To determine how many black tiles Sam will use when he tiles the floor with the repeating pattern "Blue Tile, White Tile, Black Tile," we first observe the pattern.
The repeating sequence consists of 3 tiles:
- Blue Tile
- White Tile
- Black Tile
If Sam uses 85 tiles, we first find out how many complete sets of 3 tiles fit into 85:
\[ 85 \div 3 = 28 \text{ remainder } 1 \]
This calculation tells us that Sam can complete 28 full sets of the 3-tile pattern, which accounts for:
\[ 28 \times 3 = 84 \text{ tiles} \]
In these 84 tiles, each color (Blue, White, and Black) appears equally across the 28 sets. Therefore, the number of each color tile from these complete sets is:
\[ 28 \text{ Blue Tiles, } 28 \text{ White Tiles, and } 28 \text{ Black Tiles} \]
After using 84 tiles, there is still 1 tile left to place out of the 85 total tiles. The next tile according to the repeating pattern is a Blue Tile.
Thus, the total number of each color tile after using all 85 tiles is:
- Blue Tiles: \( 28 + 1 = 29 \)
- White Tiles: \( 28 \)
- Black Tiles: \( 28 \)
From this, we can conclude that the number of black tiles Sam will use is:
\[ \boxed{28} \]
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