Difference between revisions of "2011 AMC 12A Problems/Problem 6"
Technodoggo (talk | contribs) (→Solution 3) |
m (→Solution 2) |
||
(2 intermediate revisions by 2 users not shown) | |||
Line 19: | Line 19: | ||
==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 00:18, 16 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
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.