Chapter 9, Problem 61P

To determine

(a)

The value of $X$ that makes the two alternatives equally desirable.

Answer to Problem 61P

The value of $X$ that makes the two alternatives equally desirable is $9.23\text{years}$.

Explanation of Solution

Given:

B’s first cost is $\$1200$.

Concept used:

Write the expression to calculate the equivalent uniform annual cost.

$EUAC=A\left(A/P,i,n\right)$ ...... (I)

Here, the amount is $A$, rate of interest is $i$ and time period is $n$.

Calculation:

Calculate the equivalent uniform annual cost for $A$.

Substitute, $800$ for $A$, $12\%$ for $i$ and $5$ for $n$ in Equation (I).

$\begin{array}{c}EUAC=800\left(A/P,12\%,5\right)\\ =800\left(\frac{0.12{\left(1+0.12\right)}^5}{{\left(1+0.12\right)}^5-1}\right)\\ =800\times 0.2777\\ =221.92\end{array}$

To make both the alternatives equally desirable.

$\begin{array}{l}{\left(EUAB-EUAC\right)}_A={\left(EUAB-EUAC\right)}_B\\ {\left(EUAB-EUAC\right)}_A={\left(EUAB-A\left(A/P,i,n\right)\right)}_B\end{array}$ ...... (II)

Here, equivalent uniform annual benefit is $EUAB$, equivalent uniform annual cost is $EUAC$, amount is $A$, interest rate is $i$ and time period is $n$.

Substitute, $\$230$ for $EUA{B}_A$, $\$221.92$ for $EUA{C}_A$, $\$230$ for $EUA{B}_B$

$12\%$ for $i$ and $1200$ for $A$ in Equation (II)

$\begin{array}{l}230-221.92=230-1200\left(A/P,12\%,X\right)\\ 221.92=1200\left(\frac{0.12{\left(1+0.12\right)}^X}{{\left(1+0.12\right)}^X-1}\right)\end{array}$

Calculate the value of $X$

$X=9.23\text{years}$.

Conclusion:

Therefore, the value of $X$ that makes the two alternatives equally desirable is $9.23\text{years}$.

To determine

(b)

The value of $X$ that makes the two alternatives equally desirable.

Answer to Problem 61P

The value of $X$ that makes the two alternatives equally desirable is $5.13\text{years}$.

Explanation of Solution

Given:

B’s annual benefit is $\$280$.

Concept used:

Write the expression to calculate the equivalent uniform annual cost

$EUAC=A\left(A/P,i,n\right)$ ...... (I)

Here, the amount is $A$, rate of interest is $i$ and time period is $n$.

Calculation:

Calculate the equivalent uniform annual cost for $A$.

Substitute, $800$ for $A$, $12\%$ for $i$ and $5$ for $n$ in Equation (I)

$\begin{array}{c}EUAC=800\left(A/P,12\%,5\right)\\ =800\left(\frac{0.12{\left(1+0.12\right)}^5}{{\left(1+0.12\right)}^5-1}\right)\\ =800\times 0.2777\\ =221.92\end{array}$

To make both the alternatives equally desirable,

$\begin{array}{l}{\left(EUAB-EUAC\right)}_A={\left(EUAB-EUAC\right)}_B\\ {\left(EUAB-EUAC\right)}_A={\left(EUAB-A\left(A/P,i,n\right)\right)}_B\end{array}$ ...... (II)

Here, equivalent uniform annual benefit is $EUAB$, equivalent uniform annual cost is $EUAC$, amount is $A$, interest rate is $i$ and time period is $n$.

Substitute, $\$230$ for $EUA{B}_A$, $\$221.92$ for $EUA{C}_A$, $\$280$ for $EUA{B}_B$

$12\%$ for $i$ and $1000$ for $A$ in Equation (II)

$\begin{array}{l}230-221.92=280-1000\left(A/P,12\%,X\right)\\ 271.92=1000\left(\frac{0.12{\left(1+0.12\right)}^X}{{\left(1+0.12\right)}^X-1}\right)\end{array}$

Calculate the value of $X$

$X=5.13\text{years}$.

Conclusion:

The value of $X$ that makes the two alternatives equally desirable is $5.13\text{years}$.

To determine

(c)

The value of $X$ that makes the two alternatives equally desirable.

