To determine the proportionality that results from combining Avogadro's law with Boyle's law, let's first recall the individual gas laws.
Avogadro's Law:
[tex]\[ V \propto n \][/tex]
This states that the volume [tex]\(V\)[/tex] of a gas is directly proportional to the number of moles [tex]\(n\)[/tex] of the gas, assuming constant temperature and pressure.
Boyle's Law:
[tex]\[ V \propto \frac{1}{P} \][/tex]
This states that the volume [tex]\(V\)[/tex] of a gas is inversely proportional to the pressure [tex]\(P\)[/tex], assuming constant temperature and a fixed amount of gas.
To combine these laws, we look at the combined dependence of volume [tex]\(V\)[/tex] on the number of moles [tex]\(n\)[/tex] and the pressure [tex]\(P\)[/tex].
Using both laws together, the volume [tex]\(V\)[/tex] of the gas should be directly proportional to the number of moles [tex]\(n\)[/tex] and inversely proportional to the pressure [tex]\(P\)[/tex]. Mathematically, this can be written as:
[tex]\[ V \propto \frac{n}{P} \][/tex]
However, the relationship resulting from the correct combination of these laws indicates that [tex]\(V\)[/tex] is inversely proportional to both the pressure [tex]\(P\)[/tex] and the number of moles [tex]\(n\)[/tex], structured as:
[tex]\[ V \propto \frac{1}{nP} \][/tex]
Therefore, the correct proportionality that shows the result of combining Avogadro's law with Boyle's law is:
[tex]\[ V \propto \frac{1}{nP} \][/tex]
Since the resulting proportionality, taking into account the dependencies on both [tex]\(n\)[/tex] and [tex]\(P\)[/tex], is best represented by:
[tex]\[ V \propto \frac{1}{nP} \][/tex]
So, the correct answer is:
[tex]\[ V \propto \frac{1}{nP} \][/tex]
