Bill needs 864 tiles to tile a bathroom. He can buy tiles individually for $0.40 each, and he can buy...

1. Translate the problem requirements: Bill needs exactly 864 tiles and wants to minimize cost. He has two purchasing options: individual tiles at \(\$0.40\) each, or boxes of 100 tiles at \(\$32.50\) per box. We need to find the least expensive way to get at least 864 tiles.
2. Calculate cost per tile for each option: Compare the unit cost of buying individually versus buying in boxes to understand which option is more economical.
3. Determine optimal box-and-individual combination: Since 864 is not a perfect multiple of 100, find how many complete boxes to buy and how many individual tiles are needed to reach exactly 864 tiles.
4. Calculate total minimum cost: Compute the final cost using the optimal combination of boxes and individual tiles.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding exactly what Bill needs and what options he has.

Bill needs exactly 864 tiles for his bathroom. He has two ways to buy them:

• Option 1: Buy tiles one by one at \(\$0.40\) per tile
• Option 2: Buy boxes of 100 tiles at \(\$32.50\) per box

Our goal is to find the cheapest way to get at least 864 tiles. Notice I said "at least" because if he buys boxes, he might end up with more tiles than he needs, but that's okay as long as he gets the minimum required.

Process Skill: TRANSLATE - Converting the word problem into clear mathematical requirements

2. Calculate cost per tile for each option

Before we figure out the best combination, let's see which option gives us a better price per tile.

For individual tiles: Each tile costs \(\$0.40\)

For boxes: Each box costs \(\$32.50\) and contains 100 tiles
So the cost per tile when buying boxes = \(\$32.50 ÷ 100 = \$0.325\) per tile

This tells us something important: buying in boxes is cheaper per tile (\(\$0.325\) vs \(\$0.40\)). So we should buy as many complete boxes as possible, and only buy individual tiles if we need a few extra to reach exactly 864.

3. Determine optimal box-and-individual combination

Now let's figure out the best mix of boxes and individual tiles.

First, let's see how 864 breaks down in terms of boxes of 100:
\(864 ÷ 100 = 8.64\)

This means \(864 = 8 × 100 + 64\)
So Bill needs 8 complete boxes (which give him 800 tiles) plus 64 more individual tiles.

Let's check: \(8 \text{ boxes} × 100 \text{ tiles per box} = 800 \text{ tiles}\)
\(800 + 64 = 864 \text{ tiles}\) ✓

This seems like the obvious choice, but let's double-check by considering if buying 9 complete boxes might somehow be cheaper (even though he'd have extra tiles).

\(9 \text{ boxes would give him } 9 × 100 = 900 \text{ tiles}\) (36 more than needed)
Cost of 9 boxes = \(9 × \$32.50 = \$292.50\)

Cost of 8 boxes + 64 individual tiles:
8 boxes: \(8 × \$32.50 = \$260.00\)
64 individual tiles: \(64 × \$0.40 = \$25.60\)
Total: \(\$260.00 + \$25.60 = \$285.60\)

Since \(\$285.60 < \$292.50\), the 8 boxes + 64 individual tiles approach is indeed cheaper.

Process Skill: CONSIDER ALL CASES - Checking both the obvious solution and alternatives to ensure we find the true minimum

4. Calculate total minimum cost

Using our optimal combination:

• 8 boxes at \(\$32.50\) each = \(8 × \$32.50 = \$260.00\)
• 64 individual tiles at \(\$0.40\) each = \(64 × \$0.40 = \$25.60\)
• Total minimum cost = \(\$260.00 + \$25.60 = \$285.60\)

Let's verify this gives him enough tiles:
\(8 \text{ boxes} × 100 \text{ tiles per box} + 64 \text{ individual tiles} = 800 + 64 = 864 \text{ tiles}\) ✓

4. Final Answer

The least amount Bill must pay is \(\$285.60\), which corresponds to answer choice C.

This makes sense because we used the more economical box option as much as possible (8 complete boxes) and only bought individual tiles for the remainder that couldn't fill a complete box.

Common Faltering Points

Errors while devising the approach

1. Missing the mixed-strategy consideration: Students often assume they must choose only one buying method (all individual tiles OR all boxes) rather than considering a combination of both. They might calculate the cost of buying all 864 tiles individually (\(\$345.60\)) or the cost of buying 9 complete boxes (\(\$292.50\)) without realizing that mixing 8 boxes + individual tiles for the remainder could be cheaper.

2. Misunderstanding the constraint: Students may think Bill needs exactly 864 tiles and cannot buy more, leading them to avoid the 9-box option entirely. However, the problem asks for the "least amount to buy the tiles he needs," which means buying more than 864 tiles is acceptable as long as he gets at least 864.

Errors while executing the approach

1. Division interpretation error: When calculating \(864 ÷ 100 = 8.64\), students might incorrectly round up to 9 boxes immediately without recognizing that 8.64 means "8 complete boxes plus 64 remaining tiles." This leads them to miss the mixed-strategy calculation entirely.

2. Arithmetic mistakes in cost calculations: Students may make computational errors when calculating costs, such as: \(64 × \$0.40 = \$25.60\) (might calculate as \(\$26.40\)), or \(8 × \$32.50 = \$260.00\) (might get \(\$265.00\)), or the final addition \(\$260.00 + \$25.60 = \$285.60\) (might get \(\$286.40\)).

Errors while selecting the answer

1. Selecting a sub-optimal strategy: Even after calculating multiple approaches correctly, students might select the cost of 9 complete boxes (\(\$292.50\), choice D) thinking it's "simpler" or "safer" rather than choosing the true minimum cost of \(\$285.60\) from the mixed strategy.