# Simultaneous Equation Method

In simultaneous equation method of allocation of service department costs, we establish simultaneous equations and solve them to obtain the final balances of production departments. This method accurately allocates service department costs in the given percentages.

Simultaneous equation method is best explained using an example.

## Example

In this example we use simultaneous equation method to solve the same problem we solved earlier using repeated distribution method. Following is the problem:

γ ltd. has three production departments (P, Q and R) and two service departments (X and Y). The total overheads for the departments are given below:

P\$35,000
Q\$64,000
R\$19,000
X\$22,000
Y\$38,000

The reallocation percentages of the service departments' costs are given below:

DepartmentPQRXY
X20%25%25%10%
Y25%30%30%15%

Use the simultaneous equation method to allocate the service department overheads to production departments.

Solution

Let,

 x = total overheads of department X after reallocation y = total overheads of department Y after reallocation

Then total overhead of department X will be 22,000 + 15% of department Y overhead after reallocation whereas the total overhead of department Y will be 38,000 + 10% of department X overhead after reallocation. Therefore,

 x = 22,000 + 0.15y y = 38,000 + 0.10x

Solving the above equations for x and y we get:

 Total overheads of department X after reallocation = x ≈ 28,122 Total overheads of department Y after reallocation = y ≈ 40,812

The total overheads as calculated above are allocated to production departments in specified percentages as shown below:

Department P Q R
Dept. X Reallocation 5,624 7,030 7,030
Dept. Y Reallocation 10,203 12,244 12,244