2015 AMC amc10b 真题 答案 详解

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中英双语真题

2015 AMC 10B

Problem 1
What is the value of (2 - (-2)^{-2})?
(2-(-2)^{-2}) 的值是多少?
(A) (-2) (B) (\frac{1}{16}) (C) (\frac{7}{4}) (D) (\frac{9}{4}) (E) 6

Problem 2
Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. When does she finish the third task?
Marie 连续做了3项耗时相同的任务，中间没有休息。她从下午1:00开始做第一个任务，在下午2:40完成第二个任务。她何时完成第三个任务？
(A) 3:10 PM (B) 3:30 PM (C) 4:00 PM (D) 4:10 PM (E) 4:30 PM

Problem 3
Isaac has written down one integer two times and another integer three times. The sum of the five numbers is 100, and one of the numbers is 28. What is the other number?
Isaac 把一个整数写了2遍，另一个整数写了3遍，这5个数之和为100.其中一个数是28，另一个数是多少?
(A) 8 (B) 11 (C) 14 (D) 15 (E) 18

英文真题

2015 AMC 10B Problems

Problem 1
What is the value of $2-(-2)^{-2}$?
(A) -2 (B) $\frac{1}{16} \quad$ (C) $\frac{7}{4} \quad$ (D) $\frac{9}{4} \quad$ (E) 6

Problem 2
Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. When does she finish the third task?
(A) 3:10 PM (B) 3:30 PM (C) 4:00 PM (D) 4:10 PM (E) 4:30 PM

Problem 3
Isaac has written down one integer two times and another integer three times. The sum of the five numbers is 100, and one of the numbers is 28. What is the other number?
(A) 8 (B) 11 (C) 14 (D) 15 (E) 18

Problem 4
Four siblings ordered an extra large pizza. Alex ate $\frac{1}{5}$, Beth $\frac{1}{3}$, and Cyril $\frac{1}{4}$ of the pizza. Dan got the leftovers. What is the sequence of the siblings in decreasing order of the part of pizza they consumed?
(A) Alex, Beth, Cyril, Dan (B) Beth, Cyril, Alex, Dan (C) Beth, Cyril, Dan, Alex (D) Beth, Dan, Cyril, Alex (E) Dan, Beth, Cyril, Alex

Problem 5
David, Hikmet, Jack, Marta, Rand, and Todd were in a 12-person race with 6 other people. Rand finished 6 places ahead of Hikmet. Marta finished 1 place behind Jack. David finished 2 places behind Hikmet. Jack finished 2 places behind Todd. Todd finished 1 place behind Rand. Marta finished in 6th place. Who finished in 8th place?
(A) David (B) Hikmet (C) Jack (D) Rand (E) Todd

Problem 6
Marley practices exactly one sport each day of the week. She runs three days a week but never on two consecutive days. On Monday she plays basketball and two days later golf. She swims and plays tennis, but she never plays tennis the day after running or swimming. Which day of the week does Marley swim?
(A) Sunday (B) Tuesday (C) Thursday (D) Friday (E) Saturday

Problem 7
Consider the operation "minus the reciprocal of," defined by $a \diamond b = a-\frac{1}{b}$. What is $((1 \diamond 2) \diamond 3)-(1 \diamond(2 \diamond 3))$?
(A) $-\frac{7}{30}$ (B) $-\frac{1}{6}$ (C) 0 (D) $\frac{1}{6}$ (E) $\frac{7}{30}$

Problem 8
The letter F shown below is rotated 90° clockwise around the origin, then reflected in the y-axis, and then rotated a half turn around the origin. What is the final image?

真题文字详解(英文)

Problem 1 What is the value of $2-\left(-2\right)^{-2}$?
(A) -2 (B) $\frac{1}{16}$ (C) $\frac{7}{4}$ (D) $\frac{9}{4}$ (E) 6

Solution
$2-\left(-2\right)^{-2}=2-\frac{1}{\left(-2\right)^{2}}=2-\frac{1}{4}=\frac{8}{4}-\frac{1}{4}=\boxed{\text{(C)} \frac{7}{4}}$

Problem 2
Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. When does she finish the third task?
(A) 3:10 PM (B) 3:30 PM (C) 4:00 PM (D) 4:10 PM (E) 4:30 PM

Solution
Marie finishes 2 tasks in 1 hour and 40 minutes. Therefore, one task should take 50 minutes to finish. 50 minutes after 2:40 PM is 3:30 PM, so our answer is $\boxed{\text{(B) }3:30 \text{ PM}}$.

