2010 AMC amc10a 真题 答案 详解

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中英双语真题

2010 AMC 10A
Problem 1
Mary's top book shelf holds five books with the following widths, in centimeters: (6, \frac{1}{2}, 1, 2.5), and (10). What is the average book width, in centimeters?
Mary 的顶层书架上放了 5 本书，它们的宽度（单位：厘米）分别是 6 和 10。问书本的平均宽度是多少厘米？
(A) (1) (B) (2) (C) (3) (D) (4) (E) (5)

Problem 2
Four identical squares and one rectangle are placed together to form one large square as shown. The length of the rectangle is how many times as large as its width?
4 个全等的正方形和 1 个矩形放置在一起形成一个大的正方形，如下图所示。问矩形的长度是它的宽度的几倍？
(A) (\frac{5}{4}) (B) (\frac{4}{3}) (C) (\frac{3}{2}) (D) (2) (E) (3)

Problem 3
Tyrone had 97 marbles and Eric had 11 marbles. Tyrone then gave some of his marbles to Eric so that Tyrone ended with twice as many marbles as Eric. How many marbles did Tyrone give to Eric?
Tyrone 有 97 颗玻璃球，Eric 有 11 颗玻璃球。Tyrone 给了 Eric 若干颗玻璃球，最后 Tyrone 的玻璃球颗数是 Eric 的 2 倍。问 Tyrone 给了 Eric 多少颗玻璃球？
(A) (3) (B) (13) (C) (18) (D) (25) (E) (29)

英文真题

2010 AMC 10A Problems Problem 1 Mary's top book shelf holds five books with the following widths, in centimeters: (6, \frac{1}{2}, 1, 2.5,) and (10). What is the average book width, in centimeters?
(A) (1) (B) (2) (C) (3) (D) (4) (E) (5)

Problem 2 Four identical squares and one rectangle are placed together to form one large square as shown. The length of the rectangle is how many times as large as its width?
(A) (\frac{5}{4}) (B) (\frac{4}{3}) (C) (\frac{3}{2}) (D) (2) (E) (3)

Problem 3 Tyrone had 97 marbles and Eric had 11 marbles. Tyrone then gave some of his marbles to Eric so that Tyrone ended with twice as many marbles as Eric. How many marbles did Tyrone give to Eric?
(A) (3) (B) (13) (C) (18) (D) (25) (E) (29)

Problem 4 A book that is to be recorded onto compact discs takes 412 minutes to read aloud. Each disc can hold up to 56 minutes of reading. Assume that the smallest possible number of discs is used and that each disc contains the same length of reading. How many minutes of reading will each disc contain?
(A) (50.2) (B) (51.5) (C) (52.4) (D) (53.8) (E) (55.2)

Problem 5 The area of a circle whose circumference is (24\pi) is (k\pi). What is the value of (k)?
(A) (6) (B) (12) (C) (24) (D) (36) (E) (144)

Problem 6 For positive numbers (x) and (y) the operation (x \spadesuit y) is defined as
(x \spadesuit y = x - \frac{1}{y})
What is (2 \spadesuit (2 \spadesuit 2))?
(A) (\frac{2}{3}) (B) (1) (C) (\frac{4}{3}) (D) (\frac{5}{3}) (E) (2)

真题文字详解(英文)

Problem 1 Problem 1 Mary's top book shelf holds five books with the following widths, in centimeters: $6, \frac{1}{2}, 1, 2.5,$ and $10$. What is the average book width, in centimeters?
$(A)\ 1\quad (B)\ 2\quad (C)\ 3\quad (D)\ 4\quad (E)\ 5$ Solution To find the average, we add up the widths $6, \frac{1}{2}, 1, 2.5,$ and $10$, to get a total sum of $20$. Since there are $5$ books, the average book width is $\frac{20}{5} = 4$. The answer is [D].

