当前位置：首页 → 要闻 → 正文

AMC8--2010年真题解析（英文解析+中文解析）

时间:2020-05-10 20:12:40 阅读：2498+ 来源：自媒体 编辑：艾蕾特教育

导读：原标题：AMC8 -- 2010年真题解析（英文解析+中文解析）Problem 1Answer: C Solution: Given that these are the only math teach

原标题：AMC8 -- 2010年真题解析（英文解析+中文解析）

Problem 1

Answer: C

Solution:

Given that these are the only math teachers at Euclid Middle School and we are told how many from each class are taking the AMC 8, we simply add the three numbers to find the total. 11+8+9=28.

中文解析:

参与比赛的学生总人数是： 11+8+9=28. 答案是C。

Answer: D

Solution:

Substitute a=5, b=10 into the expression for a@b to get: 5@10=(5*10)/(5+10)=50/15=10/3.

中文解析：

（5*10）/（5+10）=50/15=10/3. 答案是D。

Answer: C

Solution:

The highest price was in Month 1, which was $17. The lowest price was in Month 3, which was $10. 17 is 17/10 =170% of 10, and is 170-100=70% more than 10. Therefore, the answer is 70.

中文解析：

最高价是1月，17美元。最低价格是3月10美元。最高价比最低价多：（17-10）/10=70%。答案是C。

Answer: C

Solution:

Putting the numbers in numerical order we get the list 0,0,1,2,3,3,3,4 The mode is 3, The median is (2+3)/2=2.5. The average is 2. is The sum of all three is 3+2.5+2=7.5.

中文解析：

这组数依照从小到大的次序排列是： 0，0，1，2，3，3，3，4. 中位数Median是2.5； mode是3， mean是16/8=2. 因而mean,median,mode的和是: 2.5+3+2=7.5. 答案是C。

Answer: B

Solution:

Convert everything to the same unit. The ceiling is 240 centimeters above the floor. The combined height of Alice and the light bulb when she reaches for it is 10+150+46=206 centimeters. That means the stool's height needs to be 240-206=34.

中文解析：

房高240cm. 灯离天花板10cm, Alice身高150cm, 手臂能伸高46cm. 因而还需求： 240-10-150-46=34cm. 答案是B。

Answer: E

Solution:

An equilateral triangle has 3 lines of symmetry. A non-square rhombus has 2 lines of symmetry. A non-square rectangle has 2 lines of symmetry. An isosceles trapezoid has 1 line of symmetry. A square has 4 lines of symmetry. Therefore, the answer is E.

中文解析：

选项A的对称线有3条；选项B ,菱形的对称线有2条；选项C长方形的对称线有2条；选项D等腰三角形的对称线有1条。选项E正方形的对称线有4条。答案是E。

Answer: B

Solution:

You need 2 dimes, 1 nickel, and 4 pennies for the first 25 cents. From 26 cents to 50 cents, you only need to add 1 quarter. From 51 cents to 75 cents, you also only need to add 1 quarter. The same for 76 cents to 99 cents. Thus, the correct answer is 3 quarters +2 dimes +1 nickel +4 pennies.

中文解析：

25分以内的恣意钱数，能够正常的运用 2个1角, 1个5分，4个1分来组成。从26分到50分，只需求在此基础上添加1个25分的硬币即可。添加到75分，需求再添加一个25分硬币；添加到1元，需求再添加一个25分。因而若想表明小于1元的钱数，能够正常的运用3个25分， 2个1角，1个1分，4个1分硬币。共需求10个硬币。答案是B。

Answer: D

Solution:

Because they are both moving in the same direction, Emily is riding relative to Emerson 12-8=4 mph. Now we can look at it as if Emerson is not moving at all [on his skateboard] and Emily is riding at 4 mph. It takes her (1/2) / 4= 1/8 hour to ride the 1/2 mile to reach him, and then the same amount of time to be 1/2 mile ahead of him. This totals to 1/8+1/8=1/4 hour =15 minutes.

中文解析：

两人是相同的方向，一个速度是12，一个速度是8，其相对速度是4，因而咱们咱们能够以为Emily的速度是4， Emerson不动。当Emerson在Emily前面1/2mile的间隔时， Emily需求(1/2)/4=1/8 小时才干抵达Emerson处。然后Emily超过了Emerson，当两人的间隔还在1/2mile内时，Emily能单到Emerson。这个1/2 英里，相同需求Emily1/8小时。因而从Emerson在Emily的前方1/2英里，到Emily超出Emerson1/2英里，共需求1/8+1/8小时，即15分钟。答案是D。

Problem 9

Answer: D

Solution:

Ryan answered 0.8*25=20 problems correct on the first test, 0.9*40=36 on the second, and 0.7*10=7 on the third. This amounts to a total of 20+36+7=63 problems correct. The total number of problems is 25+40+10=75. Therefore, the percentage is 63/75=84%.

