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**Hint:**We will assume the cost price of the motorcycle as Rs. x. Then, we will find the loss of Ravish using the formula, $loss\%=\dfrac{loss}{C.P}\times 100$ and then, find the SP of the motorcycle for Ravish using the formula, loss= C.P -S.P. This SP will be the CP for Vineet. We will then add Rs. 1680 to the CP to get the total CP. Given that there will be a profit of 12.5% for Vineet, we will use the formula, $gain\%=\dfrac{gain}{C.P}\times 100$, where gain = S.P -C.P. And from here, we will get the value of x, which is the cost price of the motorcycle for Ravish.

**Complete step by step answer:**

Let us first take the cost price (C.P.) of the motorcycle to be Rs. x.

Now, it is given in the question that he sold the motorcycle at a loss% of 28%.

Now, using the formula for calculating the loss that is given in the hint, we get the following

$\begin{align}

& \Rightarrow loss\%=\dfrac{loss}{C.P}\times 100 \\

& \Rightarrow 28=\dfrac{loss}{x}\times 100 \\

& \Rightarrow loss=\dfrac{28x}{100}=\dfrac{7x}{25} \\

\end{align}$

Now. Using the loss amount, we can calculate the selling price of the motorcycle for Ravish which is as follows

\[\begin{align}

& loss=\text{ }C.P\text{ }-S.P. \\

& \dfrac{7x}{25}=x-S.P. \\

& S.P.=\dfrac{18x}{25} \\

\end{align}\]

Now, this selling price for Ravish would be the cost price for Vineet.

Now, Vineet spent Rs. 1680 for the repair work of the motorcycle, hence, the net cost price for Vineet is Rs. \[\dfrac{18x}{25}\]+1680.

(Because the repair cost will also be taken up by Vineet only)

Now, the selling price for Vineet is given as Rs. 35910 and the profit percent for his transaction is 12.5%.

Hence, we can again use the formula that is given in the hint as follows

\[\begin{align}

& \Rightarrow gain\%=\dfrac{gain}{C.P}\times 100 \\

& \Rightarrow 12.5=\dfrac{S.P.-C.P.}{\dfrac{18x}{25}+1680}\times 100 \\

& \Rightarrow 12.5=\dfrac{35910-\left( \dfrac{18x}{25}+1680 \right)}{\dfrac{18x}{25}+1680}\times 100 \\

& \Rightarrow 12.5=\dfrac{34230\times 25-18x}{18x+1680\times 25}\times 100 \\

& \Rightarrow \dfrac{125}{10}=\dfrac{34230\times 25-18x}{18x+1680\times 25}\times 100 \\

& \Rightarrow \dfrac{1}{8}=\dfrac{34230\times 25-18x}{18x+1680\times 25} \\

& \Rightarrow 68460\times 100-144x=18x+1680\times 25 \\

& \Rightarrow 68460\times 100=162x+1680\times 25 \\

& \Rightarrow x=Rs.\ 42000 \\

\end{align}\]

**Hence, the cost price of the motor cycle for Ravish is 42000.**

**Note:**In this question, it is very important to know all the formulae beforehand and also, the calculations should also be done very carefully as there are very high chances that the students might commit a mistake while solving the question and getting to a solution. Most of the students make mistakes while finding the cost price of Vineet and forget to add the extra amount he spent for repair with his cost price and this will lead to errors in the final answer.

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