2020 AMC amc12a 真题 答案 详解

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中英双语真题

2020 AMC 12A
2020 AMC 12A

Problem 1
Carlos took 70% of a whole pie. Maria took one third of the remainder. What portion of the whole pie was left?
Carlos 拿走了馅饼的 70%，Maria 拿走了剩下的三分之一。问最后剩下原来馅饼的百分之多少？
(A) 10% (B) 15% (C) 20% (D) 30% (E) 35%

Problem 2
The acronym AMC is shown in the rectangular grid below with grid lines spaced 1 unit apart. In units, what is the sum of the lengths of the line segments that form the acronym AMC?
缩略词 AMC 如下显示在直角网格里，网格线之间的距离是 1 个单位。那么形成缩略词 AMC 的所有线段长度之和是多少？
(A) 17 (B) (15 + 2\sqrt{2}) (C) (13 + 4\sqrt{2}) (D) (11 + 6\sqrt{2}) (E) 21

Problem 3
A driver travels for 2 hours at 60 miles per hour, during which her car gets 30 miles per gallon of gasoline. She is paid $0.50 per mile, and her only expense is gasoline at $2.00 per gallon. What is her net rate of pay, in dollars per hour, after this expense?
一位司机以 60 英里每小时的速度开了 2 小时，在此期间她的车每消耗 1 加仑汽油可以开 30 英里。她开 1 英里的收入是 0.5 美元，并且她的唯一支出是汽油支出，为每加仑 2 美元，在扣除掉此支出后，她的单位时间净收入是每小时多少美元？
(A) 20 (B) 22 (C) 24 (D) 25 (E) 26

英文真题

2020 AMC 12A Problems

Problem 1

Carlos took 70% of a whole pie. Maria took one third of the remainder. What portion of the whole pie was left?

(A) 10%

(B) 15%

(C) 20%

(D) 30%

(E) 35%

Problem 2

The acronym AMC is shown in the rectangular grid below with grid lines spaced 1 unit apart. In units, what is the sum of the lengths of the line segments that form the acronym AMC?

(A) 17

(B) $15 + 2\sqrt{2}$

(C) $13 + 4\sqrt{2}$

(D) $11 + 6\sqrt{2}$

(E) 21

Problem 3

A driver travels for 2 hours at 60 miles per hour, during which her car gets 30 miles per gallon of gasoline. She is paid $0.50 per mile, and her only expense is gasoline at $2.00 per gallon. What is her net rate of pay, in dollars per hour, after this expense?

(A) 20

(B) 22

(C) 24

(D) 25

(E) 26

Problem 4

How many 4-digit positive integers (that is, integers between 1000 and 9999, inclusive) having only even digits are divisible by 5?

(A) 80

(B) 100

(C) 125

(D) 200

(E) 500

Problem 5

The 25 integers from -10 to 14, inclusive, can be arranged to form a 5-by-5 square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum?

(A) 2

(B) 5

(C) 10

(D) 25

(E) 50

Problem 6

In the plane figure shown below, 3 of the unit squares have been shaded. What is the least number of additional unit squares that must be shaded so that the resulting figure has two lines of symmetry?

(A) 4

(B) 5

(C) 6

(D) 7

(E) 8

真题文字详解(英文)

Problem 1

Carlos took 70% of a whole pie. Maria took one third of the remainder. What portion of the whole pie was left?

(A) 10%

(B) 15%

(C) 20%

(D) 30%

(E) 35%

Solution 1

If Carlos took 70% of the pie, there must be $(100 - 70)\% = 30\%$ left. After Maria takes $\frac{1}{3}$ of the remaining $30\%, 1 - \frac{1}{3} = \frac{2}{3}$ of the remaining $30\%$ is left.

Therefore, the answer is $30\% \cdot \frac{2}{3} = \boxed{(C) 20\%}$.

Solution 2

Like solution 1, it is clear that there is $30\%$ of the pie remaining. Since Maria takes $\frac{1}{3}$ of the remainder, she takes $\frac{1}{3} \cdot 30\% = 10\%$, meaning that there is $30\% - 10\% = \boxed{(C) 20\%}$ left.

