Difference between revisions of "2011 AMC 12A Problems/Problem 6"
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==Solution 3== | ==Solution 3== | ||
We let <math>a</math> be the number of <math>2</math>-point shots, <math>b</math> be the number of <math>3</math>-point shots, and <math>x</math> be the number of free throws. We are looking for <math>x.</math> | We let <math>a</math> be the number of <math>2</math>-point shots, <math>b</math> be the number of <math>3</math>-point shots, and <math>x</math> be the number of free throws. We are looking for <math>x.</math> | ||
| − | We know that <math>2a=3b | + | We know that <math>2a=3b</math>, and that <math>x=a+1</math>. Also, <math>2a+3b+1x=61</math>. We can see |
| − | + | ||
| + | <cmath> | ||
| + | \begin{align*} | ||
| + | a&=x-1 \\ | ||
| + | 2a &= 2x-2 \\ | ||
| + | 3a &= 2x-2. \\ | ||
| + | \end{align*} | ||
| + | </cmath> | ||
| + | |||
| + | Plugging this into <math>2a+3b+1x=61</math>, we see | ||
| + | |||
| + | <cmath> | ||
| + | \begin{align*} | ||
| + | 2x-2+2x-2+x &= 61 \\ | ||
| + | 5x-4 &= 61 \\ | ||
| + | 5x &= 65 \\ | ||
| + | x &= \boxed{\textbf{(A) }13}. | ||
| + | \end{align*} | ||
| + | </cmath> | ||
| + | |||
~Technodoggo | ~Technodoggo | ||
| + | |||
| + | ~MrThinker | ||
==Video Solution == | ==Video Solution == | ||
Latest revision as of 23:18, 15 July 2024
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.
Solution 3
We let 
be the number of 
-point shots, 
be the number of 
-point shots, and be the number of free throws. We are looking for 
We know that 
, and that 
. Also, 
. We can see
Plugging this into , we see
~Technodoggo
~MrThinker
Video Solution
https://www.youtube.com/watch?v=6tlqpAcmbz4 ~Shreyas S
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 | |
| 2011 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 11 |
Followed by Problem 13 | |
| 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 | ||
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