# 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:

Department | Overheads |
---|---|

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:

Department | P | Q | R | X | Y |
---|---|---|---|---|---|

X | 20% | 25% | 25% | — | 10% |

Y | 25% | 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 |
---|---|---|---|

Initial Overheads | 35,000 | 64,000 | 19,000 |

Dept. X Reallocation | 5,624 | 7,030 | 7,030 |

Dept. Y Reallocation | 10,203 | 12,244 | 12,244 |

Total Overheads | 50,827 | 83,274 | 38,274 |

Written by Irfanullah Jan and last modified on