Self - doubt

Question

At the end of year $1998$, Shepard bought nine dozen goats. Henceforth, every year he added $p\%$ of the goats at the beginning of the year and sold $q\%$ of the goats at the end of the year where $p>0$ and $q>0$. If Shepard had nine dozen goats at the end of year $2002$, after making the sales for that year, which of the following is true?

• A. $ p = q$
• B. $p < q$
• C. $p > q$
• D. $p = \dfrac{q}{2}$

Answer 1

Assuming the total no. of goats are $x$

• if  percentage of adding (increase) i.e $p$ = $10\%$ & percentage of selling (decrese) percentage $q$ = $10\%$ , then at the end of the year , number of goats will be $x * \dfrac{11}{10}*\dfrac{9}{10}$ = $x*\dfrac{99}{100}$ ⇒ which is always lesser than x.

           $\color{red}{So, p = q \hspace{0.1cm}is \hspace{0.1cm}not\hspace{0.1cm} possible.}$

Now,

• if $p<q$

            Assuming $p = 10\%$ & $q = 11.11\%$,

             then at the end of the year number of goats will be $x * \dfrac{11}{10}*\dfrac{8}{9}$ = $x * \dfrac{88}{90} \color{red}{ \hspace{0.5cm}i.e<x}$

              $\color{red}{∴ p<q \hspace{0.1cm}is\hspace{0.1cm} not\hspace{0.1cm} possible.}$

• if $p>q$

            Assuming $p = 11.11\%$ & $q = 10\%$,

             then at the end of the year number of goats will be $x * \dfrac{10}{9}*\dfrac{9}{10}$ = $x  \color{blue}{\hspace{0.5cm}i.e =x}$

         $ {\color{blue} {∴ p>q\hspace{0.1cm} is \hspace{0.1cm}possible.}}$

• if $p>\dfrac{q}{2}$

           Assuming $p = 5\%$ & $q = 10\%$,

            then at the end of the year number of goats will be $x * \dfrac{21}{20}*\dfrac{9}{10}$  = $x*\dfrac{189}{200} \color{red}{\hspace{0.5cm}i.e <x}$

       $\color{red}{∴ p=\dfrac{q}{2}\hspace{0.1cm} is\hspace{0.1cm} not\hspace{0.1cm} possible.}$