Two rods,one of aluminum and another of steel,have initial lengths $ℓ_1$ and $ℓ_2$ respectively. They are joined to form a single rod of length $ℓ_1 + ℓ_2$. The coefficients of linear expansion for aluminum and steel are $\alpha_a$ and $\alpha_s$ respectively. If the increase in length of both rods is the same when the temperature is increased by $t \ ^\circ C$,find the ratio $\frac{ℓ_1}{ℓ_1 + ℓ_2}$.

• A
  $\frac{\alpha_s}{\alpha_a}$
• B
  $\frac{\alpha_a}{\alpha_s}$
• C
  $\frac{\alpha_s}{\alpha_a + \alpha_s}$
• D
  $\frac{\alpha_a}{\alpha_a + \alpha_s}$