Question $3$: $(20$ marks$)$ A consulting actuary aged $48,$ in business on his own account, agrees to sell a share of his practice to two employees aged $35$ and $30.$ After the sale, $(35)$ and $(30)$ will own $25\%$ and $20\%$ respectively, $(48)$ owning the balance. $(48)$ plans to retire at age $65$ and the agreement provides that at that time, or on his earlier death, the other surviving partners will buy his share by increasing their own shares to $50\%$ each or$,$ if only one survives $(48),$ the survivor will increase his share to $100\%.$ In the event of one of $(35)$ and $(30)$ dying before attaining age $65,$ his share will be purchased by the other. $(48)$ will repurchase their shares if both $(35)$ and $(30)$ die before he retires. $(35)$ plans to retire when he attains age $65$ and his share at that time will be purchased by $(30)$ if alive. The initial transaction values the business at $250,000$ and subsequent transactions are to be based on the same figure. $(30)$ decides to effect a policy to provide the necessary money that he will require on death or retirement of $(35)$ and $(48)$ to purchase additional shares in the business. Derive an expression in terms of whole life contingent assurances, payable on the first death, and commutation functions for the net single premium payable.