2010 AMC 8 Exam Solutions
Problems used with permission of the Mathematical Association of America. Scroll down to view solutions, print PDF solutions, view answer key, or:
1.
At Euclid Middle School the mathematics teachers are Mrs. Germain, Mr. Newton, and Mrs. Young. There are students in Mrs. Germain's class, students in Mr. Newton's class, and students in Mrs. Young's class taking the AMC this year. How many mathematics students at Euclid Middle School are taking the contest?
Solution(s):
There are students.
Therefore, the answer is C.
2.
If for positive integers, then what is
Solution(s):
Given our definition of we have
Thus, the answer is D.
3.
The graph shows the price of five gallons of gasoline during the first ten months of the year. By what percent is the highest price more than the lowest price?
Solution(s):
The highest price is and the lowest price is This means the percent is
Thus, the answer is C.
4.
What is the sum of the mean, median, and mode of the numbers
Solution(s):
The list reordered is
The median is the mean of the middle two numbers, which would be The mode is since appears the most. The mean is
Their sum is
Thus, their answer is C.
5.
Alice needs to replace a light bulb located centimeters below the ceiling in her kitchen. The ceiling is meters above the floor. Alice is meters tall and can reach centimeters above the top of her head. Standing on a stool, she can just reach the light bulb. What is the height of the stool, in centimeters?
Solution(s):
We know the height of the ceiling is cm. Subtracting out all the given values, we get Therefore, the height of the stool is 34 cm. Thus, B is the correct answer.
6.
Which of the following figures has the greatest number of lines of symmetry?
equilateral triangle
non-square rhombus
non-square rectangle
isosceles trapezoid
square
Solution(s):
First, each line of symmetry must go through the center of the shape. This would ensure that the center of the shape isn't on only one side, as that would make it asymmetric.
Next, if a line goes through any side, it must go through its midpoint, to ensure that after a reflection over this line, the same amount of the line is on both sides. Moreover, it must be perpendicular, to ensure that the line when reflected stays on itself.
Similarly, if a line goes through any corner, it must go through its angle bisector, to ensure that after a reflection over this line, the angle of the line is the same after reflection. Moreover, it must have the same side length on both sides.
Now, lets look at each of the shapes. An equilateral triangle has lines that intersect a corner or a midpoint, so it has at most symmetry lines. A non-square rhombus only has two symmetry lines, as only the lines that go through the corners work, but not the ones through the midpoints as they would not intersect perpendicularly. A non-square rectangles only has two symmetry lines as it has symmetry lines through the midpoints of opposite sides, but not through the corners since it doesn't bisect the angle. An isosceles trapezoid only has symmetry line, that goes through the midpoints of opposite sides. A square has symmetry lines, which go through opposite corners and opposite midpoints.
A square therefore has the most symmetry lines.
Thus, the answer is E.
7.
Using only pennies, nickels, dimes, and quarters, what is the smallest number of coins Freddie would need so he could pay any amount of money less than a dollar?
Solution(s):
Freddie needs pennies as this is the only way to pay cents. Next, he needs nickel or pennies to make a cents after having the starting pennies. Here, we have coins. Using nickel minimizes the coins needed. Now, to make cents, we need more cents. This can be done with a dime and nickel. Now, we have coints Now, for the last cents, we can use quarters to make the rest. This leaves us with coins.
Therefore, the answer is B.
8.
As Emily is riding her bicycle on a long straight road, she spots Emerson skating in the same direction mile in front of her. After she passes him, she can see him in her rear mirror until he is mile behind her. Emily rides at a constant rate of miles per hour, and Emerson skates at a constant rate of miles per hour. For how many minutes can Emily see Emerson?
Solution(s):
Let be how far Emily is ahead of Emerson. Emily sees Emerson if Suppose at where is in hours, that Then, Since we must find where we find the time where Since hours passed, we know that minutes passed.
Therefore, the answer is D.
9.
Ryan got of the problems correct on a -problem test, on a -problem test, and on a -problem test. What percent of all the problems did Ryan answer correctly?
Solution(s):
There were a total of problems.
On the first test, he solved problems.
On the second test, he solved problems.
On the third test, he solved problems.
Therefore, he solved a total of This means the fraction he solved is
Therefore, the answer is D.
10.
Six pepperoni circles will exactly fit across the diameter of a -inch pizza when placed. If a total of circles of pepperoni are placed on this pizza without overlap, what fraction of the pizza is covered by pepperoni?
Solution(s):
Each circle has the diameter of the large circle, so it has of the total area.
Since there are pepperoni, they take up of the area.
Therefore, the answer is B.
11.
The top of one tree is feet higher than the top of another tree. The heights of the two trees are in the ratio In feet, how tall is the taller tree?
Solution(s):
Let be the heights of bigger and smaller trees respectively. Then, and If we substitute, we get
Thus, the answer is B.
12.
Of the balls in a large bag, are red and the rest are blue. How many of the red balls must be removed so that of the remaining balls are red?
Solution(s):
If is the number of red balls, then Therefore, if is the number of blue balls, then
If there are red balls after removing balls, then there are blue balls. This means the total number of balls is This means the total number of balls decreased by
Thus, the answer is D.
13.
The lengths of the sides of a triangle in inches are three consecutive integers. The length of the shortest side is of the perimeter. What is the length of the longest side?
