Sure, let's go step by step to find the correct equation that models the situation where Duncan shares 20 percent of his sales commissions with his personal assistant.
1. Understanding the relationship:
- Duncan earns [tex]\( x \)[/tex] dollars in commissions.
- He gives [tex]\( y \)[/tex] dollars to his personal assistant.
- The problem states that [tex]\( y \)[/tex] is 20 percent of [tex]\( x \)[/tex].
2. Representing mathematically:
- 20 percent of [tex]\( x \)[/tex] can be written as [tex]\( 0.20 \times x \)[/tex].
- So, [tex]\( y = 0.20 \times x \)[/tex].
3. Analyzing the equation:
- The equation [tex]\( y = 0.20 \times x \)[/tex] correctly models the situation because it shows that [tex]\( y \)[/tex], the amount given to the assistant, is 20 percent of the total commissions [tex]\( x \)[/tex].
We need to check which of the options provided matches the equation [tex]\( y = 0.20 \times x \)[/tex].
- Option 1: [tex]\( x = \frac{y}{20} \)[/tex]
- This option does not match our derived equation because it suggests that [tex]\( x \)[/tex] is a fraction of [tex]\( y \)[/tex], not related by 20 percent.
- Option 2: [tex]\( y = 0.20 x \)[/tex]
- This option matches our derived equation perfectly. It shows [tex]\( y \)[/tex] as 20 percent of [tex]\( x \)[/tex].
- Option 3: [tex]\( x = 0.20 y \)[/tex]
- This option incorrectly suggests that the total commissions [tex]\( x \)[/tex] is 20 percent of the amount given to the assistant [tex]\( y \)[/tex], which is not the case.
- Option 4: [tex]\( y = \frac{x}{20} \)[/tex]
- This option also does not align with our derived equation. Dividing [tex]\( x \)[/tex] by 20 does not represent the relationship of 20 percent correctly.
Conclusion:
The correct equation is:
[tex]\[ y = 0.20 x \][/tex]
So, the correct answer is:
Option 2: [tex]\( y = 0.20 x \)[/tex].
