https://www.scasd.org/cms/lib/PA01000006/Centricity/Domain/1599/M1C3%20Study%20Packet%202020.pdf
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Problem 1
Answer: A
Solution:
You can just do 289/20 and round your answer to get 14.
中文解析:
比例是1:20 ,真实建筑是289feet,因此复制品的高度是: 289/20约等于14feet. 答案是A.
Problem 2
Answer: D
Solution:
By adding up the numbers in each of the 6 parentheses, we have:(2/1)*(3/2)*(4/3)*(5/4)*(6/5)*(7/6) . Using telescoping, most of the terms cancel out diagonally. We are left with 7 .
中文解析:
=(2/1)*(3/2)*(4/3)*(5/4)*(6/5)*(7/6)=7/1=7. 答案是D.
Problem 3
Answer: D
Solution:
The five numbers which cause people to leave the circle are 7, 14, 17, 21,27. The most straightforward way to do this would be to draw out the circle with the people, and cross off people as you count. The last one standing is Dan.
中文解析:
依次离开圆圈的数字分别是7, 14, 17, 21, 27,… 直接把A, B, C, D, E 按圆形排列, 依次数, 划掉出局的, 最后剩下的是D. 答案是D.
Problem 4
Answer: C
Solution:
We count 3*3=9 unit squares in the middle, and 4 small triangles each with an area of 1. Thus, the answer is 9+4=13.
中文解析:
这个12边形围起来的小正方形共有9个,其面积是9*1=9. 小三角形有8个,其面积是8*0.5=4. 因此面积一共是9+4=13. 答案是C.
Problem 5
Answer: E
Solution:
We can rewrite the given expression as
1+(3-2)+(5-4)+…+(2017-2016)+(2019-2018)=1+1+1+…+1.
The number of 1s is the same as the number of terms in 1,3,5,7, ……2017, 2019. Thus the answer is 1010.
中文解析:
把上式按下面的规则用小括号括起来:
=1+(3-2)+(5-4)+(7-6)+…+(2017-2016)+(2019-2018)=1+1+1+1+…+1+1.
除了第一个1外,还有: 2018/2=1009个1. 因此和是1010. 答案是E.
Problem 6
Answer: C
Solution:
Since Anh spends half an hour to drive 10 miles on the coastal road, his speed is 10/0.5=20 mph. His speed on the highway then is 20*3=60 mph. He drives 50 miles, so he drives for 5/6 hours, which is equal to 50 minutes. The total amount of minutes spent on his trip is 30+50=80 minutes.
中文解析:
海滨道路上花费的时间是30分钟(0.5小时),路程是10英里,因此速度是: 10/0.5=20 . 则高速路的速度是3*20=60 mph. 高速路的距离是50mile, 则时间是50/60=5/6 小时=50分钟. 答案是A.
Problem 7
Answer: B
Solution:
We use the property that the digits of a number must sum to a multiple of 9 if it are divisible by 9. This means u=7. Since we are looking for the remainder when divided by 8, we can ignore the thousands. The remainder when 187 is divided by 8 is 3.
中文解析:
因为这个数能被9整除,因此 2+0+1+8+U 是9的倍数,因此u=7. 20187除以8的余数 只需要看后三位187除以8的余数. 余数是3. 答案是B.
Problem 8
Answer: C
Solution:
The mean, or average number of days is the total number of days divided by the number of students. The total number of days is (1*1+2*3+3*2+4*6+5*8+6*4+7*2)=109. The total number of students is 1+3+2+6+8+3+2=25. Hence, 109/25=4.36.
中文解析:
有1个学生锻炼了1天, 3个学生锻炼了2天, 2个学生锻炼了3天, ….
平均值是: (1+3*2+2*3+6*4+8*5+3*6+2*7)/(1+3+2+6+8+3+2)=109/25=4.36 答案是C.
Problem 9
Answer: B
Solution:
He will place （12*2）+（14*2）=52 tiles around the border. For the inner part of the room, we have 10*14=140 square feet. Each tile takes up 4 square feet, so he will use 140/4=35 tiles for the inner part of the room. Thus, the answer is 52+35=87.
