2013 AMC 12A Problems/Problem 6

Problem

In a recent basketball game, Shenille attempted only three-point shots and two-point shots. She was successful on of her three-point shots and of her two-point shots. Shenille attempted shots. How many points did she score?

Solution

Let the number of 3-point shots attempted be . Since she attempted 30 shots, the number of 2-point shots attempted must be .

Since she was successful on , or , of her 3-pointers, and , or , of her 2-pointers, then her score must be

, which is

Alternative Solution

Since the problem doesn't specify the number of 3-point shots she attempted, it can be assumed that number doesn't matter, so let it be . Then, she must have made 2-point shots. So, her score must be:

,which is .

Additional note

It is also reasonably easy to find all possibilities for the number of two-point and three-point shots she made. Just note that both numbers of successful throws have to be integers. For " of her two-point shots" to be an integer we need the number of two-point shots to be divisible by 10. This only leaves four possibilities for the number of two-point shots: 0, 10, 20, or 30. Each of them also works for the three-point shots, and as shown above, for each of them the total number of points scored is the same.

See also

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.