The alternative to be selected.

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Answer 1

Question

Chapter 9, Problem 30P

To determine

The alternative to be selected.

Expert Solution & Answer

Answer to Problem 30P

The alternative to be selected is Alternative $B$ .

Explanation of Solution

Given:

MARR is $10%$ .

Useful life is $5$ years.

Calculation:

Write the expression to calculate the present worth of the benefit.

$P=A((1+i)n−1i(1+i)n)$ ....... (I)

Here, the annual benefit is $P$ , the present worth is $P$ , the rate of interest is $i$ and the number of period is $n$ .

Calculate the present worth benefit of the alternative $A$ .

Substitute $$158.3$ for $A$ , $10%$ for $i$ and $5$ for $n$ in Equation (I).

$P=$158.3((1+0.10)5−10.10(1+0.10)5)=$158.3(3.79)=$600$

Calculate the present worth benefit of the alternative $B$

Substitute $$138.7$ for $A$ , $10%$ for $i$ and $5$ for $n$ in Equation (I).

$P=$138.7((1+0.10)5−10.10(1+0.10)5)=$138.7(3.79)=$525.673$

Calculate the present worth benefit of the alternative $C$

Substitute $$58.3$ for $A$ , $10%$ for $i$ and $5$ for $n$ in Equation (I).

$P=$58.3((1+0.10)5−10.10(1+0.10)5)=$58.3(3.79)=$220.957$

Write the expression to calculate the benefit-cost ratio.

$B/C=BC$ ....... (II)

Here, the present worth benefit is $B$ , the present worth cost is $C$ .

For Alternative $A$ .

Substitute $$599.957$ for $B$ , $$600$ for $C$ , Equation (I).

$B/C=$600$600=1$

For Alternative $B$ .

Substitute $$525.673$ for $B$ , $$500$ for $C$ , Equation (I).

$B/C=$525.63$500=1.05126$

For Alternative $C$ .

Substitute $$220.957$ for $B$ , $$200$ for $C$ , Equation (I).

$B/C=$220.957$200=1.104$

Since all the ratios are positive hence we need to make comparison be increment method.

Calculate the $ΔCOST$ for alternative $B−C$ .

$ΔCost=(PWC−PWB)$ ...... (III)

Here the incremental cost between the alternative $B$ and $C$ is $ΔCost$ , the present worth of cost for alternative $C$ is $PWC$ and the present worth of cost for alternative $B$ is $PWB$ .

Substitute $$200$ for $PWC$ and $$500$ for $PWB$ in Equation (III).

$ΔCost=($200−$500)=−$300$

Calculate the $ΔBenefit$ for alternative $B−C$ .

$ΔBenefit=(PC−PB)$ ...... (IV)

Here the incremental benefit between the alternative $B$ and $C$ is $ΔBenefit$ , the present worth of benefit for alternative $C$ is $PC$ and the present worth of benefit for alternative $B$ is $PB$ .

Substitute $$220.957$ for $PC$ and $$525.673$ for $PB$ in Equation (IV).

$ΔBenefit=$220.957−$525.673=−304.716$

Calculate the incremental benefit-cost ratio

$Δ(B/C)=ΔBenefitΔCost$ ...... (V)

Substitute $−$304.716$ for and $−$300$ for $ΔCost$ in Equation (V).

$Δ(B/C)=−$304.716−$300=1.0572$

Since, $Δ(B/C)$ ratio greater than $1$ therefore alternative $B$ is desirable.

Calculate the $ΔCOST$ for alternative $A−B$

$ΔCost=(PWA−PWB)$ ...... (VI)

Here the incremental cost between the alternative $B$ and $A$ is $ΔCost$ , the present worth of cost for alternative $B$ is $PWB$ and the present worth of cost for alternative $A$ is $PWA$ .

Substitute $$200$ for $PWA$ and $$500$ for $PWB$ in Equation (III).

$ΔCost=($600−$500)=$100$

Calculate the $ΔBenefit$ for alternative $A−B$

$ΔBenefit=(PA−PB)$ ...... (VII)

Here the incremental benefit between the alternative $B$ and $A$ is $ΔBenefit$ , the present worth of benefit for alternative $A$ is $PA$ and the present worth of benefit for alternative $B$ is $PB$ .

