A company is conducting a biochemical experiment for the next 12 months. In the first month, the expenses are estimated to be $30,000. As the experiment progresses, the expenses are expected to increase by 5 percent each month. The company plans to pay for the experiment with a government grant, which is received in six monthly installments, starting a month after the experiment after the experiment completion date. Determine the amount of monthly installment so that the total the total of the six installments pays for all expenses incurred during the experiment. Annual nominal interest rate is 12 percent, compounded monthly.

Question

A company is conducting a biochemical experiment for the next 12 months. In the first month, the expenses are estimated to be $30,000. As the experiment progresses, the expenses are expected to increase by 5 percent each month. The company plans to pay for the experiment with a government grant, which is received in six monthly installments, starting a month after the experiment after the experiment completion date. Determine the amount of monthly installment so that the total the total of the six installments pays for all expenses incurred during the experiment. Annual nominal interest rate is 12 percent, compounded monthly.

Answer 1

To determine the amount of the monthly installment required to cover all expenses incurred during the 12-month biochemical experiment, we can break down the problem into smaller steps.

Step 1: Calculate the total expenses for the 12-month experiment.
The expenses for each month increase by 5 percent. We can calculate the expenses using the formula for compound interest:

P = P0 (1 + r)^n

Where:
P0 = Initial expense ($30,000)
r = Monthly interest rate (5% or 0.05)
n = Number of months (12)

Using the formula, we can calculate the total expenses for the 12-month experiment:

P = 30,000 (1 + 0.05)^12
P ≈ $46,325.93

So, the total expenses for the 12-month experiment is approximately $46,325.93.

Step 2: Calculate the future value of the six monthly installments.
To calculate the future value of the six monthly installments, we can use the compound interest formula again. However, since the payments are made at the end of each month, we need to adjust the formula slightly:

FV = P × [(1 + r)^n - 1] / r

Where:
FV = Future value (total of six installments)
P = Monthly installment amount
r = Monthly interest rate (12% or 0.12/12 as it is compounded monthly)
n = Number of installments (6)

We want the future value to be equal to the total expenses incurred during the experiment, which is $46,325.93. So, we can set up the equation:

46,325.93 = P × [(1 + 0.01)^6 - 1] / (0.12/12)

Simplifying the equation, we find:

P × [(1.01)^6 - 1] / (0.12/12) = 46,325.93

Step 3: Solve the equation for the monthly installment amount.
To solve for P, we can rearrange the equation:

P = 46,325.93 × (0.12/12) / [(1.01)^6 - 1]
P ≈ $7,273.49

So, the monthly installment amount needed to cover all expenses incurred during the experiment is approximately $7,273.49.