Question

two boxes contain picture of hockey and basketball players.in one box the ratio of hockey players to basketball player is 4:3 in the other box the ratio is 3:2 the boxes contain the same number of picture what could the total number of pictures be?

Answer 1

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.