Substitute $$600$ for $PA$ and $$525.673$ for $PB$ in Equation (IV).

$ΔBenefit=$600−$525.673=$74.327$

Calculate the incremental benefit-cost ratio

$Δ(B/C)=ΔBenefitΔCost$ ...... (V)

Substitute $−$74.327$ for $ΔBenefit$ and $−$100$ for $ΔCost$ in Equation (V).

$Δ(B/C)=$74.327$100=0.74327$

Since, $Δ(B/C)$ ratio less than $1$ therefore alternative $A$ is not desirable.

Hence, alternative $B$ is desirable.

In both the cases alternative $B$ is desirable.

Hence select alternative $B$ .

Conclusion:

The alternative to be selected is Alternative $B$ .

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Answer 2

Question

Chapter 9, Problem 30P

To determine

The alternative to be selected.

Expert Solution & Answer

Answer to Problem 30P

The alternative to be selected is Alternative $B$ .

Explanation of Solution

Given:

MARR is $10%$ .

Useful life is $5$ years.

Calculation:

Write the expression to calculate the present worth of the benefit.

$P=A((1+i)n−1i(1+i)n)$ ....... (I)

Here, the annual benefit is $P$ , the present worth is $P$ , the rate of interest is $i$ and the number of period is $n$ .

Calculate the present worth benefit of the alternative $A$ .

Substitute $$158.3$ for $A$ , $10%$ for $i$ and $5$ for $n$ in Equation (I).

$P=$158.3((1+0.10)5−10.10(1+0.10)5)=$158.3(3.79)=$600$

Calculate the present worth benefit of the alternative $B$

Substitute $$138.7$ for $A$ , $10%$ for $i$ and $5$ for $n$ in Equation (I).

$P=$138.7((1+0.10)5−10.10(1+0.10)5)=$138.7(3.79)=$525.673$

Calculate the present worth benefit of the alternative $C$

Substitute $$58.3$ for $A$ , $10%$ for $i$ and $5$ for $n$ in Equation (I).

$P=$58.3((1+0.10)5−10.10(1+0.10)5)=$58.3(3.79)=$220.957$

Write the expression to calculate the benefit-cost ratio.

$B/C=BC$ ....... (II)

Here, the present worth benefit is $B$ , the present worth cost is $C$ .

For Alternative $A$ .

Substitute $$599.957$ for $B$ , $$600$ for $C$ , Equation (I).

$B/C=$600$600=1$

For Alternative $B$ .

Substitute $$525.673$ for $B$ , $$500$ for $C$ , Equation (I).

$B/C=$525.63$500=1.05126$

For Alternative $C$ .

Substitute $$220.957$ for $B$ , $$200$ for $C$ , Equation (I).

$B/C=$220.957$200=1.104$

Since all the ratios are positive hence we need to make comparison be increment method.

Calculate the $ΔCOST$ for alternative $B−C$ .

$ΔCost=(PWC−PWB)$ ...... (III)

Here the incremental cost between the alternative $B$ and $C$ is $ΔCost$ , the present worth of cost for alternative $C$ is $PWC$ and the present worth of cost for alternative $B$ is $PWB$ .

Substitute $$200$ for $PWC$ and $$500$ for $PWB$ in Equation (III).

$ΔCost=($200−$500)=−$300$

Calculate the $ΔBenefit$ for alternative $B−C$ .

$ΔBenefit=(PC−PB)$ ...... (IV)

Here the incremental benefit between the alternative $B$ and $C$ is $ΔBenefit$ , the present worth of benefit for alternative $C$ is $PC$ and the present worth of benefit for alternative $B$ is $PB$ .

Substitute $$220.957$ for $PC$ and $$525.673$ for $PB$ in Equation (IV).

$ΔBenefit=$220.957−$525.673=−304.716$

Calculate the incremental benefit-cost ratio

$Δ(B/C)=ΔBenefitΔCost$ ...... (V)

Substitute $−$304.716$ for and $−$300$ for $ΔCost$ in Equation (V).

$Δ(B/C)=−$304.716−$300=1.0572$

Since, $Δ(B/C)$ ratio greater than $1$ therefore alternative $B$ is desirable.

