1. Stephanie has 1000 eggs to pack into cartons of 12 eggs. While packing the eggs, she breaks $n$ eggs. The unbroken eggs completely fill a collection of cartons with no eggs left over. If $n<12$, what is the value of $n$ ?
1. Solution 1
Since $\frac{1000}{12} \approx 83.33$, then the largest multiple of 12 less than 1000 is $83 \times 12=996$.
Therefore, if Stephanie filled 83 cartons, she would have broken 4 eggs.
If Stephanie filled fewer than 83 cartons, she would have broken more than 12 eggs. If Stephanie filled more than 83 cartons, she would have needed more than 1000 eggs.
Thus, $n=4$.
Solution 2
Since $80 \times 12=960$, then filling 80 cartons would leave $1000-960=40$ eggs left over.
Since $81 \times 12=972$, then filling 81 cartons would leave $1000-972=28$ eggs left over.
Since $82 \times 12=984$, then filling 82 cartons would leave $1000-984=16$ eggs left over.
Since $83 \times 12=996$, then filling 83 cartons would leave $1000-996=4$ eggs left over.
Since $84 \times 12=1008$, then Stephanie could not have filled 84 cartons.
Therefore, Stephanie must have filled 83 cartons and broken 4 eggs, and so $n=4$.
ANSWER: $n=4$