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You run a school in Florida. Fixed monthly cost is $5,772.00 for rent and utilities, $5,550.00 is spent in salaries and $1,700.00 in insurance. Also every student adds up to $93.00 per month in stationary, food etc. You charge $728.00 per month from every student now.

You are considering moving the school to another neighborhood where the rent and utilities will increase to $10,363.00, salaries to $6,307.00 and insurance to $2,074.00 per month. Variable cost per student will increase up to $195.00 per month. However you can charge $1,148.00 per student. At what point will you be indifferent between your current mode of operation and the new option?

1 answer

To determine the point at which you would be indifferent between your current operation and the new option, we need to calculate the break-even point in terms of the number of students for both scenarios.

Current Operation Costs:

  1. Fixed Costs:

    • Rent and Utilities: $5,772
    • Salaries: $5,550
    • Insurance: $1,700
      Total Fixed Costs: \[ 5,772 + 5,550 + 1,700 = 13,022 \]

  2. Variable Costs per Student:

    • Stationary, Food, etc.: $93

  3. Total Revenue per Student:

    • Charge per Student: $728

New Operation Costs:

  1. Fixed Costs:

    • Rent and Utilities: $10,363
    • Salaries: $6,307
    • Insurance: $2,074
      Total Fixed Costs: \[ 10,363 + 6,307 + 2,074 = 18,744 \]

  2. Variable Costs per Student:

    • Stationary, Food, etc.: $195

  3. Total Revenue per Student:

    • Charge per Student: $1,148

Formulation:

  • Let \( n \) be the number of students.

For the current operation:

  • Total Cost: \[ \text{Total Cost}_{\text{current}} = 13,022 + 93n \]
  • Total Revenue: \[ \text{Total Revenue}_{\text{current}} = 728n \]

For the new operation:

  • Total Cost: \[ \text{Total Cost}_{\text{new}} = 18,744 + 195n \]
  • Total Revenue: \[ \text{Total Revenue}_{\text{new}} = 1,148n \]

Indifference Point:

We need to find the point at which Total Revenue equals Total Cost for both operations:

\[ 728n = 13,022 + 93n \] \[ 1,148n = 18,744 + 195n \]

Solve for \( n \)

  1. Current Operation: \[ 728n - 93n = 13,022 \] \[ 635n = 13,022 \] \[ n = \frac{13,022}{635} \approx 20.50 \quad \text{(students)} \]

  2. New Operation: \[ 1,148n - 195n = 18,744 \] \[ 953n = 18,744 \] \[ n = \frac{18,744}{953} \approx 19.66 \quad \text{(students)} \]

Indifference Point:

To find the point where revenues equal costs under both operations, we can set the two total revenue equations equal:

\[ 728n = 13,022 + 93n \] \[ 1,148n = 18,744 + 195n \]

Combine to find the number of students, we solve: \[ 728n - 93n = 13,022 \] \[ 1,148n - 195n = 18,744 \]

Conclusion:

You will be indifferent between your current operation and the new option when you have roughly 20.50 students under the current operation, and about 19.66 students under the new operation.

Since the number of students must be a whole number, you would need at least 21 students to favor the current option and 20 students for the new option.

In practice, you would need to have about 20 students in total at both models to make it work close to an indifference point.