2009 AMC amc10a 真题 答案 详解

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英文真题

2009 AMC 10A Problems
Problem 1
One can holds 12 ounces of soda. What is the minimum number of cans needed to provide a gallon (128 ounces) of soda?
(A) 7 (B) 8 (C) 9 (D) 10 (E) 11

Problem 2
Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes and quarters. Which of the following could not be the total value of the four coins, in cents?
(A) 15 (B) 25 (C) 35 (D) 45 (E) 55

Problem 3
Which of the following is equal to $1+\frac{1}{1+\frac{1}{1+1}}$?
(A) $\frac{5}{4}$ (B) $\frac{3}{2}$ (C) $\frac{5}{3}$ (D) 2 (E) 3

Problem 4
Eric plans to compete in a triathlon. He can average 2 miles per hour in the $\frac{1}{4}$-mile swim and 6 miles per hour in the 3-mile run. His goal is to finish the triathlon in 2 hours. To accomplish his goal what must his average speed in miles per hour be for the 15-mile bicycle ride?
(A) $\frac{120}{11}$ (B) 11 (C) $\frac{56}{5}$ (D) $\frac{45}{4}$ (E) 12

Problem 5
What is the sum of the digits of the square of 111,111,111?
(A) 18 (B) 27 (C) 45 (D) 63 (E) 81

Problem 6
A circle of radius 2 is inscribed in a semicircle, as shown. The area inside the semicircle but outside the circle is shaded. What fraction of the semicircle's area is shaded?
(A) $\frac{1}{2}$ (B) $\frac{\pi}{6}$ (C) $\frac{2}{\pi}$ (D) $\frac{2}{3}$ (E) $\frac{3}{\pi}$

Problem 7
A carton contains milk that is 2% fat, an amount that is 40% less fat than the amount contained in a carton of whole milk. What is the percentage of fat in whole milk?
(A) $\frac{12}{5}$ (B) 3 (C) $\frac{10}{3}$ (D) 38 (E) 42

真题文字详解(英文)

Problem 1 One can holds 12 ounces of soda, what is the minimum number of cans needed to provide a gallon (128 ounces) of soda?
(A) 7 (B) 8 (C) 9 (D) 10 (E) 11

Solution 1
10 cans would hold 120 ounces, but 128 > 120, so 11 cans are required. Thus, the answer is (E) 11.

Solution 2
We want to find (\left\lceil \frac{128}{12} \right\rceil) because there are a whole number of cans.
[ \frac{128}{12}=10R8 \rightarrow 11 \rightarrow (E). ]

Problem 2 The following problem is from both the 2009 AMC 12A #4 and 2009 AMC 10A #2, so both problems redirect to this page. Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could not be the total value of the four coins, in cents?
(A) 15 (B) 25 (C) 35 (D) 45 (E) 55

Problem 3 The following problem is from both the 2009 AMC 12A #2 and 2009 AMC 10A #3, so both problems redirect to this page. Which of the following is equal to (1+\frac{1}{1+\frac{1}{1+1}})?
(A) (\frac{5}{4}) (B) (\frac{3}{2}) (C) (\frac{5}{3}) (D) 2 (E) 3

Solution
We compute:
[ 1+\frac{1}{1+\frac{1}{1+1}} = 1+\frac{1}{1+\frac{1}{1+1}} = 1+\frac{1}{1+\frac{1}{2}} = 1+\frac{1}{\frac{3}{2}} = 1+\frac{2}{3} = \frac{5}{3} ]

真题文字详解(中英双语)

Problem 1 One can holds 12 ounces of soda, what is the minimum number of cans needed to provide a gallon (128 ounces) of soda? 一个杯子可以装 12 盎司的汽水，那么需要多少个杯子才能提供一加仑（128 盎司）的汽水？ (A) 7 (B) 8 (C) 9 (D) 10 (E) 11 Solution 1 10 cans would hold 120 ounces, but 128 > 120, so 11 cans are required. Thus, the answer is (E) 11. 10 个杯子可以装 120 盎司，但是 128 > 120，所以需要 11 个杯子。因此，答案是 (E) 11。 Solution 2 We want to find \left\lceil \frac{128}{12} \right\rceil because there are a whole number of cans. 我们想要找到 \left\lceil \frac{128}{12} \right\rceil ，因为杯子的数量必须是整数。 \frac{128}{12} = 10R8 → 11 → (E). Problem 2 The following problem is from both the 2009 AMC 12A #4 and 2009 AMC 10A #2, so both problems redirect to this page. 下面的问题是来自2009年AMC 12A第4题和2009年AMC 10A第2题，所以这两个问题都重定向到这个页面。 Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could not be the total value of the four coins, in cents? 从一个装有分币、五分币、一角币和二角五分币的猪形储蓄罐中挑出四枚硬币。以下哪一项不可能是这四枚硬币的总价值（以分为单位）？ (A) 15 (B) 25 (C) 35 (D) 45 (E) 55 Problem 3 The following problem is from both the 2009 AMC 12A #2 and 2009 AMC 10A #3, so both problems redirect to this page. 这个问题既来自2009年AMC 12A第2题，也来自2009年AMC 10A第3题，因此这两个问题都重定向到这个页面。 Which of the following is equal to 1 + \frac{1}{1 + \frac{1}{1 + 1}}? 以下哪一项等于 1 + \frac{1}{1 + \frac{1}{1 + 1}}? (A) \frac{5}{4} (B) \frac{3}{2} (C) \frac{5}{3} (D) 2 (E) 3 Solution