To determine the correct equation that models the relationship between the hours worked by Amir and Mitul, we must analyze the given ratio statement: "Amir works 15 hours for every 19 hours that Mitul works."
Let:
- [tex]\( x \)[/tex] represent the number of hours that Mitul works.
- [tex]\( y \)[/tex] represent the number of hours that Amir works.
The relationship between Amir's and Mitul's working hours can be expressed as a ratio or proportion. Specifically, Amir works 15 hours for every 19 hours that Mitul works, which can be set up as:
[tex]\[ \frac{y}{x} = \frac{15}{19} \][/tex]
To transform this proportion into an equation, we can multiply both sides by [tex]\( x \)[/tex] to solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{15}{19} x
\][/tex]
So, the equation that correctly models the relationship between the hours worked by Amir ([tex]\( y \)[/tex]) and Mitul ([tex]\( x \)[/tex]) is:
[tex]\[ y = \frac{15}{19} x \][/tex]
Therefore, the correct choice is the first option:
[tex]\[ y = \frac{15}{19} x \][/tex]