中文解析：

25题对80%，即20题； 40题对90%，即36题； 10题对70%，即7道。一共答对了20+36+7=63题，总标题数是： 25+40+10=75题。正确率是： 84%。答案是D。

Problem 10

Answer: B

Solution:

The pepperoni circles' diameter is 2, since 12/6=2. From that we see that the area of the 24 circles of pepperoni is 24* Pi*1*1=24Pi. The large pizza's area is Pi*6*6. Therefore, the ratio is 24Pi/36Pi=2/3.

中文解析：

大圆的直径是12，因而小圆的直径是2. 半径是1，小圆的面积是Pi. 24个小圆的面积是24Pi。大圆的半径是6，大圆的面积是36Pi。因而标题所求的比值是： 24Pi/36Pi=2/3. 答案是B。

Problem 11

Answer: B

Solution:

Let the height of the taller tree be h and let the height of the smaller tree be h-16. Since the ratio of the smaller tree to the larger tree is 3/4, we have (h-16): h=3:4; . Solving for h gives us k64.

中文解析：

假定这两棵树的高度分别是3x和4x. 则较高的树比较低的树高x. 标题奉告高16英尺，因而x=16. 较高的那棵树的高度是4*16=64英尺。答案是B。

Problem 12

Answer: D

Solution:

Since 80 percent of the 500 balls are red, there are 400 red balls. Therefore, there must be 100 blue balls. For the 100 blue balls to be 25% or 1/4 of the bag, there must be 400 balls in the bag so 100 red balls must be removed. The answer is 100.

中文解析：

本来有500*80%=400颗赤色的球。假定拿走x颗红球后，剩余的小球是500-x个。其间有75%的红球，即（500-x）*75% 颗赤色的球。从另一个视点，本来400颗赤色的球，拿走x颗赤色后，剩余赤色的球是： 400-x. 因而咱们得到等式：（500-x）*75%=400-x. 求得x=100. 拿走的红球数是100个。答案是D。

Problem 13

Answer: E

Solution:

Let n, n+1, n+2 be the lengths of the sides of the triangle. Then the perimeter of the triangle is n+n+1+n+2=3n+3. Using the fact that the length of the smallest side is 30% of the perimeter, it follows that: n=0.3*(3n+3) => n=9. The longest side is then n+2=11.

中文解析：

假定三条边长分别是n, n+1,n+2. 则其周长是3n+3. 最对的边是周长的30%。即n=(3n+3)*30%. 求得：n=9. 最长的边是n+2,即11. 答案是E。

Problem 14

Answer: C

Solution:

First, we must find the prime factorization of 2010. 2010=2*3*5*67. We add the factors up to get 77.

中文解析：

2010的质因数序列是： 2*3*5*67. 其质因数的和是77. 答案是C。

Problem 15

Answer: C

Solution:

We do 100-30-20-15-10 to find the percent of gumdrops that are green. We find that 25% of the gumdrops are green. That means there are 120 gumdrops. If we replace half of the blue gumdrops with brown gumdrops, then 15% of the jar's gumdrops are brown. 35%*120=42.

中文解析：

30%是蓝色， 20%是棕色，15%是赤色，10%是黄色，剩余25%（1-30%-20%-15%-10%）是绿色。绿色是30个，因而总数是30/25%=120个。蓝色有120*30%=36个，棕色有120*20%=24个。把一半的蓝色换成棕色，则棕色是：24+18=42个。答案是C。

Problem 16

Answer: B

Solution:

Let the side length of the square be s, and let the radius of the circle be r. Thus we have s*s=Pi*r*r. Dividing each side by r*r, we get s*s/r*r= Pi. So, s/r=sqrt(Pi).

中文解析：

假定正方形的边长是s，圆形的半径是r. 正方形的面积是s*s，圆形的面积是Pi*r*r. 其面积持平，咱们得到： s*s=Pi*r*r. 因而(s*s)/(r*r)=Pi. s/r =sqrt(Pi) . 答案是B。

Problem 17

Answer: D

Solution:

We see that half the area of the octagon is 5. We see that the triangle area is 5-1=4. That means that 5*h/2=4 => h=8/5. QY=8/5-1=3/5. Thus, XQ=2/5. The answer is 2/3.

中文解析：

整个图形的面积是：10*1=10. PQ把图形分成了上下面积持平的两部分，下半部分的面积是5. 小正方形YABC的面积是1，因而三角形PQA 的面积应该是5-1=4. 三角形PQA的底是5，因而高是：4*2/5=1.6. qizhong AY=1,因而QY=1.6-1=0.6. 因为XY=1，其间QY=0.6，因而XQ=0.4. 则XQ/QY=0.4/0.6=2/3. 答案是D。

Problem 18

Answer: C

Solution:

AD: AB=3:2, AB=30, so, AD=45.

