Let's denote the total number of pictures in each box as \( N \).
Box 1:
- Ratio of hockey players to basketball players = 4:3.
- Let the number of hockey players be \( 4x \) and the number of basketball players be \( 3x \).
- Therefore, the total number of pictures in Box 1 is: \[ N_1 = 4x + 3x = 7x \]
Box 2:
- Ratio of hockey players to basketball players = 3:2.
- Let the number of hockey players be \( 3y \) and the number of basketball players be \( 2y \).
- Therefore, the total number of pictures in Box 2 is: \[ N_2 = 3y + 2y = 5y \]
Since the two boxes contain the same number of pictures, we set \( N_1 = N_2 \): \[ 7x = 5y \]
To find a common value for \( N \), we can express \( y \) in terms of \( x \): \[ y = \frac{7}{5}x \]
Now, we can substitute \( y \) back to find \( N \): \[ N = 7x = 5y = 5 \left(\frac{7}{5} x\right) = 7x \]
From \( 7x = 5y \), we know:
- \( 7x \) must be a multiple of 5 for \( y \) to be an integer.
- Let \( x = 5k \) for some integer \( k \) (so that \( 7x \) is a multiple of 5): \[ N = 7x = 7(5k) = 35k \]
Thus, the total number of pictures \( N \) could be any multiple of 35. Therefore, the total number of pictures can be: \[ 35, 70, 105, 140, \ldots \]
In conclusion, the total number of pictures could be any multiple of 35.