2011 AMC 12A Problem 6

Question

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

61

points. How many free throws did they make?

13

14

15

16

17

Answer: A

Concepts:system of equations substitution

Difficulty rating: 1170

Small hint:

Let

a

be the number of two-point shots; the points from two-point and three-point shots are each

2a

Big hint:

The free throws number

a + 1,

so the total is

2a + 2a + (a + 1)

Solution:

Let

a

be the number of two-point shots. The two-point shots score

2a

points, and the three-point shots score the same

2a

points. The free throws number

a + 1

and score

a + 1

points.

The total is

\begin{aligned} 2a + 2a + (a + 1) \\ = 5a + 1 = 61, \end{aligned}

so

a = 12

and the free throws number

a + 1 = 13.

Thus, the correct answer is A.