Thermal Expansion Aluminium Problem

[smashbrohamme]

A brass ring of diameter 10.00 cm at 19.9°C is heated and slipped over an aluminum rod of diameter 10.01 cm at 19.9°C. Assume the average coefficients of linear expansion are constant.
(a) To what temperature must the combination be cooled to separate the two metals?



Brass=19 x 10^-6
Aluminum=24 x 10^-6

(Change)A=γ x Ao x (change)Temperature.

This is the equation for one instance.

What is a conveniant equation to solve for both at the same time? I am stuck and I feel if I keep using that top equation I will just be guessing all day.

 

[smashbrohamme]

I came across a equation that works, but I am not sure with the theory behind it..maybe someone can explain it for me.

T= Initial temperature - (10.00-10.01)/[(10x19-10.01x24)10^-6]

Proofing equations is probably my weakest area in academia.