Chapter 14, Problem 65P

To determine

(a)

The value of $a$, $b$, $c$, $d$, $e$ and $f$.

Answer to Problem 65P

The value of $a$ is $7.52\%$.

The value of $b$ is $105$.

The value of $c$ is $9.5\%$.

The value of $d$ is $9.6\%$.

The value of $e$ is $128$.

The value of $f$ is $162$.

Explanation of Solution

Concept used:

Write the expression for annual percentage increase in ${\text{n}}^{\text{th}}$ year.

$\left({\text{Annual \% increase in n}}^{\text{th}}\text{ year}\right)=\left(\frac{{\text{MSI}}_{\text{n}}-{\text{MSI}}_{\text{n-1}}}{{\text{MSI}}_{\text{n-1}}}\right)\times 100$ ...... (I)

Here, manager’s salary index of the year is ${\text{MSI}}_{\text{n}}$ and the manager’s salary index of the previous year is ${\text{MSI}}_{\text{n-1}}$.

Write the expression for $\text{MSI}$ in any year.

${\text{MSI}}_{\text{n}}={\text{MSI}}_{\text{n-1}}+\left(\text{change in MSI}\%\left({\text{MSI}}_{\text{n-1}}\right)\right)$ ...... (II)

Calculations:

Calculate the annual percentage increase in $2009$.

Substitute $100$ for ${\text{MSI}}_{2009}$ and $93$ for ${\text{MSI}}_{2008}$ in Equation (I).

$\begin{array}{c}\text{Annual \% increase in year }2009=\left(\frac{100-93}{\text{93}}\right)\times 100\\ =7.52\%\end{array}$

The value of $a$ is $7.52\%$.

Calculate the $\text{MSI}$ in year $2010$.

Substitute $100$ for ${\text{MSI}}_{2009}$ and $5\%$ for change in $\text{MSI\%}$ in Equation (II).

$\begin{array}{c}{\text{MSI}}_{2010}=100+5\%\left(100\right)\\ =100+5\\ =105\end{array}$

The value of $b$ is $105$.

Calculate the annual percentage increase in $2011$.

Substitute $115$ for ${\text{MSI}}_{2011}$ and $105$ for ${\text{MSI}}_{2010}$ in Equation (I).

$\begin{array}{c}\text{Annual \% increase in year }2011=\left(\frac{115-105}{105}\right)\times 100\\ =9.5\%\end{array}$

The value of $c$ is $9.5\%$.

Calculate the annual percentage increase in $2012$.

Substitute $126$ for ${\text{MSI}}_{2012}$ and $115$ for ${\text{MSI}}_{2011}$ in Equation (I).

$\begin{array}{c}\text{Annual \% increase in year 2012}=\left(\frac{126-115}{115}\right)\times 100\\ =9.6\%\end{array}$

The value of $d$ is $9.6\%$.

Calculate the $\text{MSI}$ in year $2013$.

Substitute $126$ for ${\text{MSI}}_{2012}$ and $1.2\%$ for change in $\text{MSI\%}$ in Equation (II).

$\begin{array}{c}{\text{MSI}}_{2013}=126+1.2\%\left(126\right)\\ =126+2\\ =128\end{array}$

The value of $e$ is $128$.

Calculate the $\text{MSI}$ in year $2016$.

Substitute $145$ for ${\text{MSI}}_{2015}$ and $12\%$ for change in $\text{MSI\%}$ in Equation (II).

$\begin{array}{c}{\text{MSI}}_{2016}=145+12\%\left(145\right)\\ =145+17\\ =162\end{array}$

The value of $f$ is $162$.

Conclusion:

The value of $a$ is $7.52\%$.

The value of $b$ is $105$.

The value of $c$ is $9.5\%$.

The value of $d$ is $9.6\%$.

The value of $e$ is $128$.

The value of $f$ is $162$.

To determine

(b)

The base year of $\text{MSI}$.

Answer to Problem 65P

The base year of $\text{MSI}$ is $2009$ because it has an index value of $100$.

Explanation of Solution

The base value is taken as $100$ and the year in which value is $100$ is deemed to be a base year. There is always a year called as base year among all index having value of $100$.

The base year of $\text{MSI}$ is $2009$ because it has index value of $100$.

To determine

(c)

The average annual price increase in salaries from $2009$ to $2013$ and $2013$ to $2017$.

Answer to Problem 65P

The average annual price increase in salaries from $2009$ to $2013$ is $6.56\%$.

The average annual price increase in salaries from $2013$ to $2017$ is $6.22\%$.

Explanation of Solution

Calculations:

The table for average increase in salaries from $2009$ to $2013$ is shown below.

Year	MSI	Change in MSI
$2009$	$100$	$7.52\%$
$2010$	$105$	$5\%$
$2011$	$115$	$9.5\%$
$2012$	$126$	$9.5\%$
$2013$	$128$	$1.2\%$
Average		$\frac{32.82\%}{5}=6.56\%$

Table (1)

The average annual price increase in salaries from $2009$ to $2013$ is $6.56\%$.

The table for average increase in salaries from $2013$ to $2017$ is shown below.

Year	MSI	Change in MSI
$2013$	$128$	$1.2\%$
$2014$	$138$	$8.30\%$
$2015$	$145$	$5.10\%$
$2016$	$162$	$12\%$
$2017$	$170$	$4.5\%$
Average		$\frac{31.1\%}{5}=6.22\%$

Table (2)

The average annual price increase in salaries from $2013$ to $2017$ is $6.22\%$.

Conclusion:

The average annual price increase in salaries from $2009$ to $2013$ is $6.56\%$.

The average annual price increase in salaries from $2013$ to $2017$ is $6.22\%$.

To determine

(d)

The $\text{MSI}$ in year $2018$, $2019$ and $2020$.

Answer to Problem 65P

The $\text{MSI}$ in year $2018$ is $181$.

The $\text{MSI}$ in year $2019$ is $193$.

The $\text{MSI}$ in year $2020$ is $206$.

Explanation of Solution

Calculations:

Calculate the $\text{MSI}$ in year $2018$.

Substitute $170$ for ${\text{MSI}}_{2017}$ and $6.56\%$ for change in $\text{MSI\%}$ in Equation (II).

$\begin{array}{c}{\text{MSI}}_{2017}=170+\left(6.56\%\left(170\right)\right)\\ =170+11\\ =181\end{array}$

The $\text{MSI}$ in year $2018$ is $181$.

Calculate the $\text{MSI}$ in year $2019$.

Substitute $181$ for ${\text{MSI}}_{2018}$ and $6.56\%$ for change in $\text{MSI\%}$ in Equation (II).

$\begin{array}{c}{\text{MSI}}_{2019}=181+\left(6.56\%\left(181\right)\right)\\ =181+12\\ =193\end{array}$

The $\text{MSI}$ in year $2019$ is $193$.

Calculate the $\text{MSI}$ in year $2020$.

Substitute $193$ for ${\text{MSI}}_{2019}$ and $6.56\%$ for change in $\text{MSI\%}$ in Equation (II).

$\begin{array}{c}{\text{MSI}}_{2019}=193+\left(6.56\%\left(193\right)\right)\\ =193+13\\ =206\end{array}$

The $\text{MSI}$ in year $2020$ is $206$.

Conclusion:

The $\text{MSI}$ in year $2018$ is $181$.

The $\text{MSI}$ in year $2019$ is $193$.

The $\text{MSI}$ in year $2020$ is $206$.