into the Smallville Animal Shelter. Half of the adult female cats were accompanied by a litter of kittens. The average number of kittens per litter was 4. What was the total number of cats and kittens received by the shelter last year? (A) 150 (B) 200 (C) 250 (D) 300 (E) 400 Problem 6 The product of two positive numbers is 9. The reciprocal of one of these numbers is 4 times the reciprocal of the other number. What is the sum of the two numbers? (A) \frac{10}{3} (B) \frac{20}{3} (C) 7 (D) \frac{15}{2} (E) 8 Problem 7 In a bag of marbles, \frac{3}{5} of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red? (A) \frac{2}{5} (B) \frac{3}{7} (C) \frac{4}{7} (D) \frac{3}{5} (E) \frac{4}{5} Problem 8 The sums of three whole numbers taken in pairs are 12, 17, and 19. What is the middle number? (A) 4 (B) 5 (C) 6 (D) 7 (E) 8 Problem 9 A pair of six-sided dice are labeled so that one die has only even numbers (two each of 2, 4, and 6), and the other die has only odd numbers (two each of 1, 3, and 5). The pair of dice is rolled. What is the probability that the sum of the numbers on the tops of the two dice is 7? (A) \frac{1}{6} (B) \frac{1}{5} (C) \frac{1}{4} (D) \frac{1}{3} (E) \frac{1}{2}

真题文字详解(英文)

Problem 1 The following problem is from both the 2012 AMC 12A #2 and 2012 AMC 10A #1, so both problems redirect to this page. Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30 seconds. Working together, how many cupcakes can they frost in 5 minutes?
(A) 10 (B) 15 (C) 20 (D) 25 (E) 30

Solution 1
Cagney can frost one in 20 seconds, and Lacey can frost one in 30 seconds. Working together, they can frost one in
$$\frac{20 \cdot 30}{20 + 30} = \frac{600}{50} = 12 \text{ seconds. In } 300 \text{ seconds (5 minutes), they can frost (D) 25 cupcakes.}$$

Solution 2
In 300 seconds (5 minutes), Cagney will frost $$\frac{300}{20} = 15 \text{ cupcakes, and Lacey will frost } \frac{300}{30} = 10 \text{ cupcakes. Therefore, working together they will frost } 15 + 10 = (D) 25 \text{ cupcakes.}$$

Solution 3
Since Cagney frosts 3 cupcakes a minute, and Lacey frosts 2 cupcakes a minute, they together frost $3 + 2 = 5$ cupcakes a minute.
Therefore, in 5 minutes, they frost $5 \times 5 = 25 \Rightarrow (D)$

Problem 2
A square with side length 8 is cut in half, creating two congruent rectangles. What are the dimensions of one of these rectangles?
(A) 2 by 4 (B) 2 by 6 (C) 2 by 8 (D) 4 by 4 (E) 4 by 8

Solution
Cutting the square in half will bisect one pair of sides while the other side will remain unchanged. Thus, the new square is $\frac{8}{2}$ by 8, or (E) 4 by 8.

Problem 3
The following problem is from both the 2012 AMC 12A #1 and 2012 AMC 10A #3, so both problems redirect to this page. A bug crawls along a number line, starting at -2. It crawls to -6, then turns around and crawls to 5. How many units does the bug crawl altogether?
(A) 9 (B) 11 (C) 13 (D) 14 (E) 15

真题文字详解(中英双语)

Problem 1 The following problem is from both the 2012 AMC 12A #2 and 2012 AMC 10A #1, so both problems redirect to this page. 以下题目来自2012年AMC 12A第2题和2012年AMC 10A第1题，因此这两个题目都重定向到这个页面。 Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30 seconds. Working together, how many cupcakes can they frost in 5 minutes? Cagney 可以每 20 秒装饰一个纸杯蛋糕，Lacey 可以每 30 秒装饰一个纸杯蛋糕。他们一起工作，5 分钟内可以装饰多少个纸杯蛋糕？ (A) 10 (B) 15 (C) 20 (D) 25 (E) 30 Solution 1 Cagney can frost one in 20 seconds, and Lacey can frost one in 30 seconds. Working together, they can frost one in \frac{20\cdot30}{20+30}=\frac{600}{50}=12 seconds. In 300 seconds (5 minutes), they can frost (D) 25 cupcakes. Solution 2 In 300 seconds (5 minutes), Cagney will frost \frac{300}{20}=15 cupcakes, and Lacey will frost \frac{300}{30}=10 cupcakes. Therefore, working together they will frost 15 + 10 = (D) 25 cupcakes. Solution 3 Since Cagney frosts 3 cupcakes a minute, and Lacey frosts 2 cupcakes a minute, they together frost 3 + 2 = 5 cupcakes a minute. Therefore, in 5 minutes, they frost 5 × 5 = 25 ⇒ (D) Problem 2 A square with side length 8 is cut in half, creating two congruent rectangles. What are the dimensions of one of these rectangles? (A) 2 by 4 (B) 2 by 6 (C) 2 by 8 (D) 4 by 4 (E) 4 by 8 Solution Cutting the square in half will bisect one pair of sides while the other side will remain unchanged. Thus, the new square is \frac{8}{2} by 8, or (E) 4 by 8 Problem 3 The following problem is from both the 2012 AMC 12A #1 and 2012 AMC 10A #3, so both problems redirect to this page. A bug crawls along a number line, starting at -2. It crawls to -6, then turns around and crawls to 5. How many units does the bug crawl altogether? (A) 9 (B) 11 (C) 13 (D) 14 (E) 15 Solution