2007 AMC 10A Problem 10

Below is the professionally curated solution for Problem 10 of the 2007 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2007 AMC 10A solutions, or check the answer key.

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Concepts:mean linear equation

Difficulty rating: 1240

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?

$2$

$3$

$4$

$5$

$6$

Solution:

Let $N$ be the number of children and $T$ the total age of the family.

Then $20 = \dfrac{T}{N + 2}$ and $16 = \dfrac{T - 48}{N + 1}.$

These give $20N + 40 = T$ and $16N + 64 = T,$ so $20N + 40 = 16N + 64.$

Hence $4N = 24$ and $N = 6.$

Thus, the correct answer is E.

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