Question

Joey and his five brothers are ages $3,$ $5,$ $7,$ $9,$ $11,$ and $13.$ One afternoon two of his brothers whose ages sum to $16$ went to the movies, two brothers younger than $10$ went to play baseball, and Joey and the $5$ -year-old stayed home. How old is Joey?

$3$

$7$

$9$

$11$

$13$

Solution:

The age pairs that sum to $16$ are $(3,13)$ , $(5,11)$ , and $(7,9)$ .

The $5$ -year-old stayed home, so the movie pair cannot be $(5,11)$ . If $7$ and $9$ went to the movies, then the only remaining brother younger than $10$ would be the $3$ -year-old, not enough for baseball.

Thus the movie pair was $(3,13)$ , the baseball pair was $(7,9)$ , and Joey is the remaining brother, age $11$ .

Thus, D is the correct answer.

2013 AMC 10A Problem 6

Solution:

Problem 6 in Other Years