中文解析:
1-foot 的正方形需要: 16*2+10*2=52块.
中间剩下的面积是: 14*10=140. 2-foot的瓷砖的面积是2*2=4. 需要140/4=35块瓷砖.
共需要: 52+35=87块瓷砖. 答案是B.
Problem 10
Answer: C
Solution:
The sum of the reciprocals is 1/1 + 1/2 +1/4=7/4. Their average is 7/12. Taking the reciprocal of this gives 12/7.
中文解析:
harmonic mean是先计算这组数的倒数的算数平均值,然后再求倒数. 1/1, 1/2, 1/4 的算数平均值是: 7/12. 其倒数是12/7. 因此harmonic mean是12/7. 答案是C.
Problem 11
Answer: C
Solution:
The total number of ways is C(6,2)=15, if we treat Abby and Bridget as a pair and distinguishable and forget the others. The total number of ways they are adjacent = 4 (for the rows) + 3 (for the columns) Therefore, P(Abby and Bridget sitting adjacent) is 7/15.
中文解析:
第一步安排A的座位, 第二步安排B的座位.
A的座位可能位于4个角落之一,也可能位于两个中间位置之一. ,分两种情况.
第一种, A的座位位于4个角落之一的可能性是4/6, 下一步安排B的座位, B和A相邻的可能性有: 2/5.
第二种, A的座位唯一两个中间位置之一的可能性是2/6, 下一步安排B的座位, B和A相邻的可能性有: 3/5.
综合起来: 4/6*2/5 + 2/6 * 3/5 =7/15. 答案是C.
Problem 12
Answer: B
Solution:
We see that every 35 minutes the clock passes, the watch passes 30 minutes. That means that the clock is 7/6 as fast the watch, so when the car clock passes 7 hours, the watch has passed 6 hours, meaning that the time would be 6:00.
中文解析:
当实际时间过了30分钟时(从12:00 到12: 30), 她的汽车表的时间过了35分钟. 其比值时30:35=6:7. 当汽车表显示7个小时时, 根据比例关系,实际时间应该是6点. 答案是B.
Problem 13
Answer: A
Solution:
Say Lalia gets a value of x on her first 4 tests, and a value of y on her last test. Thus, 4x+y=410.
The value y has to be greater than 82, because otherwise she would receive the same score on her last test. Additionally, the greatest value for y is 98 (as y=100 would give x as a decimal), so therefore the greatest value y can be is 98. As a result, only 4 numbers work, 86, 90, 94, 98.
中文解析:
假设前4次考试的分数都是x, 最后一次考试的分数是y, 则4x+y=82*5. 即4x=410-y.
其中x,y都是正整数, 82<y<=100. 并且410-y必须是4的倍数.
y最大可以是98, 因此每次减4,y依次可以是98, 94, 90, 86. 共4种可能性. 答案是A.
Problem 14
Answer: D
Solution:
120=2*2*2*3*5. For the greatest five-digit number , 120=8*5*3*1*1. Thus, n is 85311. The sum of each digits is 8+5+3+1+1=18.
中文解析:
120=2*2*2*3*5. 为了这个5位数最大,首先让万位数上的数字最大, 120可以写成8*5*3*1*1. 因此这个5位数是85311. 其各位数字之和是: 8+5+3+1+1=18. 答案是D.
Problem 15
Answer: D
Solution:
Let the radius of the large circle be R. Then the radii of the smaller circles are R/2. The areas of the circles are directly proportional to the square of the radii, so the ratio of the area of the small circle to the large one is 1/4. This means the combined area of the 2 smaller circles is half of the larger circle, and therefore the shaded region is equal to the combined area of the 2 smaller circles, which is 1.
中文解析:
两个小圆面积合起来是1, 那么一个小圆的面积是0.5. 大圆的半径是小圆的半径的2倍,因此大圆的面积是小圆面积的4倍,大圆的面积是0.5*4=2. 阴影部分的面积是: 2-1=1. 答案是D.
