Difference between revisions of "2011 AMC 12A Problems/Problem 6"
(→See also) |
(→Solution) |
||
| Line 9: | Line 9: | ||
\textbf{(E)}\ 17 </math> | \textbf{(E)}\ 17 </math> | ||
| − | == Solution == | + | == Solution 1 == |
For the points made from two-point shots and from three-point shots to be equal, the numbers of made shots are in a <math>3:2</math> ratio. Therefore, assume they made <math>3x</math> and <math>2x</math> two- and three- point shots, respectively, and thus <math>3x+1</math> free throws. The total number of points is <cmath>2 \times (3x) + 3 \times (2x) + 1 \times (3x+1) = 15x+1</cmath> | For the points made from two-point shots and from three-point shots to be equal, the numbers of made shots are in a <math>3:2</math> ratio. Therefore, assume they made <math>3x</math> and <math>2x</math> two- and three- point shots, respectively, and thus <math>3x+1</math> free throws. The total number of points is <cmath>2 \times (3x) + 3 \times (2x) + 1 \times (3x+1) = 15x+1</cmath> | ||
Set that equal to <math>61</math>, we get <math>x = 4</math>, and therefore the number of free throws they made <math>3 \times 4 + 1 = 13 \Rightarrow \boxed{A}</math> | Set that equal to <math>61</math>, we get <math>x = 4</math>, and therefore the number of free throws they made <math>3 \times 4 + 1 = 13 \Rightarrow \boxed{A}</math> | ||
| + | |||
| + | == Solution 2 == | ||
| + | Let <math>x</math> be the number of free throws. Then the number of points scored by two-pointers is <math>2(x-1)</math> and the same goes for three-pointers because they scored the same number of points with twos and threes. Thus, our equation is <math>x+4(x-1) = 61 \Rightarrow x=13</math>, giving us <math>\boxed{(A)}</math> for an answer. | ||
== See also == | == See also == | ||
{{AMC12 box|year=2011|num-b=5|num-a=7|ab=A}} | {{AMC12 box|year=2011|num-b=5|num-a=7|ab=A}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 23:23, 29 October 2018
Contents
Problem
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 1
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 
free throws. The total number of points is ![\[2 \times (3x) + 3 \times (2x) + 1 \times (3x+1) = 15x+1\]](http://latex.artofproblemsolving.com/2/3/2/23266fed552512114a8d3bf8b07a2ec846b2faec.png)
Set that equal to , we get 
, and therefore the number of free throws they made 
Solution 2
Let 
be the number of free throws. Then the number of points scored by two-pointers is 
and the same goes for three-pointers because they scored the same number of points with twos and threes. Thus, our equation is 
, giving us 
for an answer.
See also
| 2011 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 | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
