2007 AMC 10A Problems/Problem 10
The Dunbar family consists of a mother, a father, and some children. The average age of the members of the family is , the father is years old, and the average age of the mother and children is . How many children are in the family?
Let be the number of children. Then the total ages of the family is , and the total number of people in the family is . So
Let be the number of the children and the mom. The father, who is , plus the sum of the ages of the kids and mom divided by the number of kids and mom plus (for the dad) = . This is because the average age of the entire family is This statement, written as an equation, is:
people - mom = children.
Therefore, the answer is
Let be the Mom's age.
Let the number of children be and their average age be . Their age totaled up is simply .
We have the following two equations:
, where is the family's total age and is the total number of people in the family.
The next equation is , where is the total age of the Mom and the children, and is the number of children along with the Mom.
We know the value for , so we substitute the value back in the first equation.
Earlier, we set to be the number of children. Therefore, there are children.
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