Problem 16
Answer: C
Solution:
Since the Arabic books and Spanish books have to be kept together, we can treat them both as just one book. That means we’re trying to find the number of ways you can arrange one Arabic book, one Spanish book, and three German books, which is just 5 factorial. Now we multiply this product by 2! because there are 2 factorial ways to arrange just the Arabic books, and 4! ways to arrange just the Spanish books. Multiplying all these together, we have 5!*2!*4!=5760.
中文解析：
用捆绑法， 把2本Arabic 书看做一个整体， 4本Spanish书看做一个整体。这样Arabic， Spanish, 以及3本German书做排列，是5！。此外，2本Arabic内部有2！个排列方式， 4本Spanish书有4！个排列方式。因此本题的排列方法数是：5!*2!*4!=5760. 答案是C。
Problem 17
Answer: A
Solution:
Since Ella rides 5 times as fast as Bella, Ella rides at a rate of 25/2. Together, they move 15 feet towards each other every unit. Dividing 10560 by 15 to find the number of steps Bella takes results in the answer of 704.
中文解析：
Ella的速度是Bella的5倍，Bella每步走5/2 feet, Ella虽然骑自行车，我们可以理解为Ella每步走25/2 feet. 假设走x步两人相遇，因此有等式: 5/2 *x +25/2 * x =10560. 求得x=704. 答案是A。
Problem 18
Answer: E
Solution:
We can first find the prime factorization of 23232, which is 2^6 * 3^1 * 11^2 . Now, we just add one to our powers and multiply. Therefore, the answer is (6+1)*(1+1)*(2+1)=7*2*3=42.
中文解析：
写出23，232的质因数序列：23232=2*2*2*2*2*2*3*11*11. 因此她的因子的个数是: (6+1)*(1+1)*(2+1)=42. 答案是E。
Problem 19
Answer: C
Solution:
The sign of the next row on the pyramid depends on previous row. There are two options for the previous row, – or +. There are three rows to the pyramid that depend on what the top row is. Therefore, the ways you can make a + on the top is 2*2*2=8, so the answer is 8
中文解析：
第一行（最上面的一行）是+， 则第二行可以有两种方案：两个+ 或两个-。第二行的两个+和两个-分别对应第三行两种符号安排， 即第三行有2*2=4种安排。第三行的每种符号安排，都对应第四行的两种安排，因此第四行有4*2=8种安排。答案是C。
Problem 20
Answer: A
Solution:
By similar triangles, we have ADE=1/9ABC. Similarly, we see that BEF=4/9ABC. Using this information, we get ACFE=5/9 ABC. Then, since ADE=1/9 ABC, it follows that the CDEF=4/9 ABC. Thus, the answer would be 4/9.
中文解析:
由于DE//BC， 且AE：AB=1：3， 因此三角形ADE和三角形ABC相似， 其面积的比值是 1：9.
有与EF//AC， 且BE：AB=2：3， 因此三角形BEF和三角形ABC相似，其面积的比值是4：9.
平行四边形CDEF的面积=三角形ABC的面积-三角形BEF的面积-三角形AED的面积，可以认为是9-1-4=5.
因此，题目所求四边形CDEF的面积和三角形ABC的面积的比值是：5：9. 答案是C。
Problem 21
Answer: E
Solution:
Looking at the values, we notice that 11-7=4, 9-5=4, 6-2=4. This means we are looking for a value that is four less than a multiple of 11, 9, 6. The least common multiple of (11,9,6) is 198. So the numbers that fulfill this can be written as 198k-4, where k is a positive integer. This value is only a three digit integer when k is 1,2,3,4,5, which gives 194, 392, 590, 788, 986 respectively. Thus we have 5 values, so our answer is 5.
中文解析：
假设这个数是n. n被6除的余数是2， 注意2+4=6；n被9除的余数是5，注意5+4=9；n被11除的余数是7， 注意7+4=11.