Calculate the $ΔCOST$ for alternative $A−B$

$ΔCost=(PWA−PWB)$ ...... (VI)

Here the incremental cost between the alternative $B$ and $A$ is $ΔCost$ , the present worth of cost for alternative $B$ is $PWB$ and the present worth of cost for alternative $A$ is $PWA$ .

Substitute $$200$ for $PWA$ and $$500$ for $PWB$ in Equation (III).

$ΔCost=($600−$500)=$100$

Calculate the $ΔBenefit$ for alternative $A−B$

$ΔBenefit=(PA−PB)$ ...... (VII)

Here the incremental benefit between the alternative $B$ and $A$ is $ΔBenefit$ , the present worth of benefit for alternative $A$ is $PA$ and the present worth of benefit for alternative $B$ is $PB$ .

Substitute $$600$ for $PA$ and $$525.673$ for $PB$ in Equation (IV).

$ΔBenefit=$600−$525.673=$74.327$

Calculate the incremental benefit-cost ratio

$Δ(B/C)=ΔBenefitΔCost$ ...... (V)

Substitute $−$74.327$ for $ΔBenefit$ and $−$100$ for $ΔCost$ in Equation (V).

$Δ(B/C)=$74.327$100=0.74327$

Since, $Δ(B/C)$ ratio less than $1$ therefore alternative $A$ is not desirable.

Hence, alternative $B$ is desirable.

In both the cases alternative $B$ is desirable.

Hence select alternative $B$ .

Conclusion:

The alternative to be selected is Alternative $B$ .

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Answer 3

Question

Chapter 9, Problem 30P

To determine

The alternative to be selected.

Expert Solution & Answer

Answer to Problem 30P

The alternative to be selected is Alternative $B$ .

Explanation of Solution

Given:

MARR is $10%$ .

Useful life is $5$ years.

Calculation:

Write the expression to calculate the present worth of the benefit.

$P=A((1+i)n−1i(1+i)n)$ ....... (I)

Here, the annual benefit is $P$ , the present worth is $P$ , the rate of interest is $i$ and the number of period is $n$ .

Calculate the present worth benefit of the alternative $A$ .

Substitute $$158.3$ for $A$ , $10%$ for $i$ and $5$ for $n$ in Equation (I).

$P=$158.3((1+0.10)5−10.10(1+0.10)5)=$158.3(3.79)=$600$

Calculate the present worth benefit of the alternative $B$

Substitute $$138.7$ for $A$ , $10%$ for $i$ and $5$ for $n$ in Equation (I).

$P=$138.7((1+0.10)5−10.10(1+0.10)5)=$138.7(3.79)=$525.673$

Calculate the present worth benefit of the alternative $C$

Substitute $$58.3$ for $A$ , $10%$ for $i$ and $5$ for $n$ in Equation (I).

$P=$58.3((1+0.10)5−10.10(1+0.10)5)=$58.3(3.79)=$220.957$

Write the expression to calculate the benefit-cost ratio.

$B/C=BC$ ....... (II)

Here, the present worth benefit is $B$ , the present worth cost is $C$ .

For Alternative $A$ .

Substitute $$599.957$ for $B$ , $$600$ for $C$ , Equation (I).

$B/C=$600$600=1$

For Alternative $B$ .

Substitute $$525.673$ for $B$ , $$500$ for $C$ , Equation (I).

$B/C=$525.63$500=1.05126$

For Alternative $C$ .

Substitute $$220.957$ for $B$ , $$200$ for $C$ , Equation (I).

$B/C=$220.957$200=1.104$

Since all the ratios are positive hence we need to make comparison be increment method.

Calculate the $ΔCOST$ for alternative $B−C$ .

$ΔCost=(PWC−PWB)$ ...... (III)

Here the incremental cost between the alternative $B$ and $C$ is $ΔCost$ , the present worth of cost for alternative $C$ is $PWC$ and the present worth of cost for alternative $B$ is $PWB$ .

Substitute $$200$ for $PWC$ and $$500$ for $PWB$ in Equation (III).

$ΔCost=($200−$500)=−$300$

Calculate the $ΔBenefit$ for alternative $B−C$ .

$ΔBenefit=(PC−PB)$ ...... (IV)

Here the incremental benefit between the alternative $B$ and $C$ is $ΔBenefit$ , the present worth of benefit for alternative $C$ is $PC$ and the present worth of benefit for alternative $B$ is $PB$ .

