Chapter 9, Problem 72APQ

To determine

Benefit–cost ratio.

Explanation of Solution

Initial cost (C) is $2,800,000. Additional cost (AC) for every 5 years is $200,000. Maintenance cost (M) is $20,000 per year. Decreased damage amount (B) is $310,000 per year. Interest rate is 6%. Time period (n) is 5.

Option (c):

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

$\begin{array}{l}\text{BC}=\frac{\text{B}}{\left(\text{C}\times \text{i}\right)+\text{M}+\text{AC}\left(\frac{\text{i}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)}\\ \text{1}\text{.4}=\frac{310,000}{\left(\text{2,800,000}\times \text{0}\text{.06}\right)+\text{20,0000}+\text{200,000}\left(\frac{\text{0}\text{.06}}{{\left(1+0.06\right)}^{\text{5}}-1}\right)}\\ \text{1}\text{.4}=\frac{310,000}{168,000+\text{20,0000}+\text{200,000}\left(\frac{\text{0}\text{.06}}{1.338226-1}\right)}\\ \text{1}\text{.4}=\frac{310,000}{188,000+\text{200,000}\left(\frac{\text{0}\text{.06}}{0.338226}\right)}\\ \text{1}\text{.4}=\frac{310,000}{188,000+35,479.2}\\ \text{1}\text{.4}=\frac{310,000}{188,000+35,479.2}\\ \text{1}\text{.4}=1.4\end{array}$

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

Option (a):

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

$\begin{array}{l}\text{BC}=\frac{\text{B}}{\left(\text{C}\times \text{i}\right)+\text{M}+\text{AC}\left(\frac{\text{i}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)}\\ \text{0}\text{.46}=\frac{310,000}{\left(\text{2,800,000}\times \text{0}\text{.06}\right)+\text{20,0000}+\text{200,000}\left(\frac{\text{0}\text{.06}}{{\left(1+0.06\right)}^{\text{5}}-1}\right)}\\ \text{0}\text{.46}=\frac{310,000}{168,000+\text{20,0000}+\text{200,000}\left(\frac{\text{0}\text{.06}}{1.338226-1}\right)}\\ \text{0}\text{.46}=\frac{310,000}{188,000+\text{200,000}\left(\frac{\text{0}\text{.06}}{0.338226}\right)}\\ 0.46=\frac{310,000}{188,000+35,479.2}\\ \text{0}\text{.46}=\frac{310,000}{188,000+35,479.2}\\ \text{0}\text{.46}<1.4\end{array}$

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

Option (b):

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

$\begin{array}{l}\text{BC}=\frac{\text{B}}{\left(\text{C}\times \text{i}\right)+\text{M}+\text{AC}\left(\frac{\text{i}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)}\\ \text{1}\text{.02}=\frac{310,000}{\left(\text{2,800,000}\times \text{0}\text{.06}\right)+\text{20,0000}+\text{200,000}\left(\frac{\text{0}\text{.06}}{{\left(1+0.06\right)}^{\text{5}}-1}\right)}\\ \text{1}\text{.02}=\frac{310,000}{168,000+\text{20,0000}+\text{200,000}\left(\frac{\text{0}\text{.06}}{1.338226-1}\right)}\\ \text{1}\text{.02}=\frac{310,000}{188,000+\text{200,000}\left(\frac{\text{0}\text{.06}}{0.338226}\right)}\\ \text{1}\text{.02}=\frac{310,000}{188,000+35,479.2}\\ \text{1}\text{.02}=\frac{310,000}{188,000+35,479.2}\\ \text{1}\text{.02}<1.4\end{array}$

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

Option (d):

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

$\begin{array}{l}\text{BC}=\frac{\text{B}}{\left(\text{C}\times \text{i}\right)+\text{M}+\text{AC}\left(\frac{\text{i}}{{\left(1+\text{i}\right)}^{\text{n}}-1}\right)}\\ \text{1}\text{.96}=\frac{310,000}{\left(\text{2,800,000}\times \text{0}\text{.06}\right)+\text{20,0000}+\text{200,000}\left(\frac{\text{0}\text{.06}}{{\left(1+0.06\right)}^{\text{5}}-1}\right)}\\ \text{1}\text{.96}=\frac{310,000}{168,000+\text{20,0000}+\text{200,000}\left(\frac{\text{0}\text{.06}}{1.338226-1}\right)}\\ \text{1}\text{.96}=\frac{310,000}{188,000+\text{200,000}\left(\frac{\text{0}\text{.06}}{0.338226}\right)}\\ \text{1}\text{.96}=\frac{310,000}{188,000+35,479.2}\\ \text{1}\text{.96}=\frac{310,000}{188,000+35,479.2}\\ 1.96>1.4\end{array}$

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