我们可以发现n+4 能够被6，9， 11整除。
则6，9，11的公倍数满足整除的要求。其最小公倍数是198， 三位数中的倍数有：198， 198*2， 198*3， 198*4, 198*5. 共5个数。这些公倍数-4之后即满足题目条件。因此答案是E。
Problem 22
Answer: B
Solution:
Let the area of CEF be x. Thus, the area of triangle ACD is 45+x and the area of the square is 2*(45+x)=90+2x.
CEF is similar with ABF with a 1:2 ratio, so the area of triangle ABF is 4x. Now consider trapezoid ABED. Its area is 45+4x. which is 3/4 the area of the square. We set up an equation in x: 45+4x=3/4 *(90+2x) . Solving, we get x=9. The area of square ABCD is 90+2x=90+2*9=108.
中文解析：
假设三角形CEF的面积是x. 则三角形ADC的面积是45+x, 正方形ABCD的面积是2*(45+x）,即90+2x。
三角形CEF和三角形ABF是相似三角形，由于CE：AB=1：2，因此其面积的比值是：1：4. 则三角形ABF的面积是4x.
三角形CEB是正方形ABCD的1/4. 因此面积是：1/4*(90+2x).
三角形CEB的面积+三角形ABF的面积+四边形AFED的面积=正方形ABCD的面积， 因此有等式: 1/4 *(90+2x)+4x+45=90+2x. 求得x=9. 因此正方形ABCD的面积是：90+2x=108. 答案是B。
Problem 23
Answer: D
Solution:
We can decide 2 adjacent points with 8 choices. The remaining point will have 6 choices. However, we have counted the case with 3 adjacent points twice, so we need to subtract this case once (8*6-8=40). The case with the 3 adjacent points has 8 arrangements(C(8,3)=56), so our answer is 40/56=5/7.
中文解析：
我们按照下面的额规则组成三角形：使用八边形的一条边，再任选一个其他的点（有6个点可选)。可以组成三角形的数量是：8*6=48. 但是相邻的三个点组成的三角形会被计入了2次. 因此需要48-8=40.
由八边形上任意三个点可以组成C（8，3）=56个三角形。因此题目所求的概率是：40/56=5/7. 答案是D。
Problem 24
Answer: C
Solution:
Note that EJCI is a rhombus by symmetry. Let the side length of the cube be s. By the Pythagorean theorem, EC=sqrt(3)*s and JI=sqrt(2)*s. Since the area of a rhombus is half the product of its diagonals, the area of the cross section is s*s*sqrt(6)/2. This gives R=sqrt(6)/2. Thus R*R=3/2.
中文解析：
假设正方体的边长是s。
注意到EJCI的四条边相等，是个菱形。其对角线分别是ＪＩ和CE。JI的长度等于BD的长度。BD=sqrt(2) *s。
在直角三角形ECA中， AC=sqrt(2)*s, AE =s, 根据勾股定理， AC=sqrt(3)*s.
根据菱形的面积公式: 1/2 * JI *AC =1/2 *sqrt(2)*s*sqrt(3)*s=1/2 sqrt(6) *s.
正方体的一个面的面积是s*s.
题目所求R*R=（1/4*6*s*s)/(s*s)=3/2. 答案是C。
Problem 25
Answer: E
Solution:
We compute 2^8+1=256+1=257. We’re all familiar with what 6*6*6 is, namely 216, which is too small. The smallest cube greater than it is 7*7*7. We notice that 2^18= （2^6）^3=64^3 , which therefore clearly will be the largest cube less than 2^18+1. So, the required number of cubes is 64-7+1=58.
中文解析：
2的8次方 +1 =256+1=257. 我们熟悉的6*6*6=216， 比257小， 7*7*7=343 满足条件比257大。
2的18次方可以写成2的6次方再3次方，即64的3次方。
题目所求的perfect cube 应该包含7的三次方，8的三次方，。。。。64的三次方。共有：64-7+1=58个。答案是E。