To determine the possibilities for how the gift bags can be filled with pencils and stickers, we need to ensure that the total number of pencils and stickers used adds up to 24 pencils and 36 stickers respectively for each proposed combination of bags.
We can start by defining:
- Let \( b \) = number of bags
- Let \( p \) = number of pencils per bag
- Let \( s \) = number of stickers per bag
The equations we need to satisfy for each possibility are:
- \( b \times p = 24 \) (total pencils)
- \( b \times s = 36 \) (total stickers)
Let's evaluate the possibilities:
A. 3 bags with 8 pencils and 12 stickers
- Pencils: \( 3 \times 8 = 24 \) (OK)
- Stickers: \( 3 \times 12 = 36 \) (OK)
✅ This is a valid option.
B. 3 bags with 10 pencils and 10 stickers
- Pencils: \( 3 \times 10 = 30 \) (not OK)
- Stickers: \( 3 \times 10 = 30 \) (not OK)
❌ This is not a valid option.
C. 4 bags with 6 pencils and 9 stickers
- Pencils: \( 4 \times 6 = 24 \) (OK)
- Stickers: \( 4 \times 9 = 36 \) (OK)
✅ This is a valid option.
D. 5 bags with 6 pencils and 6 stickers
- Pencils: \( 5 \times 6 = 30 \) (not OK)
- Stickers: \( 5 \times 6 = 30 \) (not OK)
❌ This is not a valid option.
E. 6 bags with 4 pencils and 6 stickers
- Pencils: \( 6 \times 4 = 24 \) (OK)
- Stickers: \( 6 \times 6 = 36 \) (OK)
✅ This is a valid option.
F. 12 bags with 2 pencils and 3 stickers
- Pencils: \( 12 \times 2 = 24 \) (OK)
- Stickers: \( 12 \times 3 = 36 \) (OK)
✅ This is a valid option.
Based on our analysis, the valid options from your selections are:
- A. 3 bags with 8 pencils and 12 stickers
- C. 4 bags with 6 pencils and 9 stickers
- E. 6 bags with 4 pencils and 6 stickers
- F. 12 bags with 2 pencils and 3 stickers
Thus, the four additional possibilities are A, C, E, and F.