2018 AMC amc10b 真题 答案 详解

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中英双语真题

2018 AMC 10B

Problem 1

Kate bakes a 20-inch by 18-inch pan of cornbread. The cornbread is cut into pieces that measure 2 inches by 2 inches. How many pieces of cornbread does the pan contain?

Kate 烤了一块 20 英寸 x 18 英寸 的玉米面包。这块玉米面包被切分成 2 英寸 x 2 英寸 的多个小块面包。问这大块玉米面包包含多少块小面包？

(A) 90 (B) 100 (C) 180 (D) 200 (E) 360

Problem 2

Sam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was 60 mph (miles per hour), and his average speed during the second 30 minutes was 65 mph. What was his average speed, in mph, during the last 30 minutes?

Sam 90 分钟内开了 96 英里。他在前 30 分钟内的平均速度为 60 英里每小时，在第二个 30 分钟内的平均速度为 65 英里每小时。则他在最后 30 分钟内的平均速度是多少英里每小时？

(A) 64 (B) 65 (C) 66 (D) 67 (E) 68

Problem 3

In the expression ( × ) + ( × ) each blank is to be filled in with one of the digits 1, 2, 3, or 4, with each digit being used once. How many different values can be obtained?

在表达式 ( × ) + ( × ) 中，从 1, 2, 3 或 4 中选 1 个数字填到其中 1 个空白处，并且每个数字只能使用 1 次。可以得到多少种不同的值？

(A) 2 (B) 3 (C) 4 (D) 6 (E) 24

英文真题

2018 AMC 10B Problems Problem 1 Kate bakes a 20-inch by 18-inch pan of cornbread. The cornbread is cut into pieces that measure 2 inches by 2 inches. How many pieces of cornbread does the pan contain? (A) 90 (B) 100 (C) 180 (D) 200 (E) 360 Problem 2 Sam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was 60 mph (miles per hour), and his average speed during the second 30 minutes was 65 mph. What was his average speed, in mph, during the last 30 minutes? (A) 64 (B) 65 (C) 66 (D) 67 (E) 68 Problem 3 In the expression (___ × ___ ) + ( ___ × ___ ) each blank is to be filled in with one of the digits 1, 2, 3, or 4, with each digit being used once. How many different values can be obtained? (A) 2 (B) 3 (C) 4 (D) 6 (E) 24 Problem 4 A three-dimensional rectangular box with dimensions X, Y, and Z has faces whose surface areas are 24, 24, 48, 48, 72, and 72 square units. What is X + Y + Z? (A) 18 (B) 22 (C) 24 (D) 30 (E) 36 Problem 5 How many subsets of {2, 3, 4, 5, 6, 7, 8, 9} contain at least one prime number? (A) 128 (B) 192 (C) 224 (D) 240 (E) 256 Problem 6 A box contains 5 chips, numbered 1, 2, 3, 4, and 5. Chips are drawn randomly one at a time without replacement until the sum of the values drawn exceeds 4. What is the probability that 3 draws are required? (A) \frac{1}{15} (B) \frac{1}{10} (C) \frac{1}{6} (D) \frac{1}{5} (E) \frac{1}{4} Problem 7 In the figure below, N congruent semicircles are drawn along a diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let A be the combined area of the small semicircles and B be the area of the region inside the large semicircle but outside the small semicircles. The ratio A : B is 1 : 18. What is N? (A) 16 (B) 17 (C) 18 (D) 19 (E) 36 Problem 8 Sara makes a staircase out of toothpicks as shown:

真题文字详解(英文)

Problem 1 The following problem is from both the 2018 AMC 12B #1 and 2018 AMC 10B #1, so both problems redirect to this page. Kate bakes a 20-inch by 18-inch pan of cornbread. The cornbread is cut into pieces that measure 2 inches by 2 inches. How many pieces of cornbread does the pan contain? (A) 90 (B) 100 (C) 180 (D) 200 (E) 360 Solution 1 The area of the pan is $20\cdot18=360$. Since the area of each piece is $2\cdot2=4$, there are $\frac{360}{4}=\textbf{(A)}\ 90$ pieces. Solution 2 By dividing each of the dimensions by 2, we get a $10\times9$ grid that makes \textbf{(A)} 90 pieces. Problem 2 The following problem is from both the 2018 AMC 12B #2 and 2018 AMC 10B #2, so both problems redirect to this page. Sam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was 60 mph (miles per hour), and his average speed during the second 30 minutes was 65 mph. What was his average speed, in mph, during the last 30 minutes? (A) 64 (B) 65 (C) 66 (D) 67 (E) 68 Solution 1 Suppose that Sam's average speed during the last 30 minutes was $x$ mph. Recall that a half hour is equal to 30 minutes. Therefore, Sam drove $60\cdot0.5=30$ miles during the first half hour, $65\cdot0.5=32.5$ miles during the second half hour, and $x\cdot0.5$ miles during the last half hour. We have $$30+32.5+x\cdot0.5=96$$ $$x\cdot0.5=33.5$$ $$x=\textbf{(D)}\ 67.$$ ~Haha0201 ~MRENTHUSIASM Solution 2 Suppose that Sam's average speed during the last 30 minutes was $x$ mph. Note that Sam's average speed during the entire trip was $\frac{96}{3/2}=64$ mph. Since Sam drove at 60 mph, 65 mph, and $x$ mph for the same duration (30 minutes each), his average speed during the entire trip was the average of 60 mph, 65 mph, and $x$ mph. We have $$\frac{60+65+x}{3}=64$$ $$60+65+x=192$$ $$x=\textbf{(D)}\ 67.$$ ~coolmath_2018 ~MRENTHUSIASM Problem 3 In the expression $(____\times____) + (____\times____)$ each blank is to be filled in with one of the digits 1, 2, 3, or 4, with each digit being used once. How many different values can be obtained? (A) 2 (B) 3 (C) 4 (D) 6 (E) 24 Solution 1 - Combinations We have $\binom{4}{2}$ ways to choose the pairs, and we have $2!$ ways for the values to be rearranged, hence $\frac{6}{2}=\textbf{(B)}\ 3$. Solution 2 We have four available numbers (1, 2, 3, 4). Because different permutations do not matter because they are all addition and multiplication, if we put 1 on the first space, it is obvious there are \textbf{(B)} 3 possible outcomes (2, 3, 4). Solution 3 There are exactly $4!$ ways to arrange the numbers and $2!2!2!$ overcounts per way due to commutativity. Therefore, the answer is $\frac{4!}{2!2!2!}=\textbf{(B)}\ 3$. Problem 4 A three-dimensional rectangular box with dimensions X, Y, and Z has faces whose surface areas are 24, 24, 48, 48, 72, and 72 square units. What is X + Y + Z?