Solution 3 (One Sentence)

We have $(100\% - 70\%) \cdot \left(1 - \frac{1}{3}\right) = 30\% \cdot \frac{2}{3} = \boxed{(C) 20\%}$ of the whole pie left.

Problem 2

The acronym AMC is shown in the rectangular grid below with grid lines spaced 1 unit apart. In units, what is the sum of the lengths of the line segments that form the acronym AMC?

(A) 17

(B) $15 + 2\sqrt{2}$

(C) $13 + 4\sqrt{2}$

(D) $11 + 6\sqrt{2}$

(E) 21

Solution 1

Each of the straight line segments has length 1 and each of the slanted line segments have length $\sqrt{2}$ (this can be deducted using $45 - 45 - 90$, pythag, trig, or just sense).

There are a total of 13 straight lines segments and 4 slanted line segments. The sum is $\boxed{(C) 13 + 4\sqrt{2}}$

Solution 2

Either count the straight or diagonals and deduce from the answers that the only answer possible is $\boxed{(C) 13 + 4\sqrt{2}}$.

Problem 3

The following problem is from both the 2020 AMC 12A #3 and 2020 AMC 10A #4, so both problems redirect to this page.

A driver travels for 2 hours at 60 miles per hour, during which her car gets 30 miles per gallon of gasoline. She is paid $0.50 per mile, and her only expense is gasoline at $2.00 per gallon. What is her net rate of pay, in dollars per hour, after this expense?

真题文字详解(中英双语)

Problem 1 Carlos took 70% of a whole pie. Maria took one third of the remainder. What portion of the whole pie was left?
(A) 10% (B) 15% (C) 20% (D) 30% (E) 35%

Solution 1 If Carlos took 70% of the pie, there must be $(100 - 70)\% = 30\%$ left. After Maria takes $\frac{1}{3}$ of the remaining $30\%, 1 - \frac{1}{3} = \frac{2}{3}$ of the remaining $30\%$ is left. Therefore, the answer is $30\% \cdot \frac{2}{3} = \boxed{\text{(C) } 20\%}$.

Solution 2 Like solution 1, it is clear that there is $30\%$ of the pie remaining. Since Maria takes $\frac{1}{3}$ of the remainder, she takes $\frac{1}{3} \cdot 30\% = 10\%$, meaning that there is $30\% - 10\% = \boxed{\text{(C) } 20\%}$ left.

Solution 3 (One Sentence) We have $(100\% - 70\%) \cdot \left(1 - \frac{1}{3}\right) = 30\% \cdot \frac{2}{3} = \boxed{\text{(C) } 20\%}$ of the whole pie left.

Problem 2 The acronym AMC is shown in the rectangular grid below with grid lines spaced 1 unit apart. In units, what is the sum of the lengths of the line segments that form the acronym AMC?

(A) 17 (B) $15 + 2\sqrt{2}$ (C) $13 + 4\sqrt{2}$ (D) $11 + 6\sqrt{2}$ (E) 21

Solution 1 Each of the straight line segments has length 1 and each of the slanted line segments has length $\sqrt{2}$ (this can be deducted using $45-45-90$, pythag, trig, or just sense). There are a total of 13 straight line segments and 4 slanted line segments. The sum is $\boxed{\text{(C) } 13 + 4\sqrt{2}}$.

Solution 2 Either count the straight or diagonals and deduce from the answers that the only answer possible is $\boxed{\text{(C) } 13 + 4\sqrt{2}}$.

Problem 3 The following problem is from both the 2020 AMC 12A #3 and 2020 AMC 10A #4, so both problems redirect to this page. A driver travels for 2 hours at 60 miles per hour, during which her car gets 30 miles per gallon of gasoline. She is paid $0.50 per mile, and her only expense is gasoline at $2.00 per gallon. What is her net rate of pay, in dollars per hour, after this expense?
(A) 20 (B) 22 (C) 24 (D) 25 (E) 26