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| A team of students is going to compete against a team of teachers in a trivia contest. The total number of students and teachers is <imath>15</imath>. Ash, a cousin of one of the students, wants to join the contest. If Ash plays with the students, the average age on that team will increase from <imath>12</imath> to <imath>14</imath>. If Ash plays with the teachers, the average age on that team will decrease from <imath>55</imath> to <imath>52</imath>. How old is Ash?
| | #redirect [[2025 AMC 12A Problems/Problem 3]] |
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| <imath>\textbf{(A)}~28\qquad\textbf{(B)}~29\qquad\textbf{(C)}~30\qquad\textbf{(D)}~32\qquad\textbf{(E)}~33</imath>
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| ==Video Solution==
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| https://youtu.be/l1RY_C20Q2M
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| ==Solution 1==
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| Let <imath>s</imath> be the number of students, <imath>t</imath> be the number of teachers, and <imath>a</imath> be Ash's age. We know that <imath>s+t=15.</imath> Also, the sum of the ages of the students is <imath>12s</imath> and the sum of the ages of the teachers are <imath>55t.</imath> Thus, we have the following equations:
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| <cmath>\frac{12s+a}{s+1}=14</cmath><cmath>\frac{55t+a}{t+1}=52.</cmath>
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| Isolating <imath>s</imath> and <imath>t,</imath> we have <imath>2s=a-14</imath> and <imath>3t=52-a.</imath> We also know that <imath>6s+6t=90,</imath> so substituting, we have <imath>3a-42+104-2a=90.</imath> Therefore, <imath>a=\boxed{\text{(A) 28}}.</imath>
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| - harshu13
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| == Solution 2 ==
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| Let <imath>s</imath> and <imath>t</imath> represent the number of students and teachers respectively. If the average age of each of the students increases by 2 when Ash joins them, then Ash’s age in terms of <imath>s</imath> is <imath>2s+14</imath>. Similarly, Ash’s age is <imath>52-3t</imath> in terms of <imath>t</imath>. Equating these two equations gives <imath>2s+3t=38</imath>, and <imath>s+t=15</imath> is also clearly true. Solving the system yields <imath>s=7</imath> and plugging this into <imath>2s+14</imath> gives Ash's age, <imath>\boxed{\textbf{(A) }28}</imath>.
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| ~ruihl123
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| ==Chinese Video Solution==
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| https://www.bilibili.com/video/BV18V2uBtEHt/
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| ~metrixgo
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| == Video Solution (Fast and Easy)==
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| https://youtu.be/ZqswJsf2Odo?si=jeXT21MqPOIEj_Rz ~ Pi Academy
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| ==Video Solution by Daily Dose of Math==
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| https://youtu.be/LN5ofIcs1kY
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| ~Thesmartgreekmathdude
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| ==See Also==
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| {{AMC10 box|year=2025|ab=A|num-b=3|num-a=5}}
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| * [[AMC 10]]
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| * [[AMC 10 Problems and Solutions]]
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| * [[Mathematics competitions]]
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| * [[Mathematics competition resources]]
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| {{MAA Notice}}
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