真题文字详解(中英双语)

Problem 1 The following problem is from both the 2018 AMC 12B #1 and 2018 AMC 10B #1, so both problems redirect to this page. 以下问题来自2018年AMC 12B #1和2018年AMC 10B #1，因此这两个问题都指向这一页。 Kate bakes a 20-inch by 18-inch pan of cornbread. The cornbread is cut into pieces that measure 2 inches by 2 inches. How many pieces of cornbread does the pan contain? 凯特烤了一个20英寸乘以18英寸的玉米面包。玉米面包被切成每块2英寸乘以2英寸的块。这个面包盘能切多少块玉米面包？ (A) 90 (B) 100 (C) 180 (D) 200 (E) 360 Solution 1 The area of the pan is 20 · 18 = 360. Since the area of each piece is 2 · 2 = 4, there are \frac{360}{4} = (A) 90 pieces. 面包盘的面积是20·18=360。因为每块面的面积是2·2=4，所以有\frac{360}{4}=(A) 90块。 Solution 2 By dividing each of the dimensions by 2, we get a 10 × 9 grid that makes (A) 90 pieces. 通过将每个尺寸除以2，我们得到一个10×9的网格，可以切出(A) 90块。 Problem 2 The following problem is from both the 2018 AMC 12B #2 and 2018 AMC 10B #2, so both problems redirect to this page. 以下问题来自2018年AMC 12B #2和2018年AMC 10B #2，因此这两个问题都指向这一页。 Sam drove 96 miles in 90 minutes. His average speed during the first 30 minutes was 60 mph (miles per hour), and his average speed during the second 30 minutes was 65 mph. What was his average speed, in mph, during the last 30 minutes? 山姆在96分钟内驾驶了90英里。他在前30分钟内的平均速度是60英里每小时，在后30分钟内的平均速度是65英里每小时。他在最后30分钟内的平均速度是多少（英里每小时）？ (A) 64 (B) 65 (C) 66 (D) 67 (E) 68 Solution 1 Suppose that Sam's average speed during the last 30 minutes was x mph. 假设山姆在最后30分钟内的平均速度是x英里每小时。 Recall that a half hour is equal to 30 minutes. Therefore, Sam drove 60 · 0.5 = 30 miles during the first half hour, 65 · 0.5 = 32.5 miles during the second half hour, and x · 0.5 miles during the last half hour. We have 30 + 32.5 + x · 0.5 = 96 x · 0.5 = 33.5 x = (D) 67 ~Haha0201~MRENTHUSIASM 回忆一下，半小时等于30分钟。因此，山姆在前一个半小时内驾驶了60·0.5=30英里，在第二个半小时内驾驶了65·0.5=32.5英里，在最后一个半小时内驾驶了x·0.5英里。我们有~Haha0201~MRENTHUSIASM Solution 2 Suppose that Sam's average speed during the last 30 minutes was x mph. 假设山姆在最后30分钟内的平均速度是x英里每小时。 Note that Sam's average speed during the entire trip was \frac{96}{3/2} = 64 mph. Since Sam drove at 60 mph, 65 mph, and x mph for the same duration (30 minutes each), his average speed during the entire trip was the average of 60 mph, 65 mph, and x mph. We have \frac{60+65+x}{3} = 64 60 + 65 + x = 192 x = (D) 67 ~coolmath_2018~MRENTHUSIASM 注意到萨姆在整个旅程中的平均速度是\frac{96}{3/2}=64英里每小时。由于萨姆以60英里每小时、65英里每小时和x英里每小时的速率行驶了相同的时间（每个30分钟），他在整个旅程中的平均速度是60英里每小时、65英里每小时和x英里每小时的平均值。我们有~coolmath_2018~MRENTHUSIASM Problem 3