Answer to Problem 61P

The value of $X$ that makes the two alternatives equally desirable is $6.73\text{years}$.

Explanation of Solution

Given:

$\text{MARR}$ is $10\%$.

Concept used:

Write the expression to calculate the equivalent uniform annual cost

$EUAC=A\left(A/P,i,n\right)$ ...... (I)

Here, the amount is $A$, rate of interest is $i$ and time period is $n$.

Calculation:

Calculate the equivalent uniform annual cost for $\text{A}$.

Substitute, $800$ for $A$, $10\%$ for $i$ and $5$ for $n$ in Equation (I)

$\begin{array}{c}EUAC=800\left(A/P,12\%,5\right)\\ =800\left(\frac{0.10{\left(1+0.10\right)}^5}{{\left(1+0.10\right)}^5-1}\right)\\ =800\times 0.2637\\ =211.03\end{array}$

To make both the alternatives equally desirable,

$\begin{array}{l}{\left(EUAB-EUAC\right)}_A={\left(EUAB-EUAC\right)}_B\\ {\left(EUAB-EUAC\right)}_A={\left(EUAB-A\left(A/P,i,n\right)\right)}_B\end{array}$ ...... (II)

Here, equivalent uniform annual benefit is $EUAB$, equivalent uniform annual cost is $EUAC$, amount is $A$, interest rate is $i$ and time period is $n$.

Substitute, $\$230$ for $EUA{B}_A$, $\$211.03$ for $EUA{C}_A$, $\$280$ for $EUA{B}_B$

$10\%$ for $i$ and $1000$ for $A$ in Equation (II)

$\begin{array}{l}230-211.03=230-1000\left(A/P,10\%,X\right)\\ 211.03=1000\left(\frac{0.10{\left(1+0.10\right)}^X}{{\left(1+0.10\right)}^X-1}\right)\end{array}$

Calculate the value of $X$

$X=6.73\text{years}$.

Conclusion:

The value of $X$ that makes the two alternatives equally desirable is $6.73\text{years}$.

To determine

(d)

The value of $X$ that makes the two alternatives equally desirable.

Answer to Problem 61P

The value of $X$ that makes the two alternatives equally desirable is $6.45\text{years}$.

Explanation of Solution

Given:

B’s first cost is $\$1200$

B’s annual benefit is $\$280$

$MARR$ is $10\%$.

Concept used:

Write the expression to calculate the equivalent uniform annual cost

$EUAC=A\left(A/P,i,n\right)$ ...... (I)

Here, the amount is $A$, rate of interest is $i$ and time period is $n$.

Calculation:

Calculate the equivalent uniform annual cost for $\text{A}$.

Substitute, $800$ for $A$, $10\%$ for $i$ and $5$ for $n$ in Equation (I)

$\begin{array}{c}EUAC=800\left(A/P,12\%,5\right)\\ =800\left(\frac{0.10{\left(1+0.10\right)}^5}{{\left(1+0.10\right)}^5-1}\right)\\ =800\times 0.2637\\ =211.03\end{array}$

To make both the alternatives equally desirable,

$\begin{array}{l}{\left(EUAB-EUAC\right)}_{\text{A}}={\left(EUAB-EUAC\right)}_{\text{B}}\\ {\left(EUAB-EUAC\right)}_{\text{A}}={\left(EUAB-A\left(A/P,i,n\right)\right)}_{\text{B}}\end{array}$ ...... (II)

Here, equivalent uniform annual benefit is $EUAB$, equivalent uniform annual cost is $EUAC$, amount is $A$, interest rate is $i$ and time period is $n$.

Substitute, $\$230$ for $EUA{B}_{\text{A}}$, $\$211.03$ for $EUA{C}_{\text{A}}$, $\$280$ for $EUA{B}_{\text{B}}$

$10\%$ for $i$ and $1200$ for $A$ in Equation (II)

$\begin{array}{l}230-211.03=280-1200\left(A/P,10\%,X\right)\\ 261.03=1200\left(\frac{0.10{\left(1+0.10\right)}^X}{{\left(1+0.10\right)}^X-1}\right)\end{array}$

Calculate the value of $X$

$X=6.45\text{years}$.

Conclusion:

The value of $X$ that makes the two alternatives equally desirable is $6.45\text{years}$.