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Vinod sold a watch to Arjun at a gain of $12\% $ and Arjun had to sell it to Manoj at a loss of $5\% $. If Manoj paid Rs. $3990$ for it. How much did Vinod pay for the watch?

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Hint:: We need to have the concept of Profit & Loss, percentage & solving the equation to solve this problem. Here there are three persons involved who are buying & selling the watch at a certain percentage of profit or loss. So, following the question move stepwise such that you can relate the amount Manoj paid to Arjun & while doing this you will have to assume some unknown terms as different prices of watch & solve them to get the ultimate answer.

Complete step by step solution:
As already given in question-
Vinod sold to Arjun with a profit of $12\% $
So, consider that his cost price is Rs. $x$.
Let, Cost Price (CP) of Vinod $ = Rs.x$.
His profit percentage $ = 12\% $.
He sold it with a selling price of Rs. $y$to Arjun.
Arjun buys the watch with Rs. $y$ and sell it to Manoj at Rs. $3990$. Manoj bought the watch at Rs. $3990$. So, Rs. $3990$is the selling price for Arjun
Selling price (SP) of Vinod $ = $Cost price (SP) of Arjun $Rs.y$.
Arjun made a loss of $5\% $.
Loss $ = 5\% $.
Selling price (SP) of Arjun $ = $Cost price (CP) of Manoj $ = Rs.3990$.
Now, find out the value of $y$ as it is a cost price $(CP)$for Arjun.
As we know,
\[C.P{\text{ }} = {\text{ }}S.P{\text{ }} + loss{\text{ }}\% {\text{ }}of{\text{ }}C.P\]
 \[
   \Rightarrow C.P = {\text{ }}S.P{\text{ }} + {\text{ }}\dfrac{{loss\% }}{{100}} \times C.P \\
   \Rightarrow C.P - \dfrac{{loss\% }}{{100}} \times C.P = S.P \\
   \Rightarrow C.P(\dfrac{{100 - loss\% }}{{100}}) = S.P \\
 \]
$ \Rightarrow CP = \dfrac{{SP \times 100}}{{100 - loss\% }}$……………… (1)
Now, using this formula of C.P derived here & put the value of $y$in formula used as cost price for Vinod
 $\therefore $We know from equation (1) –
$CP = \dfrac{{SP \times 100}}{{100 - loss\% }}$
Therefore, for Arjun putting the values of –
$CP = Rs.y$, $SP = Rs.3990$, $loss\% = 5$.
$\therefore y = \dfrac{{3990 \times 100}}{{100 - 5}} = \dfrac{{3990 \times 100}}{{95}} = Rs.4200$
$\therefore $CP of Arjun = SP of Vinod $ = Rs.4200$
Again, For Vinod –
$CP = Rs.x$, $SP = Rs.4200$, $\Pr ofit\% = 12\% $,
Similarly as equation (1) is derived , in case of profit it will be-
$CP = \dfrac{{SP \times 100}}{{100 + \Pr ofit\% }}$
$\therefore x = \dfrac{{4200 \times 100}}{{100 + 12}} = \dfrac{{4200 \times 100}}{{112}} = Rs.3750$.

$\therefore$The cost price of Vinod is $Rs.3750$or Vinod paid $Rs.3750$for the watch.

Note: To solve this type of question, it is very important to relate different selling & buying prices of different persons mentioned here first because here there’s a chance of getting confused between S.P & C.P of Vinod, Arjun & Manoj. Then note down the information given. For solving this question, one should know the basic concepts of S.P, C.P, profit, and loss. A little bit of mistake in calculation can give you the wrong solution so be extremely focused here.