Problem 3
Isaac has written down one integer two times and another integer three times. The sum of the five numbers is 100, and one of the numbers is 28. What is the other number?
(A) 8 (B) 11 (C) 14 (D) 15 (E) 18

Solution
Let the first number be $x$ and the second be $y$. We have $2x+3y=100$. We are given one of the numbers is 28. If $x$ were to be 28, $y$ would not be an integer, thus $y=28. 2x+3(28)=100$, which gives $x=\boxed{\text{(A) }8}$.

Problem 4
Four siblings ordered an extra large pizza. Alex ate $\frac{1}{5}$, Beth $\frac{1}{3}$, and Cyril $\frac{1}{4}$ of the pizza. Dan got the leftovers. What is the sequence of the siblings in decreasing order of the part of the pizza they consumed?
(A) Alex, Beth, Cyril, Dan
(B) Beth, Cyril, Alex, Dan
(C) Beth, Cyril, Dan, Alex
(D) Beth, Dan, Cyril, Alex
(E) Dan, Beth, Cyril, Alex

Solution
Let the pizza have 60 slices, since the least common multiple of $(5,3,4)=60$. Therefore, Alex ate $\frac{1}{5}\times 60=12$ slices, Beth ate $\frac{1}{3}\times 60=20$ slices, and Cyril ate $\frac{1}{4}\times 60=15$ slices. Dan must have eaten $60-(12+20+15)=13$ slices. In decreasing order, we see the answer is $\boxed{\text{(C) }Beth, Cyril, Dan, Alex}$.

Problem 5

真题文字详解(中英双语)

Problem 1 What is the value of $2-\left(-2\right)^{-2}$?
$2-\left(-2\right)^{-2}$ 的值是多少？
(A) -2 (B) $\frac{1}{16}$ (C) $\frac{7}{4}$ (D) $\frac{9}{4}$ (E) 6

Solution
$2-\left(-2\right)^{-2}=2-\frac{1}{\left(-2\right)^{2}}=2-\frac{1}{4}=\frac{8}{4}-\frac{1}{4}=\boxed{\text{(C)} \frac{7}{4}}$

Problem 2
Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at 1:00 PM and finishes the second task at 2:40 PM. When does she finish the third task?
玛丽连续做了三个耗时相等的工作，没有休息。她开始第一个任务的时间是下午1:00，完成第二个任务的时间是下午2:40。她什么时候完成第三个任务？
(A) 3:10 PM (B) 3:30 PM (C) 4:00 PM (D) 4:10 PM (E) 4:30 PM

Solution
Marie finishes 2 tasks in 1 hour and 40 minutes. Therefore, one task should take 50 minutes to finish. 50 minutes after 2:40 PM is 3:30 PM, so our answer is $\boxed{\text{(B) }3:30 \text{ PM }}$.

Problem 3
Isaac has written down one integer two times and another integer three times. The sum of the five numbers is 100, and one of the numbers is 28. What is the other number?
艾萨克写下了一个整数两次，另一个整数三次。这五个数的和是100，其中一个数是28。另一个数是多少？
(A) 8 (B) 11 (C) 14 (D) 15 (E) 18

Solution
Let the first number be $x$ and the second be $y$. We have $2x+3y=100$. We are given one of the numbers is 28. If $x$ were to be 28, $y$ would not be an integer, thus $y=28. 2x+3(28)=100$, which gives $x=\boxed{\text{(A) }8}$.

Problem 4
Four siblings ordered an extra large pizza. Alex ate $\frac{1}{5}$, Beth $\frac{1}{3}$, and Cyril $\frac{1}{4}$ of the pizza. Dan got the leftovers. What is the sequence of the siblings in decreasing order of the part of the pizza they consumed?
四个兄弟姐妹点了一份超大披萨。Alex吃了$\frac{1}{5}$，Beth吃了$\frac{1}{3}$，Cyril吃了$\frac{1}{4}$。Dan吃剩下的部分。他们按吃的披萨数量从多到少的顺序是什么？
(A) Alex, Beth, Cyril, Dan
(B) Beth, Cyril, Alex, Dan
(C) Beth, Cyril, Dan, Alex
(D) Beth, Dan, Cyril, Alex
(E) Dan, Beth, Cyril, Alex