2012 AMC amc10b 真题 答案 详解

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中英双语真题

2012 AMC 10B

Problem 1

Each third-grade classroom at Pearl Creek Elementary has 18 students and 2 rabbits. How many more students than rabbits are there in all 4 of the third-grade classrooms?

Pearl Creek 小学的每一个三年级教室都有 18 个学生和 2 只兔子，那么四个三年级教室的学生总数比兔子总数多多少个？

(A) 48 (B) 56 (C) 64 (D) 72 (E) 80

Problem 2

A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle?

一个半径为 5 的圆和一个矩形内切，如果所示，矩形的长和宽之比为 2:1。问矩形的面积是多少？

(A) 50 (B) 100 (C) 125 (D) 150 (E) 200

Problem 3

The point in the xy-plane with coordinates (1000, 2012) is reflected across the line y = 2000.

在坐标平面内，坐标为（1000，2012）的点关于直线y=2000作对称。那么对称点的坐标是多少？

(A) (998, 2012) (B) (1000, 1988) (C) (1000, 2024) (D) (1000, 4012) (E) (1012, 2012)

英文真题

2012 AMC 10B Problems Problem 1 Each third-grade classroom at Pearl Creek Elementary has 18 students and 2 rabbits. How many more students than rabbits are there in all 4 of the third-grade classrooms? (A) 48 (B) 56 (C) 64 (D) 72 (E) 80 Problem 2 A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle? (A) 50 (B) 100 (C) 125 (D) 150 (E) 200 Problem 3 The point in the xy-plane with coordinates (1000, 2012) is reflected across the line y = 2000. What are the coordinates of the reflected point? (A) (998, 2012) (B) (1000, 1988) (C) (1000, 2024) (D) (1000, 4012) (E) (1012, 2012) Problem 4 When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same with his marbles, he has 3 marbles left over. Ringo and Paul pool their marbles and place them into as many bags as possible, with 6 marbles per bag. How many marbles will be left over? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 Problem 5 Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is 10%. She leaves a 15% tip on the price of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of 27.50 dollars for dinner. What is the cost of her dinner without tax or tip in dollars? (A) 18 (B) 20 (C) 21 (D) 22 (E) 24 Problem 6 In order to estimate the value of x - y where x and y are real numbers with x > y > 0, Xiaoli rounded x up by a small amount, rounded y down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct? (A) Her estimate is larger than x - y (B) Her estimate is smaller than x - y (C) Her estimate equals x - y (D) Her estimate equals y - x (E) Her estimate is 0 Problem 7 For a science project, Sammy observed a chipmunk and a squirrel stashing acorns in holes. The chipmunk hid 3 acorns in each of the holes it dug. The squirrel hid 4 acorns in each of the holes it dug. They each hid the same number of acorns, although the squirrel needed 4 fewer holes. How many acorns did the chipmunk hide? (A) 30 (B) 36 (C) 42 (D) 48 (E) 54

真题文字详解(英文)

Problem 1 Each third-grade classroom at Pearl Creek Elementary has 18 students and 2 pet rabbits. How many more students than rabbits are there in all 4 of the third-grade classrooms?
(A) 48 (B) 56 (C) 64 (D) 72 (E) 80
Solution In each class, there are 18 - 2 = 16 more students than rabbits. So for all classrooms, the difference between students and rabbits is 16 × 4 = (C) 64.

Problem 2 A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle?
(A) 50 (B) 100 (C) 125 (D) 150 (E) 200
Solution Note that the diameter of the circle is equal to the shorter side of the rectangle. Since the radius is 5, the diameter is 2 · 5 = 10. Since the sides of the rectangle are in a 2 : 1 ratio, the longer side has length 2 · 10 = 20. Therefore the area is 20 · 10 = 200 or (E) 200.

Problem 3 The point in the x-y plane with coordinates (1000, 2012) is reflected across the line y = 2000. What are the coordinates of the reflected point?
(A) (998, 2012) (B) (1000, 1988) (C) (1000, 2024) (D) (1000, 4012) (E) (1012, 2012)
Solution The line y = 2000 is a horizontal line located 12 units beneath the point (1000, 2012). When a point is reflected about a horizontal line, only the y-coordinate will change. The x-coordinate remains the same. Since the y-coordinate of the point is 12 units above the line of reflection, the new y-coordinate will be 2000 – 12 = 1988. Thus, the coordinates of the reflected point are (1000, 1988)(B).

