2008 AMC 8 考试题目
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Used with permission of the Mathematical Association of America.
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1.
Susan had to spend at the carnival. She spent dollars on food and twice as much on rides. How many dollars did she have left to spend?
Answer: B
Solution(s):
Susan spent 2 \cdot $12 = $24 on rides. This means she spent a total of $12 + $24 = $36 at the carnival.
This means that she has $50 - $36 = $14 left to spend.
Thus, the answer is B.
2.
The ten-letter code represents the ten digits in order. What 4-digit number is represented by the code word
Answer: A
Solution(s):
The letter is in the th position, so it would be the letter The letter is in the th position, so it would be the letter The letter is in the th position, so it would be the letter The letter is in the nd position, so it would be the letter
Therefore, when putting together the word we get
Thus, the answer is A.
3.
If February is a month that contains Friday the what day of the week is February
Answer: A
Solution(s):
Since the th is a Friday, we know that the th is also a Friday. The st is days before the Friday, making it a Sunday.
Thus, the answer is A.
4.
In the figure, the outer equilateral triangle has area the inner equilateral triangle has area and the three trapezoids are congruent. What is the area of one of the trapezoids?
Answer: C
Solution(s):
Since the area of the larger triangle is and the area of the smaller triangle is Thus, the area of the other trapezoids is Since the trapezoids have a combined area of each of their areas is
Thus, the answer is C.
5.
Barney Schwinn notices that the odometer on his bicycle reads a palindrome, because it reads the same forward and backward. After riding more hours that day and the next, he notices that the odometer shows another palindrome, What was his average speed in miles per hour?
Answer: E
Solution(s):
The total distance traveled is miles. He also travelled hours. Thus, the average speed is miles per hour.
Thus, the answer is E.
6.
In the figure, what is the ratio of the area of the colored squares to the area of the uncolored squares?
Answer: D
Solution(s):
The total area of the entire square is since it is a square. The area of the colored region is making the remaining area Thus, the ratio is
Thus, the answer is D.
7.
If what is
Answer: E
Solution(s):
means
means
Therefore,
Thus, the answer is E.
8.
Candy sales from the Boosters Club from January through April are shown. What were the average sales per month in dollars?
Answer: D
Solution(s):
The total sales are
The average sales is then
Thus, the answer is D.
9.
In Tycoon Tammy invested dollars for two years. During the first year her investment suffered a loss, but during the second year the remaining investment showed a gain. Over the two-year period, what was the change in Tammy's investment?
Answer: D
Solution(s):
The loss means the investment went from to
The gain means the investment went from to \$85 \cdot 1.2 = $102. The total investment went from to $102, making a gain of
Thus, the answer is D.
10.
The average age of the people in Room A is The average age of the people in Room B is If the two groups are combined, what is the average age of all the people?
Answer: D
Solution(s):
The sum of the ages in Room A is
The sum of the ages in Room B is
The total sum is
The average is therefore
Thus, the answer is D.
11.
Each of the students in the eighth grade at Lincoln Middle School has one dog or one cat or both a dog and a cat. Twenty students have a dog and students have a cat. How many students have both a dog and a cat?
Answer: A
Solution(s):
The number of people that have both animals is equal to the number of people that own a cat plus the number of people that own a dog minus the number of people that own either.
Therefore, the number of people who own both is
Thus, the answer is A.
12.
A ball is dropped from a height of meters. On its first bounce it rises to a height of meters. It keeps falling and bouncing to of the height it reached in the previous bounce. On which bounce will it not rise to a height of meters?
Answer: C
Solution(s):
Since the height of a bounce decreases by each bounce, the height of each bounce is
After bounces, the ball bounces which is greater than
After bounces, the ball bounces which is less than
Therefore, we are under after bounces.
Thus, the answer is C.
13.
Mr. Harman needs to know the combined weight in pounds of three boxes he wants to mail. However, the only available scale is not accurate for weights less than pounds or more than pounds. So the boxes are weighed in pairs in every possible way. The results are and pounds. What is the combined weight in pounds of the three boxes?
Answer: C
Solution(s):
Let the weights be We know Adding all of this yields This makes
Thus, the answer is C.
14.
Three three and three are placed in the nine spaces so that each row and column contain one of each letter. If is placed in the upper left corner, how many arrangements are possible?
Answer: C
Solution(s):
There are rows and columns to put B in, so there are places for it. After that, there is column and row available for C, so the number of combinations is
Thus, the answer is C.
15.
In Theresa's first basketball games, she scored and points. In her ninth game, she scored fewer than points and her points-per-game average for the nine games was an integer. Similarly in her tenth game, she scored fewer than points and her points-per-game average for the games was also an integer. What is the product of the number of points she scored in the ninth and tenth games?
