Answer 1) : The density of the hot air inside the balloon can be found out by using ideal gas equation;
PV = nRT;
As n is number of moles and in gases, number of moles along with mass per mole is equal to the density of the gas.
If the moles in the gas are more the density will be more.
here, density (ρ) = mass (m) / volume (V); substituting in the ideal gas equation we get,
ρ = mP / RT
Answer 2) ρ (hot air) = ρ (cold air) X
Here according to the formula because T(hot air) >T(cold air),
So, the density of hot air greater than the density of cold air.
The relationship between the ρ (h) = ρ(c) X