2009 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.
Bridget bought a bag of apples at the grocery store. She gave half of the apples to Ann. Then she gave Cassie apples, keeping apples for herself. How many apples did Bridget buy?
Solution(s):
We can work backwards, starting with the apples that Bridget kept for herself. Adding the apples that she gave Cassie, she now has apples.
Finally, we multiply this value by since she gave half of her initial apples to Ann. so Bridget started off with apples.
Thus, E is the correct answer.
2.
On average, for every sports cars sold at the local dealership, sedans are sold. The dealership predicts that it will sell sports cars next month. How many sedans does it expect to sell?
Solution(s):
We can set up the following proportion: Cross multiplying, we get
Thus, D is the correct answer.
3.
The graph shows the constant rate at which Suzanna rides her bike. If she rides a total of a half an hour at the same speed, how many miles would she have ridden?
Solution(s):
From the graph, we can see that Suzanna rides miles in minutes. This means that in minutes, she will have ridden miles.
Thus, C is the correct answer.
4.
The five pieces shown below can be arranged to form four of the five figures below. Which figure cannot be formed?
Solution(s):
Note that option B does not have any segments in it that are blocks long. This means that it is impossible to arrange the block long piece to fit within the figure.
Thus, B is the correct answer.
5.
A sequence of numbers starts with and The fourth number of the sequence is the sum of the previous three numbers in the sequence: In the same way, every number after the fourth is the sum of the previous three numbers. What is the eighth number in the sequence?
Solution(s):
We can create the following list to find all the numbers up till the eight number denotes the -th number in the sequence.
and from the problem statement.
Thus, D is the correct answer.
6.
Steve's empty swimming pool will hold gallons of water when full. It will be filled by hoses, each of which supplies gallons of water per minute. How many hours will it take to fill Steve's pool?
Solution(s):
The hoses together fill the pool with gallons of water per minute.
To fill gallons, it will take the hoses minutes to fill the pool.
minutes is the same as hours.
Thus, A is the correct answer.
7.
The triangular plot of lies between Aspen Road, Brown Road and a railroad. Main Street runs east and west, and the railroad runs north and south. The numbers in the diagram indicate distances in miles. The width of the railroad track can be ignored. How many square miles are in the plot of land
Solution(s):
The base of is which is The altitude is as well.
Therefore, the area of is
Thus, C is the correct answer.
8.
The length of a rectangle is increased by percent and the width is decreased by percent. What percent of the old area is the new area?
Solution(s):
Let be the old width of the rectangle and be the old width. The new length is then and the new width is This means that the new area is
This shows that the new area is of the original area.
Thus, B is the correct answer.
9.
Construct a square on one side of an equilateral triangle. On one non-adjacent side of the square, construct a regular pentagon, as shown. On a non-adjacent side of the pentagon, construct a hexagon. Continue to construct regular polygons in the same way, until you construct an octagon. How many sides does the resulting polygon have?
Solution(s):
Notice that every polygon in the middle of the chain (every shape except for the triangle and octagon) contribute all but of their sides to the overall polygon. The triangle and octagon contribute all but of their sides.
Therefore, the total number of sides is
Thus, B is the correct answer.
10.
On a checkerboard composed of unit squares, what is the probability that a randomly chosen unit square does not touch the outer edge of the board?
Solution(s):
There are squares on the interior.
This means that the probability of choosing one of these squares is
Thus, D is the correct answer.
11.
The Amaco Middle School bookstore sells pencils costing a whole number of cents. Some seventh graders each bought a pencil, paying a total of dollars. Some of the sixth graders each bought a pencil, and they paid a total of dollars. How many more sixth graders than seventh graders bought a pencil?
Solution(s):
The number of seventh graders that bought a pencil is divided by the price of a pencil. Similarly, the number of sixth graders that bought a pencil is divided by the price of a pencil.
This means that the price of a pencil divides both and Prime factorizing, we get and The only numbers that divide both and are and
If cent was the price of the pencil, that means sixth graders bought pencils, which is not possible. Therefore, the price of a pencil is cents.
This means that seventh graders bought a pencil, and sixth graders bought a pencil. Therefore, more sixth graders than seventh graders bought pencils.
Thus, D is the correct answer.
12.
The two spinners shown are spun once and each lands on one of the numbered sectors. What is the probability that the sum of the numbers in the two sectors is prime?
Solution(s):
We can find the sum of the two numbers in every possible outcome.
\begin{gather*} 1 + 2 = 3 \\ 1 + 4 = 5 \\ 1 + 6 = 7 \\ 3 + 2 = 5 \\ 3 + 4 = 7 \\ 3 + 6 = 9 \\ 5 + 2 = 7 \\ 5 + 4 = 9 \\ 5 + 6 = 11. \end{gather*}
There are only outcomes where the sum is not prime (the two instances when the sum is ). Therefore, the probability that the sum is prime is
Thus, D is the correct answer.
13.
A three-digit integer contains one of each of the digits and What is the probability that the integer is divisible by
Solution(s):
Then number is equally likely to end in a or It is only divisible if the last digit is a which happens with a probability.
Thus, B is the correct answer.
