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2006 AMC 10B Problems/Problem 3: Difference between revisions

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== Problem ==
== Problem ==
A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of 34 points, and the Cougars won by a margin of 14 points. How many points did the Panthers score?  
A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of <math>34</math> points, and the Cougars won by a margin of <math>14</math> points. How many points did the Panthers score?  


<math> \mathrm{(A) \ } 10\qquad \mathrm{(B) \ } 14\qquad \mathrm{(C) \ } 17\qquad \mathrm{(D) \ } 20\qquad \mathrm{(E) \ } 24 </math>
<math> \textbf{(A) } 10\qquad \textbf{(B) } 14\qquad \textbf{(C) } 17\qquad \textbf{(D) } 20\qquad \textbf{(E) } 24 </math>


== Solution ==
== Solution ==
Let <math>x</math> be the number of points scored by the Cougars, and <math>y</math> be the number of points scored by the Panthers. The problem is asking for the value of <math>y</math>.  
Let <math>x</math> be the number of points scored by the Cougars, and <math>y</math> be the number of points scored by the Panthers. The problem is asking for the value of <math>y</math>.  
<cmath>\begin{align*}
x+y &= 34 \\
x-y &= 14 \\
2x &= 48 \\
x &= 24 \\
\end{align*}</cmath>
The answer is <math>\boxed{\textbf{(A) } 10}</math>


<math>x+y=34</math>
== Solution 2 ==
<math>c</math> is the amount the Cougars scored and <math>p</math> is the score for Panthers. Since the Cougars won by 14 points, <math>c = p + 14</math>. Using substitution,
<math>2p + 14 = 34</math>,
<math>2p = 20</math>, and then
<math>p = 10</math>.


<math>x-y=14</math>
<cmath>\begin{align*}
p &= \boxed{\textbf{(A) }10} \\
\end{align*}</cmath>


<math>2x=48</math>
-- leafy
 
<math>x=24</math>
 
<math>y=10 \Rightarrow A</math>


== See Also ==
== See Also ==
Line 21: Line 30:


[[Category:Introductory Algebra Problems]]
[[Category:Introductory Algebra Problems]]
{{MAA Notice}}

Latest revision as of 09:31, 24 February 2025

Problem

A football game was played between two teams, the Cougars and the Panthers. The two teams scored a total of $34$ points, and the Cougars won by a margin of $14$ points. How many points did the Panthers score?

$\textbf{(A) } 10\qquad \textbf{(B) } 14\qquad \textbf{(C) } 17\qquad \textbf{(D) } 20\qquad \textbf{(E) } 24$

Solution

Let $x$ be the number of points scored by the Cougars, and $y$ be the number of points scored by the Panthers. The problem is asking for the value of $y$. \begin{align*} x+y &= 34 \\ x-y &= 14 \\ 2x &= 48 \\ x &= 24 \\ \end{align*} The answer is $\boxed{\textbf{(A) } 10}$

Solution 2

$c$ is the amount the Cougars scored and $p$ is the score for Panthers. Since the Cougars won by 14 points, $c = p + 14$. Using substitution, $2p + 14 = 34$, $2p = 20$, and then $p = 10$.

\begin{align*} p &= \boxed{\textbf{(A) }10} \\ \end{align*}

-- leafy

See Also

2006 AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 2 		Followed by
Problem 4
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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