Answer: B
Solution(s):
The sum of her first scores is Since the average of the first scores is an integer, the sum of the first scores is a multiple of
Since the score is less than the sum of the scores after games is between and and is a multiple of making the sum Thus, the score of the th game is
The sum of Theresa's first scores is Since the average of the first scores is an integer, the sum of the first scores is a multiple of
Since the score is less than the sum of the scores after games is between and and is a multiple of making the sum Thus, the score of the th game is
Therefore, their product is
Thus, the answer is B.
16.
A shape is created by joining seven unit cubes, as shown. What is the ratio of the volume in cubic units to the surface area in square units?
Answer: D
Solution(s):
There are unit cubes, so the volume is
If we look at the perspective of the inner cube, we can see that there is one cube connected to each side. Furthermore, there are exposed on each of the outer cubes. This makes the surface area
This makes the ratio of volume to surface area
Thus, the answer is D.
17.
Ms.Osborne asks each student in her class to draw a rectangle with integer side lengths and a perimeter of units. All of her students calculate the area of the rectangle they draw. What is the difference between the largest and smallest possible areas of the rectangles?
Answer: D
Solution(s):
Let the length and width be respectively. The rectangle would have an area of Thus, so The area is This makes the area largest when is as close as possible to and the area is the smallest when is as far from Therefore, the largest area is when and the smallest area is when
This makes the larger area equal to and the smaller area equal to
Thus, the difference is
Thus, the answer is D.
18.
Two circles that share the same center have radii meters and meters. An aardvark runs along the path shown, starting at and ending at How many meters does the aardvark run?
Answer: E
Solution(s):
The circumference of a circle is so going a quarter of the way around is
He goes a quarter of the way around the large circle, so this part is meters. He then goes from the larger circle to the smaller circle, which is meters.
He goes a quarter of the way around the smaller circle, so this part is meters. He then goes through the diameter of the smaller circle, which is meters.
He then goes a quarter of the way around the smaller circle, so this part is meters. He finally goes from the smaller circle to the larger circle, which is meters.
The total length is
Thus, the answer is E.
19.
Eight points are spaced around at intervals of one unit around a square, as shown. Two of the points are chosen at random. What is the probability that the two points are one unit apart?
Answer: B
Solution(s):
Each dot has dots that are one unit away from it.
Therefore, regardless of the choice of the first dot, of the other dots would be within one unit, so the probability is
Thus, the answer is B.
20.
The students in Mr. Neatkin's class took a penmanship test. Two-thirds of the boys and of the girls passed the test, and an equal number of boys and girls passed the test. What is the minimum possible number of students in the class?
Answer: B
Solution(s):
Since the number of boys who passed and the number of girls who passed are the same, we can assign them the same variable. Let this number be
The number of boys in the class is and the number of girls in the class is Thus, the total number of people is
The total number of people must be a multiple of the numerator of this fraction, so the number of people must be a multiple of making the number of people
Thus, the answer is B.
21.
Jerry cuts a wedge from a -cm cylinder of bologna as shown by the dashed curve. Which answer choice is closest to the volume of his wedge in cubic centimeters?
Answer: C
Solution(s):
The total volume is where is the area of the base and is the height.
To find we take the area of the circle. It has a diameter of cm, so it has a radius of area cm. Thus, the area is Since cm, the volume of the whole thing is
Since the wedge is half of the volume of the cylinder, the volume of the wedge is
To estimate this, we can find that so
Thus, the answer is C.
22.
For how many positive integer values of are both and three-digit whole numbers?
Answer: A
Solution(s):
Let We know and are both three digit numbers.
Thus, so Every integer in this range has that creates a valid so there are valid numbers.
Thus, the answer is A.
23.
In square and What is the ratio of the area of to the area of square
Answer: C
Solution(s):
Let the side length be Note that the total area is
Since we know
Since we know
This makes the area of the area of and the area of Thus, the area of
Thus, the answer is C.
24.
Ten tiles numbered through are turned face down. One tile is turned up at random, and a die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square?
Answer: C
Solution(s):
If the rolled number was then the tile can be yielding combinations.
If the rolled number was then the tile can be yielding combinations.
If the rolled number was then the tile can be yielding combination.
If the rolled number was then the tile can be yielding combinations.
If the rolled number was then the tile can be yielding combination.
If the rolled number was then the tile can be yielding combination.
The total number of combinations is There are combinations each with equal likelihood, so the probability is
Thus, the answer is C.
25.
Margie's winning art design is shown. The smallest circle has radius 2 inches, with each successive circle's radius increasing by 2 inches. Which of the following is closest to the percent of the design that is dark-colored?
Answer: A
Solution(s):
The largest circle is of radius so the entire design has area of
Each dark region is the area of its circle minus the area of the previous circle. The largest dark area would be of area the next area would be and the smallest area would be Therefore, the combined dark area would be
This fraction would be which is approximately
Thus, the answer is A.