Problem 2 Problem 2 Four identical squares and one rectangle are placed together to form one large square as shown. The length of the rectangle is how many times as large as its width?
$(A)\ \frac{5}{4}\quad (B)\ \frac{4}{3}\quad (C)\ \frac{3}{2}\quad (D)\ 2\quad (E)\ 3$ Solution 1 Let the length of the small square be $x$, intuitively, the length of the big square is $4x$. It can be seen that the width of the rectangle is $3x$. Thus, the length of the rectangle is $4x/3x = 4/3$ times as large as the width. The answer is [B].
Solution 2 We can say the area of one small square is $x^2$, so $\frac{1}{4}$ of the area of the large square is $4x^2$ so the area of the large square is $16x^2$, so each side is $4x$ so the length of the rectangle is $4x$ and the width of the rectangle is $4x - x = 3x$ so $\frac{4x}{3x} = \frac{4}{3}$.
Solution 3 Let the side length of one of the squares equal $1$. Then, the width of the rectangle will be $4$, and since the width of the rectangle is the same as the length of the entire shape, the length of the rectangle is $4-1=3$. The ratio between the two is therefore $\frac{4}{3}$, so our answer is [B].

Problem 3 Problem 3 Tyrone had $97$ marbles and Eric had $11$ marbles. Tyrone then gave some of his marbles to Eric so that Tyrone ended with twice as many marbles as Eric. How many marbles did Tyrone give to Eric?
$(A)\ 3\quad (B)\ 13\quad (C)\ 18\quad (D)\ 25\quad (E)\ 29$ Solution 1 Let $x$ be the number of marbles Tyrone gave to Eric. Then, $97-x = 2\cdot(11+x)$. Solving for $x$ yields $75 = 3x$ and $x = 25$. The answer is [D].
Solution 2

真题文字详解(中英双语)

Problem 1 Problem 1 Mary's top book shelf holds five books with the following widths, in centimeters: (6, \frac{1}{2}, 1, 2.5), and (10). What is the average book width, in centimeters?
(A) (1) (B) (2) (C) (3) (D) (4) (E) (5) Solution To find the average, we add up the widths (6, \frac{1}{2}, 1, 2.5), and (10), to get a total sum of (20). Since there are (5) books, the average book width is (\frac{20}{5} = 4). The answer is [D].

Problem 2 Problem 2 Four identical squares and one rectangle are placed together to form one large square as shown. The length of the rectangle is how many times as large as its width?
(A) (\frac{5}{4}) (B) (\frac{4}{3}) (C) (\frac{3}{2}) (D) (2) (E) (3) Solution 1 Let the length of the small square be (x), intuitively, the length of the big square is (4x). It can be seen that the width of the rectangle is (3x). Thus, the length of the rectangle is (\frac{4x}{3x} = \frac{4}{3}) times as large as the width. The answer is [B].
Solution 2 We can say the area of one small square is (x^2), so (\frac{1}{4}) of the area of the large square is (4x^2) so the area of the large square is (16x^2), so each side is (4x) so the length of the rectangle is (4x) and the width of the rectangle is (4x - x = 3x) so (\frac{4x}{3x} = \frac{4}{3}).
Solution 3 Let the side length of one of the squares equal (1). Then, the width of the rectangle will be (4), and since the width of the rectangle is the same as the length of the entire shape, the length of the rectangle is (4-1=3). The ratio between the two is therefore (\frac{4}{3}), so our answer is [B].

Problem 3 Problem 3 Tyrone had (97) marbles and Eric had (11) marbles. Tyrone then gave some of his marbles to Eric so that Tyrone ended with twice as many marbles as Eric. How many marbles did Tyrone give to Eric?
(A) (3) (B) (13) (C) (18) (D) (25) (E) (29) Solution 1 Let (x) be the number of marbles Tyrone gave to Eric. Then, (97-x = 2 \cdot (11+x)). Solving for (x) yields (75 = 3x) and (x = 25). The answer is [D].
Solution 2