Difference between revisions of "2011 AMC 12A Problems/Problem 6"
Technodoggo (talk | contribs) |
|||
Line 16: | Line 16: | ||
== Solution 2 == | == 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. | 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. | ||
+ | |||
+ | ==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 know that <math>2a=3b,</math> and that <math>x=a+1.</math> We know that <math>2a+3b+1x=61.</math> We can see: | ||
+ | \begin{align*} | ||
+ | a&=x-1 \ | ||
+ | 2a&=2x-2 \ | ||
+ | 3b&=2x-2 \ | ||
+ | 2a+3b+x&=2x-2+2x-2+x \ | ||
+ | &=5x-4 \ | ||
+ | &=61 \ | ||
+ | 5x&=65 \ | ||
+ | x&=\boxed{\text{(A)}~13}. \ | ||
+ | \end{align*} | ||
==Video Solution == | ==Video Solution == |
Revision as of 18:33, 10 August 2023
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
We know that
We can see:
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.