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2012 AMC 12A Problems and Solution

Problem 1

A bug crawls along a number line, starting at

. It crawls to

, then turns around

and crawls to . How many units does the bug crawl altogether?

Solution

Crawling from -2 to -6 takes it a distance of 4 units. Crawling from -6 to 5 takes it a distance of 11 units. Add 4 and 11 to get

Problem 2

Cagney can frost a cupcake every

seconds and Lacey can frost a cupcake every

seconds. Working together, how many cupcakes can they frost in minutes?

Solution

Cagney can frost one in 20 seconds, and Lacey can frost one in 30 seconds. Working together, they can frost one in minutes), they can frost

.

seconds. In 300 seconds (5

Problem 3

A box centimeters high, centimeters wide, and centimeters long can hold grams of clay. A second box with twice the height, three times the width, and the same length as the first box can hold

grams of clay. What is ?

Solution

The first box has volume

, and the second has volume

. The second has a volume that is times

greater, so it holds

grams.

Problem 4

In a bag of marbles, of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red?

Solution

Assume that there are 5 total marbles in the bag. The actual number does not matter, since all we care about is the ratios, and the only operation performed on the marbles in the bag is doubling.

There are 3 blue marbles in the bag and 2 red marbles. If you double the amount of red marbles, there will still be 3 blue marbles but now there will be 4 red marbles.

Thus, the answer is .

Problem 5

A fruit salad consists of blueberries, raspberries, grapes, and cherries. The fruit salad has a total of

pieces of fruit. There are twice as many raspberries as

blueberries, three times as many grapes as cherries, and four times as many cherries as raspberries. How many cherries are there in the fruit salad?

Solution

Assume that there are 5 total marbles in the bag. The actual number does not matter, since all we care about is the ratios, and the only operation performed on the marbles in the bag is doubling.

There are 3 blue marbles in the bag and 2 red marbles. If you double the amount of red marbles, there will still be 3 blue marbles but now there will be 4 red marbles.

Thus, the answer is .

Problem 6

The sums of three whole numbers taken in pairs are middle number?

,

, and

. What is the

Solution

Let the three numbers be equal to , , and . We can now write three equations:

Adding these equations together, we get that

and

Substituting the original equations into this one, we find

Therefore, our numbers are 12, 7, and 5. The middle number is