Marta and Carmela have capital balances of P$560,000$ and P$450,000$ respectively. Both decided to admit Sophia into their partnership. He invested enough cash to have a $20\%$ interest in the partnership. The profit and loss ratio of the old partners is $3$:$2$ respectively. After the admission of Sophia, the capital balance of Carmela amounted to $495,000.$

How much cash was invested by Sophia?