2005 AMC amc12a 真题 答案 详解

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英文真题

2005 AMC 12A Problems
Problem 1
Two is 10% of x and 20% of y. What is x - y?
(A) 1 (B) 2 (C) 5 (D) 10 (E) 20

Problem 2
The equations 2x + 7 = 3 and bx - 10 = -2 have the same solution. What is the value of b?
(A) -8 (B) -4 (C) 2 (D) 4 (E) 8

Problem 3
A rectangle with diagonal length x is twice as long as it is wide. What is the area of the rectangle?
(A) (\frac{1}{4}x^2) (B) (\frac{2}{5}x^2) (C) (\frac{1}{2}x^2) (D) (x^2) (E) (\frac{3}{2}x^2)

Problem 4
A store normally sells windows at $100 each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How much will they save if they purchase the windows together rather than separately?
(A) 100 (B) 200 (C) 300 (D) 400 (E) 500

Problem 5
The average (mean) of 20 numbers is 30, and the average of 30 other numbers is 20. What is the average of all 50 numbers?
(A) 23 (B) 24 (C) 25 (D) 26 (E) 27

Problem 6
Josh and Mike live 13 miles apart. Yesterday, Josh started to ride his bicycle toward Mike's house. A little later Mike started to ride his bicycle toward Josh's house. When they met, Josh had ridden for twice the length of time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden when they met?
(A) 4 (B) 5 (C) 6 (D) 7 (E) 8

Problem 7
Square EFGH is inside the square ABCD so that each side of EFGH can be extended to pass through a vertex of ABCD. Square ABCD has side length √50 and BE = 1. What is the area of the inner square EFGH?
(A) 25 (B) 32 (C) 36 (D) 40 (E) 42

真题文字详解(英文)

Problem 1 Two is 10% of x and 20% of y. What is (x - y)?

(A) 1
(B) 2
(C) 5
(D) 10
(E) 20

Solution
(2 = \frac{1}{10}x \Rightarrow x = 20), (2 = \frac{1}{5}y \Rightarrow y = 10), (x - y = 20 - 10 = 10) (D).

See also

Problem 2 The following problem is from both the 2005 AMC 12A #2 and 2005 AMC 10A #3, so both problems redirect to this page.

The equations (2x + 7 = 3) and (bx - 10 = -2) have the same solution. What is the value of b?
(A) -8
(B) -4
(C) 2
(D) 4
(E) 8

Solution
(2x + 7 = 3 \Rightarrow x = -2), (-2b - 10 = -2 \Rightarrow -2b = 8 \Rightarrow b =) (B) -4

Problem 3 A rectangle with diagonal length (x) is twice as long as it is wide. What is the area of the rectangle?
(A) (\frac{1}{4}x^2)
(B) (\frac{2}{5}x^2)
(C) (\frac{1}{2}x^2)
(D) (x^2)
(E) (\frac{3}{2}x^2)

Solution
Let (w) be the width, so the length is (2w). By the Pythagorean Theorem, (w^2 + 4w^2 = x^2 \Rightarrow \frac{x}{\sqrt{5}} = w). The area of the rectangle is ((w)(2w) = 2w^2 = 2\left(\frac{x}{\sqrt{5}}\right)^2 = \frac{2}{5}x^2) (B).

See also

Problem 4 A store normally sells windows at $100 each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How much will they save if they purchase the windows together rather than separately?
(A) 100
(B) 200
(C) 300
(D) 400
(E) 500

真题文字详解(中英双语)

Problem 1 Two is 10% of x and 20% of y. What is (x - y) ?
(A) 1 (B) 2 (C) 5 (D) 10 (E) 20

Solution
(2 = \frac{1}{10}x \Rightarrow x = 20), (2 = \frac{1}{5}y \Rightarrow y = 10), (x - y = 20 - 10 = 10(D)).

See also

Problem 2 The following problem is from both the 2005 AMC 12A #2 and 2005 AMC 10A #3, so both problems redirect to this page.
The equations (2x + 7 = 3) and (bx - 10 = -2) have the same solution. What is the value of (b) ?
(A) -8 (B) -4 (C) 2 (D) 4 (E) 8

Solution
(2x + 7 = 3 \Rightarrow x = -2), (-2b - 10 = -2 \Rightarrow -2b = 8 \Rightarrow b = (B)-4)

Problem 3 A rectangle with diagonal length (x) is twice as long as it is wide. What is the area of the rectangle?
(A) (\frac{1}{4}x^2) (B) (\frac{2}{5}x^2) (C) (\frac{1}{2}x^2) (D) (x^2) (E) (\frac{3}{2}x^2)

Solution
Let (w) be the width, so the length is (2w). By the Pythagorean Theorem, (w^2 + 4w^2 = x^2 \Rightarrow \frac{x}{\sqrt{5}} = w). The area of the rectangle is ((w)(2w) = 2w^2 = 2\left(\frac{x}{\sqrt{5}}\right)^2 = \frac{2}{5}x^2 (B)).

See also

Problem 4 A store normally sells windows at $100 each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How much will they save if they purchase the windows together rather than separately?
(A) 100 (B) 200 (C) 300 (D) 400 (E) 500

Solution