Question

BMB Inc. purchased two pumps for use at its nanofiltration water conditioning plant. The cost of the pumps was $950 each. If the chemical cost is $11 per day, assuming 30 days per month (i.e. 360 days a year) and a 3-year pump life,

1. a) What would be the future worth of total cost at the end of 3 years at an interest rate of 12% per annum compounded monthly?
2. b) What would be the future worth of total cost at the end of 3 years if the interest rate of 12% per annum compounded continuously?

Answer 1

Given in the question,

• Cost of the pump at the start of 3 year period = $950
  • there are two pumps
• Daily cost of chemical = $11
• Interest rate per annum = 12%

a) The costs are compounded monthly.

No. of months in a 3 year period = 36

Let's see at what point of time the costs are incurred,

• At the start of 3 year period, since there are two pumps, 2x$950 = $1900
• At the end of first month, there would be cost of chemical, (additionally incurred during the month), 30x$11 = $330
• At the end of second month, there would be cost of chemical, (additionally incurred during the month), 30x$11 = $330
• ....and so on till the 36th month

The formula for future cost (FC) in terms of present cost (PC) and interest rate (r) is :

Hence, summing all the present costs in bullet points above, total future cost (TFC)

Notice that :

• from the second term onwards, it forms a geometric series
• there are a total of 37 terms
• if annual interest rate is 12%, the monthly interest rate is 1%

Therefore,

b) The formula for continuous compounding is,

where r is the interest rate and t is the time period

Here, we take r to be the annual interest rate, and hence t will be accordingly calculated in years

The PC of chemicals for a year would be = 360 x $11 = $3960

Putting all the above information together,

Notice that we have clubbed the costs in a year to the start of the year, rather than at the end.