14.
Austin and Temple are miles apart along Interstate Bonnie drove from Austin to her daughter's house in Temple, averaging miles per hour. Leaving the car with her daughter, Bonnie rode a bus back to Austin along the same route and averaged miles per hour on the return trip. What was the average speed for the round trip, in miles per hour?
Solution(s):
The trip from Austin to Temple took hours. The trip from Temple to Austin took hours. This means that the total time for the round trip was hours.
The total distance of the round trip was miles. Therefore, the average speed for the round trip was miles per hour.
Thus, B is the correct answer.
15.
A recipe that makes servings of hot chocolate requires squares of chocolate, cup sugar, cup water and cups milk. Jordan has squares of chocolate, cups of sugar, lots of water, and cups of milk. If he maintains the same ratio of ingredients, what is the greatest number of servings of hot chocolate he can make?
Solution(s):
We need to find which of the ingredients is the limiting factor.
With squares of chocolate, Jordan can only make times what the recipe requires.
With cups of sugar, Jordan can only make times the required amount.
With cups of milk, Jordan can only make times what is needed.
Therefore, the number of servings is limited by the amount of milk. The amount of servings Jordan can make is
Thus, D is the correct answer.
16.
How many -digit positive integers have digits whose product equals
Solution(s):
The only triples of integers less than that multiply to are
The triples with distinct numbers can be rearranged to form distinct -digit positive integers. The other triple can be arranged to form distinct -digit positive integers.
This leaves a total of integers.
Thus, D is the correct answer.
17.
The positive integers and are the two smallest positive integers for which the product of and is a square and the product of and is a cube. What is the sum of and
Solution(s):
For a number to be a perfect square, every exponent in the prime factorization must be even. For it to be a cube, the exponents must be divisible by
We can factor to get For to be a perfect square and to be minimized, must have one factor of and one factor of Therefore, we can let
For to be a cube, must have one factor of and two factors of Therefore, we can let suggesting
Thus, B is the correct answer.
18.
The diagram represents a -foot-by--foot floor that is tiled with -square-foot light tiles and dark tiles. Notice that the corners have dark tiles. If a -foot-by--foot floor is to be tiled in the same manner, how many dark tiles will be needed?
Solution(s):
Looking at the example given, note that there are dark tiles. This is because there are rows that contain dark tiles each.
For a -foot-by--foot floor, there are going to be rows with dark tiles in each row. This will give us dark tiles.
Thus, C is the correct answer.
19.
Two angles of an isosceles triangle measure and What is the sum of the three possible values of
Solution(s):
All the following possibilities are shown below.
In the first scenario, we get by the properties of the isosceles triangle.
In the second scenario, we get that from which we get that
From the third scenario, we get that from which we get that
The sum of these values yields
Thus, D is the correct answer.
20.
How many non-congruent triangles have vertices at three of the eight points in the array shown below?
Solution(s):
We can label to points to find all the unique triangles.
All the possible distinct triangles are
Any other triangle will be congruent to one of the listed above.
Thus, D is the correct answer.
21.
Andy and Bethany have a rectangular array of numbers with rows and columns. Andy adds the numbers in each row. The average of his sums is Bethany adds the numbers in each column. The average of her sums is What is the value of
Solution(s):
Each person includes every number in the array in their sums exactly once. This means that before they divide in the average, their sums are the exact same. This means that Therefore,
Thus, D is the correct answer.
22.
How many whole numbers between and do not contain the digit
Solution(s):
We can case on the number of digits.
There are one digit numbers excluding
There are two digit numbers that lack the digit
There are three digit numbers that do not include
This yields a total of numbers that do not contain the digit
Thus, D is the correct answer.
23.
On the last day of school, Mrs. Wonderful gave jelly beans to her class. She gave each boy as many jelly beans as there were boys in the class. She gave each girl as many jelly beans as there were girls in the class. She brought jelly beans, and when she finished, she had six jelly beans left. There were two more boys than girls in her class. How many students were in her class?
Solution(s):
Let be the number of boys in the class and be the number of girls. From the problem, we get that
If each boy gets jelly beans, then Mrs. Wonderful will give out a total of jelly beans to all the boys. Similarly, she will give out jelly beans to all the girls.
Therefore, Since cannot be negative, we get that This means that so
Thus, B is the correct answer.
24.
The letters and represent digits.
If and what digit does represent?
Solution(s):
Since we get that Using this and the second equation, we also get that This follows from carrying over a to resulting in Therefore,
Since we also know that Then
Thus, E is the correct answer.
25.
A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cut is foot from the top face. The second cut is foot below the first cut, and the third cut is foot below the second cut. From the top to the bottom the pieces are labeled and The pieces are then glued together end to end as shown in the second diagram. What is the total surface area of this solid in square feet?
Solution(s):
We can look at this shape from all directions to add up the surface area from each view to get the total surface area.
Looking at it from the ends gives us the face of which has an area of
Each side has the same area as if we were to stack up all the pieces into a unit cube and take the area of one side. This would yield
The top and bottom each have unit squares, for a total of from each view.
Adding this all up gives us a total surface area of
Thus, E is the correct answer.