We calculate the combined area of semicircle by putting together semicircle AB and CD to get a circle with radius 15. Thus, the area is 225Pi. The area of the rectangle is 30*45=1350. We calculate the ratio: 1350 : 225Pi=6 : PI

中文解析：

AD： AB=3:2， AB=30，因而AD=(30/2）*3=45. AB是半圆的直径，因而圆的半径是15. 两个半圆的面积合起来便是一个圆的面积，一个圆的面积是： Pi* 15*15=225Pi. 长方形ABCD的面积是：45*30. 因而标题所求的比值是：（45*30）: (225*Pi）,即6:Pi. 答案是C。

Problem 19

Answer: C

Solution:

Since triangle ACD is isosceles, CB bisects AD. Thus AB=BD=8. From the Pythagorean Theorem, CB=6.Thus the area between the two circles is 100Pi-36Pi=64Pi.

中文解析：

大圆的半径是AC，长度是10，大圆的面积是：Pi*10*10. 在直角三角形ABC中， AC=10， AB=16/2=8，依据勾股定理，可得CB=6. 因而小圆的半径是6，小圆的面积是Pi*6*6. 两个圆之间的面积是：100Pi-36Pi=64Pi. 答案是C。

Problem 20

Answer: A

Solution:

Let x be the number of people wearing both a hat and a glove. Since the number of people wearing a hat or a glove must be whole numbers, the number of people in the room must be a multiple of LCM(4,5)=20. Thus, there are 20*2/5=8 people wearing gloves, 20*3/4=15 people wearing hat. 8+15-x=20 =>x=3.

中文解析：

戴手套，戴帽子的人数都必须是整数，因而球5和4的最小公倍数，是20. 至少戴手套的是20*（2/5）=8人，戴帽子的是20*（3/4）=15人。在房间总人数是20人的前提下，戴手套的有8人，戴帽子的有15人，则既戴帽子又戴手套的有：8+15-20=3人。答案是A。

Problem 21

Answer: C

Solution:

Let x be the number of pages in the book. After the first day, Hui had 4x/5-12 pages left to read. After the second, she had 3/4*(4x/5-12)-15=3x/5-24 left. After the third, she had 2/3*(3x/5-24)-18=2x/5-34 left. This is equivalent to 62. 2x/5-34=62 thus x=240.

中文解析：

假定这本书有n页，第四天只剩余62页， 62+18=80， 80页是第三天没读书之前的页数的2/3，则第三天没读书之前，有120页。 120+15=135， 135页是第二天没读书之前的页数的3/4，则第二天没读书之前有180页。 180+12=192， 192页是第一天没读书之前的页数的4/5，则第一天没读书之前的页数是215页。即这本书本来有240页。答案是C。

Problem 22

Answer: E

Solution:

The result must hold for any three-digit number with hundreds digit being 2 more than the units digit. 301 is such a number. evaluating, we get 301-103=198. Thus, the units digit in the final result is 8.

中文解析：

假定这个3位数的个位数是n,则百位数是n+2, 这个三位数回转后，个位数是n+2. 原数-新数，其个位数是 n-(n+2）=-2，不行减，从十位数借位，效果个位数应该是8. 答案是E。

或许咱们假定这个三位数是321. 回转后是123. 321-123的个位数是8.

Problem 23

Answer: B

Solution:

By the Pythagorean Theorem, the radius of the larger circle turns out to be sqrt(1*1+1*1)=sqrt(2). Therefore, the area of the larger circle is 2Pi. Using the coordinate plane given, we find that the radius of each of the two semicircles to be 1. So, the area of the two semicircles is Pi*1. Finally, the ratio of the combined areas of the two semicircles to the area of circle O is 1/2.

中文解析：

由上图可知，半圆POQ的半径是1，在直角三角形QTO中， OT是半径也是1，TQ=1，依据勾股定理， QO是根号2. 即大圆的半径是根号2. 两个半径为1的半圆的面积合起来是Pi。大圆的面积是： 2Pi。标题所求的面积的比值是Pi: 2Pi=1/2. 答案是B。

Problem 24

Answer: A

Solution:

Since all of the exponents are multiples of four, we can simplify the problem by taking the fourth root of each number. evaluating we get 10^2=100, 5^3=125, 2^6=64. Since 64<100<125. it follows that 2^24<10^8<5^12.

中文解析：

10^8=(10^2)4. (10的8次方等于10的2次方，然后再4次方). 5^12=(5^3)^4. (5的12次方等于5的三次方，再4次方）。 2^24=（2^6）^4. (2的24次方等于2的6次方，然后再4次方）。这样咱们只需比较10^2， 5^3,2^6的巨细。 10的2次方是100， 5的3次方是125， 2的6次方是64. 巨细次序是64<100<125, 即 2^6 < 10^2 <5^3, 所以 2^24 < 10^8 <5^12. 答案是A。

Problem 25

Answer: E

Solution:

Use recursive approach. The number of ways to climb one stair is 1. There are 2 ways to climb two stairs: 1,1 or 2. For 3 stairs, there are 4 ways: (1,1,1),（1,2）,(2,1),(3). For four stairs, they could have hopped straight from the 1st, done a double from #2, or used a single step from #3. The ways to get to each of these steps are 1+2+4=7 ways to get to step 4. The pattern can then be extended: 4 steps: 1+2+4=7 ways. 5 steps: 2+4+7=13 ways. 6 steps:4+7+13=24 ways.

中文解析：