Question

An engineer has a fluctuating budget for the maintenance of a particular machine. During each of the first 5 years, $500 per year will be budgeted. During the second 5 years, $1000 per year will be budgeted. In addition, $2000 will be budgeted for an overhaul of the machine at the end of the fourth year, and again at the end of the eighth year. What unifrom annual expenditure would be equivalent, if interest rate is 10% per year?

Answer 1

First of all we need to calculate the Present Worth (PW) of all the costs as follows:-

According to question:-

For first 5 years $500 is budgeted

For second 5 years or we can say that for period( 6-10 year) $1000 is budgeted

Also at the end of 4th and 8th year an overhaul cost is budgeted which is $2000

So we have present worth as follows:-

P = 500 ( P/A,10%,5) + 2000( P/F,10%,4) + 1000(P/A,10%,5) * ( P/F,10%,5) + 2000( P/F,10%,8)

Note:-

The formula for ( P/A,i%,n) is :-

{ ( 1+i )ⁿ - 1 } / { i ( 1+i )ⁿ }

And the formula for ( P/F,i%,n) is:-

( 1+i )^-n

Calculating the above equation using the formulas we have:-

P = 500 ( P/A,10%,5) + 2000( P/F,10%,4) + 1000(P/A,10%,5) * ( P/F,10%,5) + 2000( P/F,10%,8)

P = 500 ( 3.78 ) + 2000 ( 0.683) + 1000(3.78) * ( 0.62) + 2000(0.466)

P= 1890+ 1366 + 2343.6 + 932

= 6531.6

Equivalent uniform annual amount = $6531.6 ( A/P,10%,10)

Note = The formula for ( A/P,i%,n) is:-

{ i(1+i)ⁿ } / { (1+i)ⁿ - 1 }

Applying formula and equating value above we get:-

Equivalent uniform annual amount = $6531.6 ( A/P,10%,10)

= $6531.6 ( 0.162 )

= $ 1058.12