Substitute $$220.957$ for $PC$ and $$525.673$ for $PB$ in Equation (IV).

$ΔBenefit=$220.957−$525.673=−304.716$

Calculate the incremental benefit-cost ratio

$Δ(B/C)=ΔBenefitΔCost$ ...... (V)

Substitute $−$304.716$ for and $−$300$ for $ΔCost$ in Equation (V).

$Δ(B/C)=−$304.716−$300=1.0572$

Since, $Δ(B/C)$ ratio greater than $1$ therefore alternative $B$ is desirable.

Calculate the $ΔCOST$ for alternative $A−B$

$ΔCost=(PWA−PWB)$ ...... (VI)

Here the incremental cost between the alternative $B$ and $A$ is $ΔCost$ , the present worth of cost for alternative $B$ is $PWB$ and the present worth of cost for alternative $A$ is $PWA$ .

Substitute $$200$ for $PWA$ and $$500$ for $PWB$ in Equation (III).

$ΔCost=($600−$500)=$100$

Calculate the $ΔBenefit$ for alternative $A−B$

$ΔBenefit=(PA−PB)$ ...... (VII)

Here the incremental benefit between the alternative $B$ and $A$ is $ΔBenefit$ , the present worth of benefit for alternative $A$ is $PA$ and the present worth of benefit for alternative $B$ is $PB$ .

Substitute $$600$ for $PA$ and $$525.673$ for $PB$ in Equation (IV).

$ΔBenefit=$600−$525.673=$74.327$

Calculate the incremental benefit-cost ratio

$Δ(B/C)=ΔBenefitΔCost$ ...... (V)

Substitute $−$74.327$ for $ΔBenefit$ and $−$100$ for $ΔCost$ in Equation (V).

$Δ(B/C)=$74.327$100=0.74327$

Since, $Δ(B/C)$ ratio less than $1$ therefore alternative $A$ is not desirable.

Hence, alternative $B$ is desirable.

In both the cases alternative $B$ is desirable.

Hence select alternative $B$ .

Conclusion:

The alternative to be selected is Alternative $B$ .

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Answer 4

Question

Chapter 9, Problem 30P

To determine

The alternative to be selected.

Expert Solution & Answer

Answer to Problem 30P

The alternative to be selected is Alternative $B$ .

Explanation of Solution

Given:

MARR is $10%$ .

Useful life is $5$ years.

Calculation:

Write the expression to calculate the present worth of the benefit.

$P=A((1+i)n−1i(1+i)n)$ ....... (I)

Here, the annual benefit is $P$ , the present worth is $P$ , the rate of interest is $i$ and the number of period is $n$ .

Calculate the present worth benefit of the alternative $A$ .

Substitute $$158.3$ for $A$ , $10%$ for $i$ and $5$ for $n$ in Equation (I).

$P=$158.3((1+0.10)5−10.10(1+0.10)5)=$158.3(3.79)=$600$

Calculate the present worth benefit of the alternative $B$

Substitute $$138.7$ for $A$ , $10%$ for $i$ and $5$ for $n$ in Equation (I).

$P=$138.7((1+0.10)5−10.10(1+0.10)5)=$138.7(3.79)=$525.673$

Calculate the present worth benefit of the alternative $C$

Substitute $$58.3$ for $A$ , $10%$ for $i$ and $5$ for $n$ in Equation (I).

$P=$58.3((1+0.10)5−10.10(1+0.10)5)=$58.3(3.79)=$220.957$

Write the expression to calculate the benefit-cost ratio.

$B/C=BC$ ....... (II)

Here, the present worth benefit is $B$ , the present worth cost is $C$ .

For Alternative $A$ .

Substitute $$599.957$ for $B$ , $$600$ for $C$ , Equation (I).

$B/C=$600$600=1$

For Alternative $B$ .

Substitute $$525.673$ for $B$ , $$500$ for $C$ , Equation (I).

$B/C=$525.63$500=1.05126$

For Alternative $C$ .

Substitute $$220.957$ for $B$ , $$200$ for $C$ , Equation (I).

$B/C=$220.957$200=1.104$

Since all the ratios are positive hence we need to make comparison be increment method.

Calculate the $ΔCOST$ for alternative $B−C$ .

