Two metal spheres having capacitances C_1 and C_2 hold some initial...

When two conducting spheres are brought into contact, charge flows between them until their electrostatic potentials become equal.

Let the common potential of the spheres after contact be $V$.

The final charge on the first sphere is $Q_1 = C_1 V$.

The final charge on the second sphere is $Q_2 = C_2 V$.

Taking the ratio of the final charges, we get:

$\dfrac{Q_1}{Q_2} = \dfrac{C_1 V}{C_2 V} = \dfrac{C_1}{C_2}$