2013 AMC 12A Problems/Problem 6
Contents
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
Solution 2
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 attempted 2-point shots. So, her score must be:
,which is .
Solution 3
(similar to Solution 1, however a slightly more obvious way)
Say that
x = # of 2-pt shots
y = # of 3-pt shots
Because the total number of shots is ,
However, Shenille was only successful on of the 3-pt shots, and of the 2-pt shots, so
= #number of successful shots
For each successful shot, there is an associated number of points with it.
Therefore, = her score
this evaluates to = her score
is already determined to be 30, so her score is
~amuppalla
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.
Video Solution
https://youtu.be/CCjcMVtkVaQ ~sugar_rush
See also
2013 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
2013 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 5 |
Followed by Problem 7 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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