Letter Corporation has two service departments $($A and B$)$ that provide service to each otherand to two operating departments $($X and Y$).$ A provides $20\%$ of its service to B$,30\%$ of its service to X$,$ and $50\%$ of its service to Y$.$ B provides $10\%$ of its service to A$,45\%$ of its service to X$,$ and $45\%$ of its service to Y$.$ Operating costs in A are $$300,000.$ Operating costs in B are $$180,000.$

Cost allocations are made starting with department A$.$ No distinction is made between variable and fixed costs.

a$.$ If service department costs are allocated using the direct method, what is the total amount of service cost that will be allocated to X$?$

b$.$ If service department costs are allocated using the step$-$down method, what is the total amount of service cost that will be allocated to Y$?$