16. Dakota purchased a doormat measuring $$ \frac{1}{2} $$ yard by $$ \frac{2}{3} $$ yard for the back door step. If the step is $$ \frac{1}{4} $$ square yard, will the mat fit? Explain.

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To determine if the doormat will fit on the step, we need to calculate the area of the doormat and compare it to the area of the step.

1. **Calculate the area of the doormat:**
   The area $$ A $$ of a rectangle is given by the formula:
   $$
   A = \text{length} \times \text{width}
   $$
   Here, the length of the doormat is $$ \frac{1}{2} $$ yard and the width is $$ \frac{2}{3} $$ yard. Therefore, the area of the doormat is:
   $$
   A_d = \frac{1}{2} \times \frac{2}{3}
   $$

2. **Calculate the area of the step:**
   The area of the step is given as $$ \frac{1}{4} $$ square yard.

3. **Compare the areas:**
   We will check if the area of the doormat $$ A_d $$ is less than or equal to the area of the step.

Let's perform the calculations step by step.

First, we will calculate the area of the doormat.

$$
A_d = \frac{1}{2} \times \frac{2}{3}
$$

Now, let's calculate this.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{1}{2}\times \frac{2}{3}$$
- step1: Reduce the numbers:
  $$1\times \frac{1}{3}$$
- step2: Multiply:
  $$\frac{1}{3}$$
The area of the doormat is calculated as follows:

$$
A_d = \frac{1}{2} \times \frac{2}{3} = \frac{1}{3} \text{ square yard}
$$

Now, we compare this with the area of the step, which is $$ \frac{1}{4} $$ square yard.

4. **Comparison of areas:**
   We need to check if $$ \frac{1}{3} $$ is less than or equal to $$ \frac{1}{4} $$.

To compare these fractions, we can convert them to a common denominator or compare their decimal values.

- The decimal value of $$ \frac{1}{3} $$ is approximately $$ 0.33 $$.
- The decimal value of $$ \frac{1}{4} $$ is $$ 0.25 $$.

Since $$ 0.33 > 0.25 $$, we conclude that:

$$
\frac{1}{3} > \frac{1}{4}
$$

5. **Conclusion:**
The area of the doormat $$ \frac{1}{3} $$ square yard is greater than the area of the step $$ \frac{1}{4} $$ square yard. Therefore, the doormat will not fit on the step.
The doormat does not fit on the step because its area ($$ \frac{1}{3} $$ square yard) is larger than the step's area ($$ \frac{1}{4} $$ square yard).

16. Dakota purchased a doormat measuring \( \frac{1}{2} \) yard by \( \frac{2}{3} \) yard for the back door step. If the step is \( \frac{1}{4} \) square yard, will the mat fit? Explain.

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