AMC8 2004 8, Problem 9

The average of the five numbers in a list is $54$. The average of the first two numbers is $48$. What is the average of the last three numbers?

Solution: Let the $5$ numbers be $a, b, c, d$, and $e$. Thus $\frac{a+b+c+d+e}{5}=54$ and $a+b+c+d+e=270$. Since $\frac{a+b}{2}=48$, $a+b=96$. Substituting back into our original equation, we have $96+c+d+e=270$ and $c+d+e=174$. Dividing by $3$ gives the average of $\fbox{58}$.

Answer: 58