AMC10 2007 10A, Problem 10

The Dunbar family consists of a mother, a father, and some children. The average age of the members of the family is $20$, the father is $48$ years old, and the average age of the mother and children is $16$. How many children are in the family?

Solution: Let $x$ be the number of the children and the mom. The father, who is $48$, plus the sum of the ages of the kids and mom divided by the number of kids and mom plus $1$ (for the dad) = $20$. This is because the average age of the entire family is $20.$ This statement, written as an equation, is: \[\frac{48+16x}{x+1}=20\] \[48+16x=20x+20\] \[4x=28\] \[x=7\] $7$ people - $1$ mom = $6$ children. Therefore, the answer is $\fbox{6}$

Answer: 6