By installing some elaborate inspection equipment
on its assembly line, the Robot Corp. can avoid hir-
ing an extra worker who would have earned $36,000
a year in wages and an additional $9500 a year in
employee benefits. The inspection equipment has a
6-year useful life and no salvage value. Use a nom-
inal 18% interest rate in your calculations. How
much can Robot afford to pay for the equipment if
the wages and worker benefits were to have been
paid
(a) At the end of each year
(b) Monthly
Explain why the answer in (b) is larger.
I need help with finding the present worth.
The formula I have is:
36000(P/A,18%,6) + 9,500(P/A,18%,6)
Am I going in the right direction??
Why did the robot go broke?
Because it had to hire an expensive inspector-bot instead of a regular worker!
But let's calculate how much it can afford to pay for the inspection equipment.
(a) To find the present worth using the formula you provided, we'll use the annual payment factor (P/A). So it's:
36000(P/A,18%,6) + 9500(P/A,18%,6)
(b) Now, if we were to pay the wages and benefits monthly, the formula would change to use the monthly payment factor (P/G). So it would be:
36000(P/G,18%/12,6*12) + 9500(P/G,18%/12,6*12)
Now, why is the answer in (b) larger? Because when we break down the annual payments into monthly payments, the interest rate is divided by 12 (since there are 12 months in a year). This reduces the interest rate for each monthly payment, making them smaller compared to annual payments. So, the present worth of the monthly payments is higher.
Keep in mind that this calculation assumes an 18% nominal interest rate and no salvage value for the inspection equipment.
Yes, you're going in the right direction. To calculate the present worth, you need to determine the cost savings from not hiring an additional worker, which includes both wages and employee benefits. Given that the wages are $36,000 per year and employee benefits are $9,500 per year, you have correctly identified the formula to calculate the present worth using the Present Worth of an Annuity factor (P/A).
To find the present worth, you need to multiply each of the annual costs by the respective Present Worth of an Annuity factor for a 6-year period at an interest rate of 18%.
(a) If the wages and worker benefits were paid at the end of each year, the formula would be:
Present Worth = $36,000 (P/A, 18%, 6) + $9,500 (P/A, 18%, 6)
(b) If the wages and worker benefits were paid monthly, you would need to adjust the formula to calculate the present worth on a monthly basis. The formula would be:
Present Worth = ($36,000/12) (P/A, 18%/12, 6*12) + ($9,500/12) (P/A, 18%/12, 6*12)
In this case, the answer in (b) would be larger because you are dividing the annual costs by 12 to convert them into monthly costs. As a result, you are calculating the present worth at a more frequent interval (monthly) compared to the annual interval in (a). Therefore, the present worth will be larger when calculated on a monthly basis.
Yes, you are on the right track to find the present worth of the inspection equipment. The formula you have mentioned is correct for finding the present worth or the equivalent annual cost of the equipment. Let's break it down step by step:
The present worth formula you mentioned, 36000(P/A,18%,6) + 9,500(P/A,18%,6), is used to calculate the equivalent annual cost by considering the wages and employee benefits saved each year for a period of 6 years.
To find the present worth when the wages and worker benefits are paid at the end of each year, we need to calculate the present worth factor (P/A,18%,6) first. The present worth factor is calculated using the formula:
P/A, i, n = (1 - (1 + i)^(-n)) / i
Here:
- P/A, i, n represents the present worth factor with interest rate 'i' and period 'n'.
- 'i' is the nominal interest rate, which is given as 18%, but we need to convert it to a decimal form (i = 0.18).
- 'n' is the number of periods, which in this case is 6 years.
By substituting values into the formula, we can calculate the present worth factor for 6 years and an interest rate of 18%:
P/A, 0.18, 6 = (1 - (1 + 0.18)^(-6)) / 0.18
Simplifying the equation will give you the value of the present worth factor.
Once you have the present worth factor, you can calculate the present worth or the equivalent annual cost by multiplying each annual cost (wages and employee benefits) by the present worth factor and summing them up:
Present Worth = 36000(P/A,18%,6) + 9,500(P/A,18%,6)
To calculate the present worth when the wages and worker benefits are paid monthly, you need to use a similar approach. However, instead of calculating the present worth factor for 6 years, you would need to adjust the present worth factor for monthly payments over 6 years.
The reason the answer in part (b) is larger when wages and benefits are paid monthly is because the present worth factor for monthly payments is greater than the present worth factor for annual payments. This is due to the compounding effect of monthly payments, which allows for more frequent savings on wages and benefits.
To find the present worth when wages and benefits are paid monthly, you would calculate the present worth factor for monthly payments using the formula:
P/A, i, n = (1 - (1 + i)^(-12*n))/(i*12)
Here, all the variables have the same meaning, except 'n' represents the number of years. By substituting the values and calculating the present worth factor, you can then proceed to calculate the present worth using the same formula as before but with the updated present worth factor.
I hope this explanation helps you in finding the present worth for both scenarios.