URGENT ! PLEASE SOLVE BELOW CHEMISTRY QUESTION. Answer is 1 atmosphere?

Suppose that gas M in a flask with volume Vₘ, and gas N in a flask with volume Vₙ.

For a fixed amount of gas at constant temperature, P₁V₁ = P₂V₂
Then, P₂ = P₁ × (V₁/V₂)

After the two flasks are connected :
Partial pressure of gas M = (1 atm) × [Vₘ/(Vₘ + Vₙ)] = Vₘ/(Vₘ + Vₙ) atm
Partial pressure of gas N = (1 atm) × [Vₙ/(Vₘ + Vₙ)] = Vₙ/(Vₘ + Vₙ) atm

Total pressure after the two flasks are connected
= (Partial pressure of gas M) + (Partial pressure of gas N)
= [Vₘ/(Vₘ + Vₙ)] + [Vₙ/(Vₘ + Vₙ)] atm
= (Vₘ + Vₙ)/(Vₘ + Vₙ)] atm
= 1 atm

let's use an example
let's say we have

flask 1
.. V = 1L
.. P = 1atm
.. T = 298.15K
.. n = PV/(RT) = 0.04087 moles

flask 2
.. V = 2L
.. P = 1atm
.. T = 298.15K
.. n = PV/(RT) = 0.08174 moles

when you combine... assuming T remains constant !!!... then
.. V = 3 L
.. T = 298.15K
.. n = 0.04087 + 0.08174 = 0.12262 mol
.. P = nRT/V = 1atm

*********
why does that work?
.. gas 1... P1V1 = n1RT1
.. gas 2... P2V2 = n2RT2
.. combined.. V3 = V1 + V2.... n3 = n2 + n1.... T3 = T2 = T1 = T.... (let's drop the 1,2,3 for simplicity)

so we can write
.. n1 = P1V1 / (RT)
.. n2 = P2V2 / (RT)
.. n3 = P1V1 / (RT) + P2V2 / (RT)

then
.. P3 = n3 R T / V3 = n3 R T / (V1 + V2)
.. P3 = [P1V1 / (RT) + P2V2 / (RT)] * R * T / (V1 + V2)

then R's and T's cancel leaving
.. P3 = (P1V1 + P2V2) / (V1 + V2)

since P1 = P2 = 1atm...
.. P3 = P1 * (V1 + V2) / (V1 + V2)
.. P3 = P1 = P2 = 1atm

lkag