The Richards Company purchased a machine for $5,000 down and $300 a month payable at the end of each of the next 36 months. How would the cash price of the machine be calculated, assuming the annual interest rate is given?

The cash price would be calculated by adding the down payment to the present value of an annuity formula.

Explanation

To calculate the full cash value of the machine, we need to use the present value of an annuity formula:

P = A (1 - (1 + i) ^-n) / i

Where:

P = Present value A = Value of the annuity i = interest rate n = number of compounding periods

In this case P is the full cash price of the machine, A is the annuity or the $300 monthly payment, i is the interest rate that we are given, and n is equal to 36 months:

P = 300 (1 - (1 + i) ^-n) / i

The only value that we have not plugged in yet is the down payment. This payment is added to the formula before the annuity value, because it is an additional payment that needs to be taken into account when calculating for the full cash price:

P = 5,000 + 300 (1 - (1 + i) ^-n) / i