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.

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.