To determine how much trail mix Atenea gives to her brother, we need to first find out how much trail mix she uses to make the seven snack bags and then subtract that from the total amount she has.
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Convert the total amount of trail mix to an improper fraction: \[ 2 \frac{5}{6} = \frac{2 \times 6 + 5}{6} = \frac{12 + 5}{6} = \frac{17}{6} \]
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Calculate the total weight of the seven snack bags: Each bag weighs \( \frac{1}{3} \) pound, so for seven bags: \[ 7 \times \frac{1}{3} = \frac{7}{3} \]
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Subtract the weight of the snack bags from the total trail mix amount: Now, we need to subtract \( \frac{7}{3} \) from \( \frac{17}{6} \). First, we need a common denominator. The least common denominator for 3 and 6 is 6.
Convert \( \frac{7}{3} \) to sixths: \[ \frac{7}{3} = \frac{7 \times 2}{3 \times 2} = \frac{14}{6} \]
Now subtract: \[ \frac{17}{6} - \frac{14}{6} = \frac{17 - 14}{6} = \frac{3}{6} = \frac{1}{2} \]
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Result: Atenea gives \( \frac{1}{2} \) pound of trail mix to her brother.
Thus, the final answer is: \[ \text{Atenea gives } \frac{1}{2} \text{ pound of trail mix to her brother.} \]