Solution(s):
Let be the smallest length. Then, all the side lengths are This would make the perimeter equal to Since then This makes so which makes the longest side length
Thus, the answer is E.
14.
What is the sum of the prime factors of
Solution(s):
First, the primes are factors of since it is a multiple of Dividing by is Then, is a factor of since the digit sum of is a multiple of Dividing this by yields the prime number This means the prime factors are which makes their sum
Thus, the correct answer is C.
15.
A jar contains different colors of gumdrops. are blue, are brown, are red, are yellow, and the other gum drops are green. If half of the blue gumdrops are replaced with brown gumdrops, how many gumdrops will be brown?
Solution(s):
Since we have percentages for every color except green, the percent of green is minus the sum of the other colors. This would make the percent of green equal to
Since we know jelly beans is we know that the total number of jelly beans is
This means there are brown jelly beans to start and blue jelly beans. If half of the blue jelly beans are turned to brown, then more brown jelly beans are added. Therefore, we have brown jelly beans.
Thus, the answer is C.
16.
A square and a circle have the same area. What is the ratio of the side length of the square to the radius of the circle?
Solution(s):
Let be the side length of the square and let be the radius of the circle. Then, since they have the same areas, This means so
Thus, the answer is B.
17.
The diagram shows an octagon consisting of unit squares. The portion below is a unit square and a triangle with base If bisects the area of the octagon, what is the ratio
Solution(s):
Since bisects the area, the area under the line is Removing the square on the right makes the bottom a triangle of base with area Let the base of this triangle be
The area being means
Therefore, and This would make
Thus, the answer is D.
18.
A decorative window is made up of a rectangle with semicircles at either end. The ratio of to is And is 30 inches. What is the ratio of the area of the rectangle to the combined area of the semicircles?
Solution(s):
Combining the semicircles would make a circle of diameter This would make the radius equal to Therefore, the combined area of the semicircles is
Side and The area of the rectangle is therefore The ratio of the area of the rectangle to the area of the semicircles is
Thus, the answer is C.
19.
The two circles pictured have the same center Chord is tangent to the inner circle at is and chord has length What is the area between the two circles?
Solution(s):
The area between the two circles is the area of the larger circle minus the area of the smaller circle. This would be By the Pythagorean Theorem, we can get Therefore, we need to find
SInce is half of we get This makes
Thus, the area is C.
20.
In a room, of the people are wearing gloves, and of the people are wearing hats. What is the minimum number of people in the room wearing both a hat and a glove?
Solution(s):
Since our room has of the people wearing gloves, the number of people must be a multiple of Since our room has of the people wearing hats, the number of people must be a multiple of Therefore, the people in the room must be a multiple of
Now, we can also use the following formula by the principle of inclusion exclusion: Fraction of people wearing both = Fraction of people wearing gloves + Fraction of people wearing hats - Fraction of people wearing either.
This makes our desired fraction equal to Fraction of people who wear either. If we wish to minimize the number of who wear both, we maximize the number of people who wear both, to make it Therefore, the fraction of people that wear both is
Since our number is a (positive) multiple of we have the number of people wearing both as if we choose to have just people.
Therefore, A is the correct answer.
21.
Hui is an avid reader. She bought a copy of the best seller Math is Beautiful. On the first day, Hui read of the pages plus more, and on the second day she read of the remaining pages plus pages. On the third day she read of the remaining pages plus pages. She then realized that there were only pages left to read, which she read the next day. How many pages are in this book?
Solution(s):
The pages left after the third day is Before reading the last pages, she had pages left. This is of the pages remaining, so she had pages left before the third day.
Before reading the pages, she had pages left. This is of the pages remaining, so she had pages left before the second day.
Before reading the pages, she had pages left. This is of the pages remaining, so she had pages left before the first day, making the book pages.
Thus, the answer is C.
22.
The hundreds digit of a three-digit number is more than the units digit. The digits of the three-digit number are reversed, and the result is subtracted from the original three-digit number. What is the units digit of the result?
Solution(s):
Let the units digit be and let the tens digit be This makes the hundreds digit be This makes the number equal to and the reversed number is This makes the difference equal to This makes the units digit
Therefore, the units digit is E.
23.
Semicircles and pass through the center What is the ratio of the combined areas of the two semicircles to the area of circle
Solution(s):
The area of each of the smaller circles have an area of Each of them have a radius of so their combined area is
Next, the radius of the larger circle is equal to the length of which is equal to Its area is
This means the ratio is
Thus, the answer is B.
24.
What is the correct ordering of the three numbers, and
2^{24} < 10^8 < 5^{12}
2^{24} < 5^{12} < 10^8
5^{12} < 2^{24} < 10^8
10^8 < 5^{12} < 2^{24}
10^8 < 2^{24} < 5^{12}
Solution(s):
First, we get
Next, we get This means
Thus, the answer is A.
25.
Everyday at school, Jo climbs a flight of stairs. Jo can take the stairs or at a time. For example, Jo could climb then then In how many ways can Jo climb the stairs?
Solution(s):
Lets count first the number of ways to climb a flight of stairs. She must land on the last stair, and for each of the other stairs, she can either step on it or not step on it. This means there are total combinations. Now, we must exclude the ways that include stepping more than steps. This would mean we climb or stairs in one step. To do this, we can do it in the following ways: This means we have cases to exlude, so we have total valid cases.
Thus, the answer is E.