Question

7 - 74 three mutually exclusive alternatives are being considered. all have a 10 - year useful life. if the marr is 25%, which alternative is preferred? a initial cost $40,000 uniform annual benefit $10,000 b initial cost $50,000 uniform annual benefit $13,000 c initial cost $55,000 uniform annual benefit $14,000

Explanation:

Step1: Recall Benefit - Cost Ratio formula

For a project with initial cost $P$ and uniform - annual benefit $A$ over $n$ years at a minimum - acceptable rate of return (MARR) $i$, the Benefit - Cost Ratio (BCR) is given by $BCR=\frac{A(P/A,i,n)}{P}$, where $(P/A,i,n)=\frac{(1 + i)^n-1}{i(1 + i)^n}$. Here, $n = 10$ years and $i=0.25$. Calculate $(P/A,0.25,10)=\frac{(1 + 0.25)^{10}-1}{0.25\times(1 + 0.25)^{10}}=\frac{9.313225746 - 1}{0.25\times9.313225746}=\frac{8.313225746}{2.328306437}\approx3.57$.

Step2: Calculate BCR for each alternative

Alternative A

$P = 40000$, $A = 10000$. $BCR_A=\frac{10000\times3.57}{40000}=\frac{35700}{40000}=0.8925$.

Alternative B

$P = 50000$, $A = 13000$. $BCR_B=\frac{13000\times3.57}{50000}=\frac{46410}{50000}=0.9282$.

Alternative C

$P = 55000$, $A = 14000$. $BCR_C=\frac{14000\times3.57}{55000}=\frac{49980}{55000}\approx0.9087$.

Step3: Select the best alternative

Since the alternatives are mutually - exclusive, we choose the alternative with the highest BCR. But in this case, all BCRs are less than 1. However, among them, $BCR_B$ is the highest.

Answer:

B. Alternative B