To find the area of each tile using the given fractional side lengths, we'll first convert each fraction to its decimal equivalent and then use these values to calculate the area of the tiles.

Step 1: Convert Fractional Lengths to Decimal Numbers

1. Convert $\frac{31}{8}$ to a decimal: $\frac{31}{8} = 31 \div 8 = 3.875 \text{ inches}$

2. Convert $3 \frac{3}{16}$ to a decimal:

   • First, convert the mixed number to an improper fraction: $3 \frac{3}{16} = \frac{3 \times 16 + 3}{16} = \frac{48 + 3}{16} = \frac{51}{16}$
   • Then, convert to a decimal: $\frac{51}{16} = 51 \div 16 = 3.1875 \text{ inches}$

3. Convert $2 \frac{1}{4}$ to a decimal:

   • First, convert the mixed number to an improper fraction: $2 \frac{1}{4} = \frac{2 \times 4 + 1}{4} = \frac{8 + 1}{4} = \frac{9}{4}$
   • Then, convert to a decimal: $\frac{9}{4} = 9 \div 4 = 2.25 \text{ inches}$

Step 2: Calculate the Area of Each Tile

We'll assume each given length represents the side of a square tile. The area $A$ of a square is given by: $A = \text{side} \times \text{side}$

1. Area of the first tile with side length $\frac{31}{8}$: $A = 3.875 \times 3.875 = 15.015625 \text{ square inches}$

2. Area of the second tile with side length $3 \frac{3}{16}$: $A = 3.1875 \times 3.1875 = 10.16015625 \text{ square inches}$

3. Area of the third tile with side length $2 \frac{1}{4}$: $A = 2.25 \times 2.25 = 5.0625 \text{ square inches}$

Summary of Results

• First tile area: $15.015625$ square inches
• Second tile area: $10.16015625$ square inches
• Third tile area: $5.0625$ square inches

These areas are calculated using the converted decimal side lengths.

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Would you like to discuss any of these calculations in more detail, or do you have any additional questions?

8 Follow-Up Questions

1. How would you convert a complex fraction like $\frac{47}{32}$ into a decimal?
2. What is the difference in area if the length is rounded up or down to the nearest $\frac{1}{16}$ inch?
3. How do you calculate the perimeter of each tile using the side lengths?
4. If a rectangular tile had these dimensions, how would you calculate the area?
5. What if Warren used a different unit, such as centimeters? How would that change the calculations?
6. Can you express the calculated areas back in fractional form?
7. How accurate are these area calculations for practical tiling work?
8. What are the implications if Warren incorrectly converts a length to a decimal?

Tip

When performing arithmetic operations with decimals derived from fractions, consider using extra decimal places for intermediate steps to maintain accuracy in your final results.