To determine the multiplicative rate of change for the given values in the table, let's analyze the changes in [tex]\( y \)[/tex] as [tex]\( x \)[/tex] increases.
The given values are:
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 2 \)[/tex]
- When [tex]\( x = 2 \)[/tex], [tex]\( y = \frac{2}{5} \)[/tex]
- When [tex]\( x = 3 \)[/tex], [tex]\( y = \frac{2}{25} \)[/tex]
- When [tex]\( x = 4 \)[/tex], [tex]\( y = \frac{2}{125} \)[/tex]
First, we'll calculate the rate of change between each pair of consecutive [tex]\( y \)[/tex]-values.
1. From [tex]\( y_1 = 2 \)[/tex] to [tex]\( y_2 = \frac{2}{5} \)[/tex]
[tex]\[ \text{Rate of Change}_1 = \frac{\frac{2}{5}}{2} = \frac{2}{5} \times \frac{1}{2} = \frac{2}{10} = 0.2 \][/tex]
2. From [tex]\( y_2 = \frac{2}{5} \)[/tex] to [tex]\( y_3 = \frac{2}{25} \)[/tex]
[tex]\[ \text{Rate of Change}_2 = \frac{\frac{2}{25}}{\frac{2}{5}} = \frac{2}{25} \times \frac{5}{2} = \frac{10}{50} = 0.2 \][/tex]
3. From [tex]\( y_3 = \frac{2}{25} \)[/tex] to [tex]\( y_4 = \frac{2}{125} \)[/tex]
[tex]\[ \text{Rate of Change}_3 = \frac{\frac{2}{125}}{\frac{2}{25}} = \frac{2}{125} \times \frac{25}{2} = \frac{50}{250} = 0.2 \][/tex]
Since the multiplicative rate of change is consistent and equal to [tex]\( 0.2 \)[/tex], we can now return to the provided options to match this rate of change:
[tex]\[
\frac{1}{5} = 0.2, \quad \frac{2}{5} = 0.4, \quad 2 = 2.0, \quad 5 = 5.0
\][/tex]
Therefore, the multiplicative rate of change is:
[tex]\[ \boxed{\frac{1}{5}} \][/tex]
