2011 AMC 12A Problems and Solution
Problem 1
A cell phone plan costs plus
messages and talked for
dollars each month, plus cents per text message sent,
hours. In January Michelle sent
text
hours. How much did she have to pay?
cents for each minute used over
Solution
The base price of Michelle's cell phone plan is texting, or dollars. She talked for
dollars. If she sent
text cents for
hours
messages and it costs cents per text, then she must have spent
hours, but
amount of time that she has to pay an additional amount for.
minutes. Since the price for phone calls is amount Michelle has to pay for phone calls is Adding
dollars
dollars
.
will give us the
cents per minute, the additional
cents, or dollars.
Problem 2
There are coins placed flat on a table according to the figure. What is the
order of the coins from top to bottom?
Solution
By inspection, the answer is (rigor?)
.
Problem 3
A small bottle of shampoo can hold can hold buy?
milliliters of shampoo, whereas a large bottle
milliliters of shampoo. Jasmine wants to buy the minimum number of
small bottles necessary to completely fill a large bottle. How many bottles must she
Solution
To find how many small bottles we need, we can simply divide simplifies to have to round up to
by
. This
, we
Since the answer must be an integer greater than bottles, or
Problem 4
At an elementary school, the students in third grade, fourth grade, and fifth grade run an average of
,
, and
minutes per day, respectively. There are twice as
many third graders as fourth graders, and twice as many fourth graders as fifth graders. What is the average number of minutes run per day by these students?
Solution
Let us say that there are fifth graders. According to the given information, there must be
fourth graders and
third graders. The average time run by each
student is equal to the total amount of time run divided by the number of students. This gives us
Problem 5
Last summer
of the birds living on Town Lake were geese,
were swans,
were herons, and swans were geese?
were ducks. What percent of the birds that were not
Solution
To simplify the problem, let us say that there were a total of of birds that are not swans is percentage is just
. The number of geese is
birds. The number . Therefore the
Problem 6
The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was points. How many free throws did they make?
Solution
For the points made from two-point shots and from three-point shots to be equal, the numbers of made shots are in a
ratio. Therefore, assume they made
and
two- and three- point shots, respectively, and thus number of points is Set that equal to made
, we get
free throws. The total
, and therefore the number of free throws they
Problem 7
A majority of the
students in Ms. Demeanor's class bought pencils at the school
bookstore. Each of these students bought the same number of pencils, and this number was greater than . The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was
. What was the cost of a pencil in cents?