$ΔCost=(PWC−PWB)$ ...... (III)

Here the incremental cost between the alternative $B$ and $C$ is $ΔCost$ , the present worth of cost for alternative $C$ is $PWC$ and the present worth of cost for alternative $B$ is $PWB$ .

Substitute $$200$ for $PWC$ and $$500$ for $PWB$ in Equation (III).

$ΔCost=($200−$500)=−$300$

Calculate the $ΔBenefit$ for alternative $B−C$ .

$ΔBenefit=(PC−PB)$ ...... (IV)

Here the incremental benefit between the alternative $B$ and $C$ is $ΔBenefit$ , the present worth of benefit for alternative $C$ is $PC$ and the present worth of benefit for alternative $B$ is $PB$ .

Substitute $$220.957$ for $PC$ and $$525.673$ for $PB$ in Equation (IV).

$ΔBenefit=$220.957−$525.673=−304.716$

Calculate the incremental benefit-cost ratio

$Δ(B/C)=ΔBenefitΔCost$ ...... (V)

Substitute $−$304.716$ for and $−$300$ for $ΔCost$ in Equation (V).

$Δ(B/C)=−$304.716−$300=1.0572$

Since, $Δ(B/C)$ ratio greater than $1$ therefore alternative $B$ is desirable.

Calculate the $ΔCOST$ for alternative $A−B$

$ΔCost=(PWA−PWB)$ ...... (VI)

Here the incremental cost between the alternative $B$ and $A$ is $ΔCost$ , the present worth of cost for alternative $B$ is $PWB$ and the present worth of cost for alternative $A$ is $PWA$ .

Substitute $$200$ for $PWA$ and $$500$ for $PWB$ in Equation (III).

$ΔCost=($600−$500)=$100$

Calculate the $ΔBenefit$ for alternative $A−B$

$ΔBenefit=(PA−PB)$ ...... (VII)

Here the incremental benefit between the alternative $B$ and $A$ is $ΔBenefit$ , the present worth of benefit for alternative $A$ is $PA$ and the present worth of benefit for alternative $B$ is $PB$ .

Substitute $$600$ for $PA$ and $$525.673$ for $PB$ in Equation (IV).

$ΔBenefit=$600−$525.673=$74.327$

Calculate the incremental benefit-cost ratio

$Δ(B/C)=ΔBenefitΔCost$ ...... (V)

Substitute $−$74.327$ for $ΔBenefit$ and $−$100$ for $ΔCost$ in Equation (V).

$Δ(B/C)=$74.327$100=0.74327$

Since, $Δ(B/C)$ ratio less than $1$ therefore alternative $A$ is not desirable.

Hence, alternative $B$ is desirable.

In both the cases alternative $B$ is desirable.

Hence select alternative $B$ .

Conclusion:

The alternative to be selected is Alternative $B$ .

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Answer 5

Chapter 9, Problem 30P

To determine

The alternative to be selected.

Expert Solution & Answer

Answer to Problem 30P

The alternative to be selected is Alternative $B$ .

Explanation of Solution

Given:

MARR is $10%$ .

Useful life is $5$ years.

Calculation:

Write the expression to calculate the present worth of the benefit.

$P=A((1+i)n−1i(1+i)n)$ ....... (I)

Here, the annual benefit is $P$ , the present worth is $P$ , the rate of interest is $i$ and the number of period is $n$ .

Calculate the present worth benefit of the alternative $A$ .

Substitute $$158.3$ for $A$ , $10%$ for $i$ and $5$ for $n$ in Equation (I).

$P=$158.3((1+0.10)5−10.10(1+0.10)5)=$158.3(3.79)=$600$

Calculate the present worth benefit of the alternative $B$

Substitute $$138.7$ for $A$ , $10%$ for $i$ and $5$ for $n$ in Equation (I).

$P=$138.7((1+0.10)5−10.10(1+0.10)5)=$138.7(3.79)=$525.673$

Calculate the present worth benefit of the alternative $C$

Substitute $$58.3$ for $A$ , $10%$ for $i$ and $5$ for $n$ in Equation (I).

$P=$58.3((1+0.10)5−10.10(1+0.10)5)=$58.3(3.79)=$220.957$

Write the expression to calculate the benefit-cost ratio.

