2009 AMC amc12b 真题 答案 详解

---

英文真题

2009 AMC 12B Problems Problem 1 Each morning of her five-day workweek, Jane bought either a $0.50 muffin or a $0.75 bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy? (A) 1 (B) 2 (C) 3 (D) 4 (E) 5 Problem 2 Paula the painter had just enough paint for 30 identically sized rooms. Unfortunately, on the way to work, three cans of paint fell off her truck, so she had only enough paint for 25 rooms. How many cans of paint did she use for the 25 rooms? (A) 10 (B) 12 (C) 15 (D) 18 (E) 25 Problem 3 Twenty percent less than 60 is one-third more than what number? (A) 16 (B) 30 (C) 32 (D) 36 (E) 48 Problem 4 A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths 15 and 25 meters. What fraction of the yard is occupied by the flower beds? (A) \frac{1}{8} (B) \frac{1}{6} (C) \frac{1}{5} (D) \frac{1}{4} (E) \frac{1}{3} Problem 5 Kiana has two older twin brothers. The product of their ages is 128. What is the sum of their three ages? (A) 10 (B) 12 (C) 16 (D) 18 (E) 24 Problem 6 By inserting parentheses, it is possible to give the expression 2 × 3 + 4 × 5 several values. How many different values can be obtained? (A) 2 (B) 3 (C) 4 (D) 5 (E) 6 Problem 7 In a certain year the price of gasoline rose by 20% during January, fell by 20% during February, rose by 25% during March, and fell by x% during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the nearest integer, what is x? (A) 12 (B) 17 (C) 20 (D) 25 (E) 35 Problem 8 When a bucket is two-thirds full of water, the bucket and water weigh a kilograms. When the bucket is one-half full of water the total weight is b kilograms. In terms of a and b, what is the total weight in kilograms when the bucket is full of water? (A) \frac{2}{3}a+\frac{1}{3}b (B) \frac{3}{2}a-\frac{1}{2}b (C) \frac{3}{2}a+b (D) \frac{3}{2}a+2b (E) 3a-2b

真题文字详解(英文)

Problem 1 The following problem is from both the 2009 AMC 10B #1 and 2009 AMC 12B #1, so both problems redirect to this page. Each morning of her five-day workweek, Jane bought either a $50$-cent muffin or a $75$-cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy?
(A) $1$ (B) $2$ (C) $3$ (D) $4$ (E) $5$

Solution 1 (Observations: Replacements) If Jane bought one more bagel but one fewer muffin, then her total cost for the week would increase by $25$ cents. If Jane bought $1$ bagel, then she bought $4$ muffins. Her total cost for the week would be $75\cdot 1+50\cdot 4=275$ cents, or $2.75$ dollars. Clearly, she bought one more bagel but one fewer muffin at a total cost of $3.00$ dollars. Therefore, she bought (B) $2$ bagels.

Solution 2 (Observations: Answer Choices) If Jane bought $1$ bagel, then she bought $4$ muffins. Her total cost for the week would be $75\cdot 1+50\cdot 4=275$ cents, or $2.75$ dollars. So, (A) is incorrect. If Jane bought $2$ bagels, then she bought $3$ muffins. Her total cost for the week would be $75\cdot 2+50\cdot 3=300$ cents, or $3.00$ dollars. So, (B) $2$ is correct. For completeness, we will check (C), (D), and (E) too. If you decide to use this solution on the real test, then you will not need to do that, as you want to save more time. If Jane bought $3$ bagels, then she bought $2$ muffins. Her total cost for the week would be $75\cdot 3+50\cdot 2=325$ cents, or $3.25$ dollars. So, (C) is incorrect. If Jane bought $4$ bagels, then she bought $1$ muffin. Her total cost for the week would be $75\cdot 4+50\cdot 1=350$ cents, or $3.50$ dollars. So, (D) is incorrect. If Jane bought $5$ bagels, then she bought $0$ muffins. Her total cost for the week would be $75\cdot 5+50\cdot 0=375$ cents, or $3.75$ dollars. So, (E) is incorrect.

