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We are given with the data of profit percentage and the selling price cost.

It is given that,

A coal merchant makes a profit of \[20\% \] by selling firewood at \[25Rs\] per quintal.

Hence, we can conclude that,

Selling Price $SP = 25Rs$

And the profit percentage $ = 20\% $

As we know that we always calculate profit or loss in relation with the Selling Price.

So $20\% $ of profit means that he gains that percent of profit on one quintal if he sold at $25Rs$ and with Cost Price as $CP$ say.

Therefore, $\dfrac{{SP - CP}}{{CP}} = $ Profit Percentage or $\dfrac{{SP - CP}}{{CP}} \times 100 = $ Profit

We know Profit percentage is $20\% $ and the selling price is $25Rs$. Substitute these values in the above equation. We get,

$ \Rightarrow \dfrac{{25 - CP}}{{CP}} = 20\% $

Percentage means per cent that is per $100$. So, we can write $20\% $ as $\dfrac{{20}}{{100}}$.

Now the equation will become as follows:

$ \Rightarrow \dfrac{{25 - CP}}{{CP}} = \dfrac{{20}}{{100}}$

Now, we need to simplify the above equation in order to get the $CP$ value.

$ \Rightarrow \dfrac{{25 - CP}}{{CP}} = \dfrac{{20}}{{100}}$

Now, do the cross multiplication.

\[

\Rightarrow 5 \times \left( {25 - CP} \right) = CP \\

\Rightarrow 125 - 5CP = CP \\

\]

In order to find out the value of $CP$, let us move all its terms to one side.

\[

\Rightarrow 6CP = 125 \\

\Rightarrow CP = \dfrac{{125}}{6} \\

\Rightarrow CP = 20.83Rs \\

\]

As per now, we got the value of Cost Price and that is $20.83Rs$.

Now, in the question we are asked about the profit percentage when the selling price is \[25.50Rs\].

So, in the above-mentioned formula, now we need to substitute $SP = 25.50$ and the Cost Price with $CP = 20.83$

\[ \Rightarrow \dfrac{{SP - CP}}{{CP}} \times 100 = \dfrac{{25.50 - 20.83}}{{20.83}} \times 100\]

\[ = \dfrac{{4.67}}{{20.83}} \times 100\]

Now taking the approximate division we will get the next step as follows: \[ \Rightarrow \dfrac{{SP - CP}}{{CP}} \times 100 = 0.2241 \times 100\]

$ = 22.41$

It means when the merchant sells the firewood at \[25.50Rs\] per quintal, he will get $22.41\% $ of profit.