$B/C=BC$ ....... (II)

Here, the present worth benefit is $B$ , the present worth cost is $C$ .

For Alternative $A$ .

Substitute $$599.957$ for $B$ , $$600$ for $C$ , Equation (I).

$B/C=$600$600=1$

For Alternative $B$ .

Substitute $$525.673$ for $B$ , $$500$ for $C$ , Equation (I).

$B/C=$525.63$500=1.05126$

For Alternative $C$ .

Substitute $$220.957$ for $B$ , $$200$ for $C$ , Equation (I).

$B/C=$220.957$200=1.104$

Since all the ratios are positive hence we need to make comparison be increment method.

Calculate the $ΔCOST$ for alternative $B−C$ .

$ΔCost=(PWC−PWB)$ ...... (III)

Here the incremental cost between the alternative $B$ and $C$ is $ΔCost$ , the present worth of cost for alternative $C$ is $PWC$ and the present worth of cost for alternative $B$ is $PWB$ .

Substitute $$200$ for $PWC$ and $$500$ for $PWB$ in Equation (III).

$ΔCost=($200−$500)=−$300$

Calculate the $ΔBenefit$ for alternative $B−C$ .

$ΔBenefit=(PC−PB)$ ...... (IV)

Here the incremental benefit between the alternative $B$ and $C$ is $ΔBenefit$ , the present worth of benefit for alternative $C$ is $PC$ and the present worth of benefit for alternative $B$ is $PB$ .

Substitute $$220.957$ for $PC$ and $$525.673$ for $PB$ in Equation (IV).

$ΔBenefit=$220.957−$525.673=−304.716$

Calculate the incremental benefit-cost ratio

$Δ(B/C)=ΔBenefitΔCost$ ...... (V)

Substitute $−$304.716$ for and $−$300$ for $ΔCost$ in Equation (V).

$Δ(B/C)=−$304.716−$300=1.0572$

Since, $Δ(B/C)$ ratio greater than $1$ therefore alternative $B$ is desirable.

Calculate the $ΔCOST$ for alternative $A−B$

$ΔCost=(PWA−PWB)$ ...... (VI)

Here the incremental cost between the alternative $B$ and $A$ is $ΔCost$ , the present worth of cost for alternative $B$ is $PWB$ and the present worth of cost for alternative $A$ is $PWA$ .

Substitute $$200$ for $PWA$ and $$500$ for $PWB$ in Equation (III).

$ΔCost=($600−$500)=$100$

Calculate the $ΔBenefit$ for alternative $A−B$

$ΔBenefit=(PA−PB)$ ...... (VII)

Here the incremental benefit between the alternative $B$ and $A$ is $ΔBenefit$ , the present worth of benefit for alternative $A$ is $PA$ and the present worth of benefit for alternative $B$ is $PB$ .

Substitute $$600$ for $PA$ and $$525.673$ for $PB$ in Equation (IV).

$ΔBenefit=$600−$525.673=$74.327$

Calculate the incremental benefit-cost ratio

$Δ(B/C)=ΔBenefitΔCost$ ...... (V)

Substitute $−$74.327$ for $ΔBenefit$ and $−$100$ for $ΔCost$ in Equation (V).

$Δ(B/C)=$74.327$100=0.74327$

Since, $Δ(B/C)$ ratio less than $1$ therefore alternative $A$ is not desirable.

Hence, alternative $B$ is desirable.

In both the cases alternative $B$ is desirable.

Hence select alternative $B$ .

Conclusion:

The alternative to be selected is Alternative $B$ .

Answer 6

Chapter 9, Problem 30P

To determine

The alternative to be selected.

Expert Solution & Answer

Answer to Problem 30P

The alternative to be selected is Alternative $B$ .

Explanation of Solution

Given:

MARR is $10%$ .

Useful life is $5$ years.

Calculation:

Write the expression to calculate the present worth of the benefit.

$P=A((1+i)n−1i(1+i)n)$ ....... (I)

Here, the annual benefit is $P$ , the present worth is $P$ , the rate of interest is $i$ and the number of period is $n$ .

Calculate the present worth benefit of the alternative $A$ .

Substitute $$158.3$ for $A$ , $10%$ for $i$ and $5$ for $n$ in Equation (I).