Problem 4 When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same with his marbles, he has 3 marbles leftover. Ringo and Paul pool their marbles and place them into as many bags as possible, with 6 marbles per bag. How many marbles will be leftover?
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5
Solution 1 In total, there were 3 + 4 = 7 marbles left from both Ringo and Paul. We know that 7 ≡ 1 (mod 6). This means that there would be 1 marble leftover, or (A).
Solution 2 (modulo) Let r be the number of marbles Ringo has and let p be the number of marbles Paul has. we have the following equations:
r ≡ 4 mod 6
p ≡ 3 mod 6
Adding these equations we get:
p + r ≡ 7 mod 6
We know that 7 ≡ 1 mod 6 so therefore:
p + r ≡ 7 ≡ 1 mod 6 → p + r ≡ 1 mod 6
Thus when Ringo and Paul pool their marbles, they will have (A) 1 marble left over.

Problem 5 Anna enjoys dinner at a restaurant in Washington, D.C., where the sales tax on meals is 10%. She leaves a 15% tip on the price of her meal before the sales tax is added, and the tax is calculated on the pre-tip amount. She spends a total of $27.50 dollars for dinner. What is the cost of her dinner without tax or tip in dollars?
(A) 18 (B) 20 (C) 21 (D) 22 (E) 24

真题文字详解(中英双语)

Problem 1 Each third-grade classroom at Pearl Creek Elementary has 18 students and 2 pet rabbits. How many more students than rabbits are there in all 4 of the third-grade classrooms?
珍珠溪小学每个三年级教室有18名学生和2只宠物兔子。所有4个三年级教室里，学生比兔子多多少？
(A) 48 (B) 56 (C) 64 (D) 72 (E) 80

Solution In each class, there are 18 - 2 = 16 more students than rabbits. So for all classrooms, the difference between students and rabbits is 16 × 4 = (C) 64
在每个班级里，有18-2=16更多学生而不是兔子。所以对于所有教室，学生和兔子之间的差异是16×4=(C) 64

Problem 2 A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle?
一个半径为5的圆内切于一个矩形，如图所示。矩形的长宽比为2:1。矩形的面积是多少？
(A) 50 (B) 100 (C) 125 (D) 150 (E) 200

Solution Note that the diameter of the circle is equal to the shorter side of the rectangle. Since the radius is 5, the diameter is 2 · 5 = 10. Since the sides of the rectangle are in a 2 : 1 ratio, the longer side has length 2 · 10 = 20. Therefore the area is 20 · 10 = 200 or (E) 200
注意圆的直径等于矩形较短的一边。由于半径是5，直径是2·5=10。由于矩形的边长在2:1的比例中，较长的一边长度是2·10=20。因此面积是20·10=200或(E) 200

Problem 3 The point in the xy-plane with coordinates (1000, 2012) is reflected across the line y = 2000. What are the coordinates of the reflected point?
xy平面中坐标为(1000, 2012)的点关于直线y=2000进行反射。反射点的坐标是什么？
(A) (998, 2012) (B) (1000, 1988) (C) (1000, 2024) (D) (1000, 4012) (E) (1012, 2012)

Solution The line y = 2000 is a horizontal line located 12 units beneath the point (1000, 2012). When a point is reflected about a horizontal line, only the y-coordinate will change. The x-coordinate remains the same. Since the y-coordinate of the point is 12 units above the line of reflection, the new y-coordinate will be 2000 - 12 = 1988. Thus, the coordinates of the reflected point are (1000, 1988). (B)
第y=2000条线是一条水平线，位于点(1000, 2012)下方12个单位。当一点关于一条水平线反射时，只有y坐标会改变。x坐标保持不变。由于点的y坐标高于反射线的12个单位，新的y坐标将是2000-12=1988。因此，反射点的坐标是(1000, 1988)。 (B)

Problem 4 When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same with his marbles, he has 3 marbles left over. Ringo and Paul pool their marbles and place them into as many bags as possible, with 6 marbles per bag. How many marbles will be leftover?
当Ringo把他的弹珠放入每袋装6个弹珠的袋子中时，还剩下4个弹珠。当保罗用同样的方法处理他的弹珠时，他剩下3个弹珠。Ringo和Paul把他们的弹珠放在一起，并尽可能多地放入每袋装6个弹珠的袋子中。会有多少个弹珠剩下？
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

Solution 1 In total, there were 3 + 4 = 7 marbles left from both Ringo and Paul. We know that 7 ≡ 1 (mod 6). This means that there would be 1 marble leftover, or (A)
Solution 2 (modulo)