2007 AMC amc10a 真题 答案 详解

英文真题

2007 AMC 10A 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 Define a@b = ab - b^2 and a#b = a + b - ab^2. What is 6@2/6#2? (A) -1/2 (B) -1/4 (C) 1/8 (D) 1/4 (E) 1/2 Problem 3 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 4 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 5 A school store sells 7 pencils and 8 notebooks for $4.15. It also sells 5 pencils and 3 notebooks for $1.77. How much do 16 pencils and 10 notebooks cost? (A) $1.76 (B) $5.84 (C) $6.00 (D) $6.16 (E) $6.32 Problem 6 At Euclid High School, the number of students taking the AMC 10 was 60 in 2002, 66 in 2003, 70 in 2004, 76 in 2005, 78 in 2006, and is 85 in 2007. Between what two consecutive years was there the largest percentage increase? (A) 2002 and 2003 (B) 2003 and 2004 (C) 2004 and 2005 (D) 2005 and 2006 (E) 2006 and 2007 Problem 7 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 $10500 for both taxes. How many dollars was his inheritance? (A) 30000 (B) 32500 (C) 35000 (D) 37500 (E) 40000 Problem 8 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 9 Real numbers a and b satisfy the equations 3^a = 81^(b+2) and 125^b = 5^(a-3). What is ab? (A) -60 (B) -17 (C) 9 (D) 12 (E) 60 Problem 10 The Dunbar family consists of a mother, a father, and some children. The average age of the members of the family is 20, the father is 48 years old, and the average age of the mother and children is 16. How many children are in the family? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6

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 Define $a@b = ab - b^2$ and $a#b = a + b - ab^2$. What is $\frac{6@2}{6#2}$? (A) $-\frac{1}{2}$ (B) $-\frac{1}{4}$ (C) $\frac{1}{8}$ (D) $\frac{1}{4}$ (E) $\frac{1}{2}$ Solution $6@2$ must be equal to $62-2^2$ which is 8. $6#2$ is equal to $6+2-62^2$ which is $8-24=-16$. Therefore $\frac{6@2}{6#2}$ must be equal to $\frac{8}{-16}=-\frac{1}{2}$. Therefore the solution is A. See also Problem 3 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 volume of the brick is $40\times 20\times 10=8000$. Thus the water volume rose $8000=100\times 40\times h \Longrightarrow h=2$ (D). See also Problem 4 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 Let the two consecutive odd integers be $a,a+2$. Then $a+2=3a$, so $a=1$ and their sum is 4 (A). See also Problem 5 The school store sells 7 pencils and 8 notebooks for $4.15. It also sells 5 pencils and 3 notebooks for $1.77. How much do 16 pencils and 10 notebooks cost? (A)$1.76 (B)$5.84 (C)$6.00 (D)$6.16 (E)$6.32

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. 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 Define $a@b = ab - b^2$ and $a#b = a + b - ab^2$. What is $\frac{6@2}{6#2}$? (A) $-\frac{1}{2}$ (B) $-\frac{1}{4}$ (C) $\frac{1}{8}$ (D) $\frac{1}{4}$ (E) $\frac{1}{2}$ Solution $6@2$ must be equal to $62-2^2$ which is $8$. $6#2$ is equal to $6+2-62^2$ which is $8-24=-16$. Therefore $\frac{6@2}{6#2}$ must be equal to $\frac{8}{-16}=-\frac{1}{2}$. Therefore the solution is A. See also Problem 3 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 volume of the brick is $40\times 20\times 10=8000$. Thus the water volume rose $8000=100\times 40\times h\Longrightarrow h=2$ (D). See also Problem 4 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 Let the two consecutive odd integers be $a,a+2$. Then $a+2=3a$, so $a=1$ and their sum is $4$ (A). See also Problem 5 The school store sells 7 pencils and 8 notebooks for $4.15. It also sells 5 pencils and 3 notebooks for $1.77. How much do 16 pencils and 10 notebooks cost? (A)$1.76 (B)$5.84 (C)$6.00 (D)$6.16 (E)$6.32