P(getting admitted to A $\cup$ getting admitted to B)=P(getting admitted to A)+P(getting admitted to B)-P(getting admitted to A $\cap$ getting admitted to B)=$1-\frac{3}{5} \times \frac{2}{5} =\frac{19}{25}$ . This quantity has to deducted as it is being double counted in the sample space.
Therefore, P(getting admitted to A given he is admitted to atleast one of the colleges)=$\frac{\frac{3}{5}}{\frac{19}{25}}=\frac{3}{5} \times \frac{25}{19}=\frac{15}{19}$