To help Warren measure the tiles accurately using his tape measure, we'll need to convert the decimal side lengths of the tiles to fractions with a denominator of 16 (i.e., to the nearest $\frac{1}{16}$ inch). After that, we'll calculate the area of the tiles using these fractional side lengths.

Step 1: Converting Decimal Values to Fractions

Given a decimal value, the first task is to convert it to a fraction with a denominator of 16. The procedure involves the following steps:

1. Multiply the decimal value by 16 to find the numerator.
2. Round the result to the nearest whole number.
3. Express this as a fraction over 16.

For example, if you have a decimal value of $0.625$:

• Multiply by 16: $0.625 \times 16 = 10$
• Round to the nearest whole number: $10$ (which is already a whole number)
• The fraction is $\frac{10}{16}$, which can be simplified to $\frac{5}{8}$.

Step 2: Calculating the Area

Once you have the side lengths in fractions, you can calculate the area of a tile as follows:

• If the tile is square, the area $A$ is given by: $A = \text{side} \times \text{side}$
• If the tile is rectangular, the area $A$ is given by: $A = \text{length} \times \text{width}$

Example Problem

Let's say Warren measures a tile and finds its side length to be $2.75$ inches.

1. Convert the decimal to a fraction:

   • Multiply by 16: $2.75 \times 16 = 44$
   • Fraction: $\frac{44}{16}$, which simplifies to $\frac{11}{4}$ inches.

2. Calculate the area:

   • If it's a square tile: $A = \left(\frac{11}{4}\right) \times \left(\frac{11}{4}\right) = \frac{121}{16} \text{ square inches.}$
   • If it's a rectangular tile with another side length measured as $1.5$ inches: $A = \left(\frac{11}{4}\right) \times \left(\frac{24}{16}\right) = \frac{11 \times 24}{4 \times 16} = \frac{264}{64} = \frac{33}{8} \text{ square inches.}$

This procedure will allow Warren to work with his tape measure effectively by converting any decimal values he encounters to a fractional form compatible with the tape's markings.

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Would you like any further details on this process or have any specific decimal values you need to convert?

8 Follow-Up Questions

1. What are some common decimal values that Warren might encounter and need to convert?
2. How would you handle decimals that are exactly halfway between two sixteenths (e.g., 0.4375)?
3. What if a decimal rounds to a fraction greater than $\frac{16}{16}$?
4. How can Warren simplify fractions after conversion?
5. What is the area of a square tile with a decimal side length of 3.125 inches?
6. How does the process change if the tape measure was marked in $\frac{1}{8}$ inches instead?
7. Can the area calculation method be used for any polygonal shape?
8. What are the implications of rounding errors when calculating the area?

Tip

When converting decimals to fractions for measurements, always double-check your rounding to ensure the accuracy of your final measurement.