$P=$158.3((1+0.10)5−10.10(1+0.10)5)=$158.3(3.79)=$600$

Calculate the present worth benefit of the alternative $B$

Substitute $$138.7$ for $A$ , $10%$ for $i$ and $5$ for $n$ in Equation (I).

$P=$138.7((1+0.10)5−10.10(1+0.10)5)=$138.7(3.79)=$525.673$

Calculate the present worth benefit of the alternative $C$

Substitute $$58.3$ for $A$ , $10%$ for $i$ and $5$ for $n$ in Equation (I).

$P=$58.3((1+0.10)5−10.10(1+0.10)5)=$58.3(3.79)=$220.957$

Write the expression to calculate the benefit-cost ratio.

$B/C=BC$ ....... (II)

Here, the present worth benefit is $B$ , the present worth cost is $C$ .

For Alternative $A$ .

Substitute $$599.957$ for $B$ , $$600$ for $C$ , Equation (I).

$B/C=$600$600=1$

For Alternative $B$ .

Substitute $$525.673$ for $B$ , $$500$ for $C$ , Equation (I).

$B/C=$525.63$500=1.05126$

For Alternative $C$ .

Substitute $$220.957$ for $B$ , $$200$ for $C$ , Equation (I).

$B/C=$220.957$200=1.104$

Since all the ratios are positive hence we need to make comparison be increment method.

Calculate the $ΔCOST$ for alternative $B−C$ .

$ΔCost=(PWC−PWB)$ ...... (III)

Here the incremental cost between the alternative $B$ and $C$ is $ΔCost$ , the present worth of cost for alternative $C$ is $PWC$ and the present worth of cost for alternative $B$ is $PWB$ .

Substitute $$200$ for $PWC$ and $$500$ for $PWB$ in Equation (III).

$ΔCost=($200−$500)=−$300$

Calculate the $ΔBenefit$ for alternative $B−C$ .

$ΔBenefit=(PC−PB)$ ...... (IV)

Here the incremental benefit between the alternative $B$ and $C$ is $ΔBenefit$ , the present worth of benefit for alternative $C$ is $PC$ and the present worth of benefit for alternative $B$ is $PB$ .

Substitute $$220.957$ for $PC$ and $$525.673$ for $PB$ in Equation (IV).

$ΔBenefit=$220.957−$525.673=−304.716$

Calculate the incremental benefit-cost ratio

$Δ(B/C)=ΔBenefitΔCost$ ...... (V)

Substitute $−$304.716$ for and $−$300$ for $ΔCost$ in Equation (V).

$Δ(B/C)=−$304.716−$300=1.0572$

Since, $Δ(B/C)$ ratio greater than $1$ therefore alternative $B$ is desirable.

Calculate the $ΔCOST$ for alternative $A−B$

$ΔCost=(PWA−PWB)$ ...... (VI)

Here the incremental cost between the alternative $B$ and $A$ is $ΔCost$ , the present worth of cost for alternative $B$ is $PWB$ and the present worth of cost for alternative $A$ is $PWA$ .

Substitute $$200$ for $PWA$ and $$500$ for $PWB$ in Equation (III).

$ΔCost=($600−$500)=$100$

Calculate the $ΔBenefit$ for alternative $A−B$

$ΔBenefit=(PA−PB)$ ...... (VII)

Here the incremental benefit between the alternative $B$ and $A$ is $ΔBenefit$ , the present worth of benefit for alternative $A$ is $PA$ and the present worth of benefit for alternative $B$ is $PB$ .

Substitute $$600$ for $PA$ and $$525.673$ for $PB$ in Equation (IV).

$ΔBenefit=$600−$525.673=$74.327$

Calculate the incremental benefit-cost ratio

$Δ(B/C)=ΔBenefitΔCost$ ...... (V)

Substitute $−$74.327$ for $ΔBenefit$ and $−$100$ for $ΔCost$ in Equation (V).

$Δ(B/C)=$74.327$100=0.74327$

Since, $Δ(B/C)$ ratio less than $1$ therefore alternative $A$ is not desirable.

Hence, alternative $B$ is desirable.

In both the cases alternative $B$ is desirable.

Hence select alternative $B$ .

Conclusion:

The alternative to be selected is Alternative $B$ .

Answer 7

Chapter 9, Problem 30P

To determine

The alternative to be selected.

