Chapter 9, Problem 71APQ

To determine

Benefit–cost ratio.

Explanation of Solution

The first cost (C) is $700,000 and the maintenance cost (M) is $25,000 per year. The initial property value (P) is $400,000 and increased property value (IP) is $600,000. The interest rate is 6% and time period (n) is 10 years.

Option (b):

The benefit–cost ratio (BC) can be calculated as follows:

$\begin{array}{l}\text{BC}=\frac{\text{IP}-\text{P}}{\left(\text{C}\left(\frac{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)+\text{M}\right)}\\ \text{1}\text{.7}=\frac{600,000-400,000}{\left(\text{700,000}\left(\frac{\text{0}\text{.06}{\left(1+0.06\right)}^{10}}{{\left(1+\text{0}\text{.06}\right)}^{\text{10}}-1}\right)+25,000\right)}\\ \text{1}\text{.7}=\frac{200,000}{\left(\text{700,000}\left(\frac{\text{0}\text{.06}\left(1.790848\right)}{1.790848-1}\right)+25,000\right)}\\ \text{1}\text{.7}=\frac{200,000}{\left(\text{700,000}\left(\frac{0.107451}{0.790848}\right)+25,000\right)}\\ \text{1}\text{.7}=\frac{200,000}{\left(\text{700,000}\left(\frac{0.107451}{0.790848}\right)+25,000\right)}\\ \text{1}\text{.7}=\frac{200,000}{\left(\text{700,000}\left(0.135868\right)+25,000\right)}\\ \text{1}\text{.7}=\frac{200,000}{\left(95,107.6+25,000\right)}\\ 1.7=1.7\end{array}$

The calculated BC value is nearly equal to the given value. Thus, option ‘b’ is correct.

Option (a):

The benefit–cost ratio (BC) can be calculated as follows:

$\begin{array}{l}\text{BC}=\frac{\text{IP}-\text{P}}{\left(\text{C}\left(\frac{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)+\text{M}\right)}\\ \text{1}\text{.3}=\frac{600,000-400,000}{\left(\text{700,000}\left(\frac{\text{0}\text{.06}{\left(1+0.06\right)}^{10}}{{\left(1+\text{0}\text{.06}\right)}^{\text{10}}-1}\right)+25,000\right)}\\ \text{1}\text{.3}=\frac{200,000}{\left(\text{700,000}\left(\frac{\text{0}\text{.06}\left(1.790848\right)}{1.790848-1}\right)+25,000\right)}\\ \text{1}\text{.3}=\frac{200,000}{\left(\text{700,000}\left(\frac{0.107451}{0.790848}\right)+25,000\right)}\\ \text{1}\text{.3}=\frac{200,000}{\left(\text{700,000}\left(\frac{0.107451}{0.790848}\right)+25,000\right)}\\ \text{1}\text{.3}=\frac{200,000}{\left(\text{700,000}\left(0.135868\right)+25,000\right)}\\ \text{1}\text{.3}=\frac{200,000}{\left(95,107.6+25,000\right)}\\ 1.3<1.7\end{array}$

The calculated BC value is greater than the given value. Thus, option ‘a’ is incorrect.

Option (c):

The benefit–cost ratio (BC) can be calculated as follows:

$\begin{array}{l}\text{BC}=\frac{\text{IP}-\text{P}}{\left(\text{C}\left(\frac{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)+\text{M}\right)}\\ \text{2}\text{.1}=\frac{600,000-400,000}{\left(\text{700,000}\left(\frac{\text{0}\text{.06}{\left(1+0.06\right)}^{10}}{{\left(1+\text{0}\text{.06}\right)}^{\text{10}}-1}\right)+25,000\right)}\\ \text{2}\text{.1}=\frac{200,000}{\left(\text{700,000}\left(\frac{\text{0}\text{.06}\left(1.790848\right)}{1.790848-1}\right)+25,000\right)}\\ \text{2}\text{.1}=\frac{200,000}{\left(\text{700,000}\left(\frac{0.107451}{0.790848}\right)+25,000\right)}\\ \text{2}\text{.1}=\frac{200,000}{\left(\text{700,000}\left(\frac{0.107451}{0.790848}\right)+25,000\right)}\\ \text{2}\text{.1}=\frac{200,000}{\left(\text{700,000}\left(0.135868\right)+25,000\right)}\\ \text{2}\text{.1}=\frac{200,000}{\left(95,107.6+25,000\right)}\\ 2.1>1.7\end{array}$

The calculated BC value is less than the given value. Thus, option ‘c’ is incorrect.

Option (d):

The benefit–cost ratio (BC) can be calculated as follows:

$\begin{array}{l}\text{BC}=\frac{\text{IP}-\text{P}}{\left(\text{C}\left(\frac{\text{i}{\left(1+\text{i}\right)}^{\text{n}}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)+\text{M}\right)}\\ \text{2}\text{.8}=\frac{600,000-400,000}{\left(\text{700,000}\left(\frac{\text{0}\text{.06}{\left(1+0.06\right)}^{10}}{{\left(1+\text{0}\text{.06}\right)}^{\text{10}}-1}\right)+25,000\right)}\\ \text{2}\text{.8}=\frac{200,000}{\left(\text{700,000}\left(\frac{\text{0}\text{.06}\left(1.790848\right)}{1.790848-1}\right)+25,000\right)}\\ \text{2}\text{.8}=\frac{200,000}{\left(\text{700,000}\left(\frac{0.107451}{0.790848}\right)+25,000\right)}\\ \text{2}\text{.8}=\frac{200,000}{\left(\text{700,000}\left(\frac{0.107451}{0.790848}\right)+25,000\right)}\\ \text{2}\text{.8}=\frac{200,000}{\left(\text{700,000}\left(0.135868\right)+25,000\right)}\\ \text{2}\text{.8}=\frac{200,000}{\left(95,107.6+25,000\right)}\\ 2.8>1.7\end{array}$

The calculated BC value is less than the given value. Thus, option ‘d’ is incorrect.