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2025 AMC 10A Problems/Problem 4: Difference between revisions

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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?
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?


Note: I remember that the total of students and teachers was 13, and Ash playing on the teachers' will decrease from 60 to 55, if you wrote this problem, please change it.
 
~ someone


<imath>\textbf{(A)}~28\qquad\textbf{(B)}~29\qquad\textbf{(C)}~30\qquad\textbf{(D)}~32\qquad\textbf{(E)}~33</imath>
<imath>\textbf{(A)}~28\qquad\textbf{(B)}~29\qquad\textbf{(C)}~30\qquad\textbf{(D)}~32\qquad\textbf{(E)}~33</imath>

Revision as of 21:12, 6 November 2025

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 $15$. 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 $12$ to $14$. If Ash plays with the teachers, the average age on that team will decrease from $55$ to $52$. How old is Ash?


$\textbf{(A)}~28\qquad\textbf{(B)}~29\qquad\textbf{(C)}~30\qquad\textbf{(D)}~32\qquad\textbf{(E)}~33$

Solution 1

Let $s$ be the number of students, $t$ be the number of teachers, and $a$ be Ash's age. We know that $s+t=15.$ Also, the sum of the ages of the students is $12s$ and the sum of the ages of the teachers are $55t.$ Thus, we have the following equations: \[\frac{12s+a}{s+1}=14\]\[\frac{55t+a}{t+1}=52.\] Isolating $s$ and $t,$ we have $2s=a-14$ and $3t=52-a.$ We also know that $6s+6t=90,$ so substituting, we have $3a-42+104-2a=90.$ Therefore, $a=\boxed{\text{(A) 28}}.$ - harshu13

Video Solution (Fast and Easy)

https://youtu.be/ZqswJsf2Odo?si=jeXT21MqPOIEj_Rz ~ Pi Academy

Video Solution by Daily Dose of Math

https://youtu.be/LN5ofIcs1kY

~Thesmartgreekmathdude

See Also

2025 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 3 		Followed by
Problem 5
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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