Answer 8

Expert Solution & Answer

Answer to Problem 30P

The alternative to be selected is Alternative $B$ .

Answer 9

Expert Solution & Answer

Answer 10

Expert Solution & Answer

Answer 11

Explanation of Solution

Given:

MARR is $10%$ .

Useful life is $5$ years.

Calculation:

Write the expression to calculate the present worth of the benefit.

$P=A((1+i)n−1i(1+i)n)$ ....... (I)

Here, the annual benefit is $P$ , the present worth is $P$ , the rate of interest is $i$ and the number of period is $n$ .

Calculate the present worth benefit of the alternative $A$ .

Substitute $$158.3$ for $A$ , $10%$ for $i$ and $5$ for $n$ in Equation (I).

$P=$158.3((1+0.10)5−10.10(1+0.10)5)=$158.3(3.79)=$600$

Calculate the present worth benefit of the alternative $B$

Substitute $$138.7$ for $A$ , $10%$ for $i$ and $5$ for $n$ in Equation (I).

$P=$138.7((1+0.10)5−10.10(1+0.10)5)=$138.7(3.79)=$525.673$

Calculate the present worth benefit of the alternative $C$

Substitute $$58.3$ for $A$ , $10%$ for $i$ and $5$ for $n$ in Equation (I).

$P=$58.3((1+0.10)5−10.10(1+0.10)5)=$58.3(3.79)=$220.957$

Write the expression to calculate the benefit-cost ratio.

$B/C=BC$ ....... (II)

Here, the present worth benefit is $B$ , the present worth cost is $C$ .

For Alternative $A$ .

Substitute $$599.957$ for $B$ , $$600$ for $C$ , Equation (I).

$B/C=$600$600=1$

For Alternative $B$ .

Substitute $$525.673$ for $B$ , $$500$ for $C$ , Equation (I).

$B/C=$525.63$500=1.05126$

For Alternative $C$ .

Substitute $$220.957$ for $B$ , $$200$ for $C$ , Equation (I).

$B/C=$220.957$200=1.104$

Since all the ratios are positive hence we need to make comparison be increment method.

Calculate the $ΔCOST$ for alternative $B−C$ .

$ΔCost=(PWC−PWB)$ ...... (III)

Here the incremental cost between the alternative $B$ and $C$ is $ΔCost$ , the present worth of cost for alternative $C$ is $PWC$ and the present worth of cost for alternative $B$ is $PWB$ .

Substitute $$200$ for $PWC$ and $$500$ for $PWB$ in Equation (III).

$ΔCost=($200−$500)=−$300$

Calculate the $ΔBenefit$ for alternative $B−C$ .

$ΔBenefit=(PC−PB)$ ...... (IV)

Here the incremental benefit between the alternative $B$ and $C$ is $ΔBenefit$ , the present worth of benefit for alternative $C$ is $PC$ and the present worth of benefit for alternative $B$ is $PB$ .

Substitute $$220.957$ for $PC$ and $$525.673$ for $PB$ in Equation (IV).

$ΔBenefit=$220.957−$525.673=−304.716$

Calculate the incremental benefit-cost ratio

$Δ(B/C)=ΔBenefitΔCost$ ...... (V)

Substitute $−$304.716$ for and $−$300$ for $ΔCost$ in Equation (V).

$Δ(B/C)=−$304.716−$300=1.0572$

Since, $Δ(B/C)$ ratio greater than $1$ therefore alternative $B$ is desirable.

Calculate the $ΔCOST$ for alternative $A−B$

$ΔCost=(PWA−PWB)$ ...... (VI)

Here the incremental cost between the alternative $B$ and $A$ is $ΔCost$ , the present worth of cost for alternative $B$ is $PWB$ and the present worth of cost for alternative $A$ is $PWA$ .

Substitute $$200$ for $PWA$ and $$500$ for $PWB$ in Equation (III).

$ΔCost=($600−$500)=$100$

Calculate the $ΔBenefit$ for alternative $A−B$

$ΔBenefit=(PA−PB)$ ...... (VII)

Here the incremental benefit between the alternative $B$ and $A$ is $ΔBenefit$ , the present worth of benefit for alternative $A$ is $PA$ and the present worth of benefit for alternative $B$ is $PB$ .