2007 AMC amc12a 真题 答案 详解

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英文真题

2007 AMC 12A Problems

Problem 1

One ticket to a show costs $20 at full price. Susan buys 4 tickets using a coupon that gives her a 25% discount. Pam buys 5 tickets using a coupon that gives her a 30% discount. How many more dollars does Pam pay than Susan?

(A) 2 (B) 5 (C) 10 (D) 15 (E) 20

Problem 2

An aquarium has a rectangular base that measures 100 cm by 40 cm and has a height of 50 cm. It is filled with water to a height of 40 cm. A brick with a rectangular base that measures 40 cm by 20 cm and a height of 10 cm is placed in the aquarium. By how many centimeters does the water rise?

(A) 0.5 (B) 1 (C) 1.5 (D) 2 (E) 2.5

Problem 3

The larger of two consecutive odd integers is three times the smaller. What is their sum?

(A) 4 (B) 8 (C) 12 (D) 16 (E) 20

Problem 4

Kate rode her bicycle for 30 minutes at a speed of 16 mph, then walked for 90 minutes at a speed of 4 mph. What was her overall average speed in miles per hour?

(A) 7 (B) 9 (C) 10 (D) 12 (E) 14

Problem 5

Last year Mr. Jon Q. Public received an inheritance. He paid 20% in federal taxes on the inheritance, and paid 10% of what he had left in state taxes. He paid a total of $10,500 for both taxes. How many dollars was his inheritance?

(A) 30000 (B) 32500 (C) 35000 (D) 37500 (E) 40000

Problem 6

Triangles ABC and ADC are isosceles with AB = BC and AD = DC. Point D is inside triangle ABC, angle ABC measures 40 degrees, and angle ADC measures 140 degrees. What is the degree measure of angle BAD?

(A) 20 (B) 30 (C) 40 (D) 50 (E) 60

Problem 7

Let a, b, c, d, and e be five consecutive terms in an arithmetic sequence, and suppose that a + b + c + d + e = 30. Which of a, b, c, d, or e can be found?

(A) a (B) b (C) c (D) d (E) e

Problem 8

A star-polygon is drawn on a clock face by drawing a chord from each number to the fifth number counted clockwise from that number. That is, chords are drawn from 12 to 5, from 5 to 10, from 10 to 3, and so on, ending back at 12. What is the degree measure of the angle at each vertex in the star polygon?

(A) 20 (B) 24 (C) 30 (D) 36 (E) 60

真题文字详解(英文)

Problem 1 The following problem is from both the 2007 AMC 12A #1 and 2007 AMC 10A #1, so both problems redirect to this page. One ticket to a show costs $20 at full price. Susan buys 4 tickets using a coupon that gives her a 25% discount. Pam buys 5 tickets using a coupon that gives her a 30% discount. How many more dollars does Pam pay than Susan? (A) 2 (B) 5 (C) 10 (D) 15 (E) 20 Official Solution Answer: (C) Susan pays $(4)(0.75)(20)=60$ dollars. Pam pays $(5)(0.70)(20)=70$ dollars, so she pays $70-60=10$ more dollars than Susan. See also Problem 2 An aquarium has a rectangular base that measures 100 cm by 40 cm and has a height of 50 cm. It is filled with water to a height of 40 cm. A brick with a rectangular base that measures 40 cm by 20 cm and a height of 10 cm is placed in the aquarium. By how many centimeters does the water rise? (A) 0.5 (B) 1 (C) 1.5 (D) 2 (E) 2.5 Solution The brick has volume $8000cm^3$. The base of the aquarium has area $4000cm^2$. For every inch the water rises, the volume increases by $4000cm^3$; therefore, when the volume increases by $8000cm^3$, the water level rises $2cm\Rightarrow$ D See also Problem 3 The larger of two consecutive odd integers is three times the smaller. What is their sum? (A) 4 (B) 8 (C) 12 (D) 16 (E) 20 Solution 1 Let $n$ be the smaller term. Then $n+2=3n\Rightarrow 2n=2\Rightarrow n=1$ Thus, the answer is $1+(1+2)=4(A)$ Solution 2 By trial and error, 1 and 3 work. 1+3=4. See also Problem 4 Kate rode her bicycle for 30 minutes at a speed of 16 mph, then walked for 90 minutes at a speed of 4 mph. What was her overall average speed in miles per hour? (A) 7 (B) 9 (C) 10 (D) 12 (E) 14 Solution

真题文字详解(中英双语)

Problem 1 The following problem is from both the 2007 AMC 12A #1 and 2007 AMC 10A #1, so both problems redirect to this page. 以下问题来自2007年AMC 12A #1和2007年AMC 10A #1，因此这两个问题都指向这一页。

One ticket to a show costs $20 at full price. Susan buys 4 tickets using a coupon that gives her a 25% discount. Pam buys 5 tickets using a coupon that gives her a 30% discount. How many more dollars does Pam pay than Susan? 一张演出票的全价是$20美元。苏珊用给她提供25%折扣的优惠券买了4张票。帕姆用给她提供30%折扣的优惠券买了5张票。帕姆比苏珊多付了多少钱？
(A) 2 (B) 5 (C) 10 (D) 15 (E) 20

Official Solution Answer: (C)Susan pays $(4)(0.75)(20)=60$ dollars. Pam pays $(5)(0.70)(20)=70$ dollars, so she pays $70-60=10$ more dollars than Susan.
Answer：(C)苏珊支付了$(4)(0.75)(20)=60$美元。帕姆支付了$(5)(0.70)(20)=70$美元，因此她比苏珊多支付了$70-60=10$美元。

See also Problem 2 An aquarium has a rectangular base that measures 100 cm by 40 cm and has a height of 50 cm. It is filled with water to a height of 40 cm. A brick with a rectangular base that measures 40 cm by 20 cm and a height of 10 cm is placed in the aquarium. By how many centimeters does the water rise? 一个水族箱有一个100厘米乘以40厘米的长方形底座，高度为50厘米。它被水填满到40厘米的高度。一个底座为40厘米乘以20厘米、高度为10厘米的矩形砖块放入水族箱中。水位上升了多少厘米？
(A) 0.5 (B) 1 (C) 1.5 (D) 2 (E) 2.5

Solution The brick has volume $8000cm^3$. The base of the aquarium has area $4000cm^2$. For every inch the water rises, the volume increases by $4000cm^3$; therefore, when the volume increases by $8000cm^3$, the water level rises $2cm\Rightarrow$ [D] 这块砖的体积是$8000cm^3$。水族箱底部的面积是$4000cm^2$。水位每上升一英寸，体积就增加$4000cm^3$；因此，当体积增加$8000cm^3$时，水位上升$2cm\Rightarrow$ [D]

See also Problem 3 The larger of two consecutive odd integers is three times the smaller. What is their sum? 两个连续奇数中较大的那个是较小那个的三倍。它们的和是多少？
(A) 4 (B) 8 (C) 12 (D) 16 (E) 20

Solution 1 Let $n$ be the smaller term. Then $n+2=3n\Rightarrow 2n=2\Rightarrow n=1$ 设$n$为较小项。那么$n+2=3n\Rightarrow 2n=2\Rightarrow n=1$