Solution 3 (Arithmetic) In this solution, all amounts are in the unit of cents. Note that the amount spent on muffins must end in either $00$ or $50$, and the amount spent on bagels must end in one of $00$, $25$, $50$, or $75$. Furthermore, the number of muffins bought and the number of bagels bought must sum to $5$. We have two possible cases: 1. The amounts spent on muffins and bagels both end in $00$. The number of muffins bought must be even, and the number of bagels bought must be a multiple of $4$. In this case, there are no solutions. 2. The amounts spent on muffins and bagels both end in $50$. The number of muffins bought must be odd, and the number of bagels bought must be $2$ more than a multiple of $4$. In this case, the only solution is $3$ muffins and $2$ bagels, from which the answer is (B) $2$.

See also Problem 2 The following problem is from both the 2009 AMC 10B #3 and 2009 AMC 12B #2, so both problems redirect to this page. Paula the painter had just enough paint for $30$ identically sized rooms. Unfortunately, on the way to work, three cans of paint fell off her truck, so she had only enough paint for $25$ rooms. How many cans of paint did she use for the $25$ rooms? (A) $10$ (B) $12$ (C) $15$ (D) $18$ (E) $25$

真题文字详解(中英双语)

Problem 1 The following problem is from both the 2009 AMC 10B #1 and 2009 AMC 12B #1, so both problems redirect to this page. 以下问题来自2009年AMC 10B #1和2009年AMC 12B #1，因此这两个问题都指向这一页。

Each morning of her five-day workweek, Jane bought either a 50-cent muffin or a 75-cent bagel. Her total cost for the week was a whole number of dollars. How many bagels did she buy? 在她五天的上班日里，简每天要么买一个0元的松饼，要么买一个1元的贝果。她一周的总花费是一整数美元。她买了多少个贝果？

(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

Solution 1 (Observations: Replacements)

解法 1（观察：替换）

If Jane bought one more bagel but one fewer muffin, then her total cost for the week would increase by 25 cents. 如果简多买一个贝果但少买一个松饼，那么她一周的总花费会增加0.25美元。

If Jane bought 1 bagel, then she bought 4 muffins. Her total cost for the week would be 75 · 1 + 50 · 4 = 275 cents, or 2.75 dollars. Clearly, she bought one more bagel but one fewer muffin at a total cost of 3.00 dollars. Therefore, she bought (B) 2 bagels.

如果简买了1个面包圈，那么她买了4个松饼。她一周的总花费将是75·1+50·4=275分钱，或2.75美元。显然，她多买了一个面包圈但少买了一个松饼，总花费为3.00美元。因此，她买了(B) 2个面包圈。

Solution 2 (Observations: Answer Choices)

解法 2（观察：答案选项）

• If Jane bought 1 bagel, then she bought 4 muffins. Her total cost for the week would be 75 · 1 + 50 · 4 = 275 cents, or 2.75 dollars. So, (A) is incorrect.

• 如果简买了1个面包圈，那么她买了4个松饼。她一周的总花费将是75·1+50·4=275分钱，或2.75美元。所以，（A）是错误的。

• If Jane bought 2 bagels, then she bought 3 muffins. Her total cost for the week would be 75 · 2 + 50 · 3 = 300 cents, or 3.00 dollars. So, (B) 2 is correct.

• 如果简买了2个面包圈，那么她买了3个松饼。她一周的总花费将是75·2+50·3=300分钱，或3.00美元。所以，（B）2 是正确的。

For completeness, we will check (C), (D), and (E) too. If you decide to use this solution on the real test, then you will not need to do that, as you want to save more time. 为了完整起见，我们也将检查(C)，(D)，和(E)。如果你决定在真实考试中使用这个解决方案，那么你就不需要那样做了，因为你想要节省更多时间。

• If Jane bought 3 bagels, then she bought 2 muffins. Her total cost for the week would be 75 · 3 + 50 · 2 = 325 cents, or 3.25 dollars. So, (C) is incorrect.

• 如果简买了3个面包圈，那么她买了2个松饼。她一周的总成本将是75·3+50·2=325分钱，或3.25美元。所以，（C）是错误的。

• If Jane bought 4 bagels, then she bought 1 muffin. Her total cost for the week would be 75 · 4 + 50 · 1 = 350 cents, or 3.50 dollars. So, (D) is incorrect.

• 如果简买了4个面包圈，那么她买了1个松饼。她一周的总成本将是75·4+50·1=350分钱，或3.50美元。所以，（D）是错误的。

• If Jane bought 5 bagels, then she bought 0 muffins. Her total cost for the week would be 75 · 5 + 50 · 0 = 375 cents, or 3.75 dollars. So, (E) is incorrect.

• 如果简买了5个面包圈，那么她买了0个松饼。她一周的总成本将是75·5+50·0