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**Hint:**In the given problem, we are asked to find the cost of $15$ soaps where the cost of some specific quantity (dozen here) is given to us. We know that a dozen implies a group of $12$ things. Therefore, in order to find the cost of $15$soaps, first we will find the cost of a single soap and then we will multiply that cost with $15$ to get the required answer. This method is called a unitary method.

**Complete step by step solution:**In this problem, it is given that the cost of a dozen soaps is Rs. $153.80$. A dozen implies a group of $12$ soaps. Therefore, we can say that the cost of $12$ soaps is Rs. $153.80$.

Now in order to calculate the cost of $1$ soap, we will divide $153.80$ by $12$. Therefore, we get

cost of $1$ soap $ = $ Rs. $\dfrac{{153.80}}{{12}}$.

Now in order to find the cost of $15$ soaps, we need to multiply the cost of $1$ soap with $15$. Therefore, we get cost of $15$ soaps $ = \left( {\dfrac{{153.80}}{{12}}} \right) \times 15$

$ \Rightarrow $ Cost of $15$ soaps $ = \left( {\dfrac{{153.80}}{4}} \right) \times 5$

$ \Rightarrow $ Cost of $15$ soaps $ = 38.45 \times 5$

$ \Rightarrow $ Cost of $15$ soaps $ = $ Rs. $192.25$

Therefore, if the cost of a dozen soaps is Rs. $153.80$ then the cost of $15$ such soaps is Rs. $192.25$.

**Note:**In unitary method, first we will find the value of a single object and then we can find the required value by multiplying the single object value. In the given problem, we can use a direct method to find the required cost. If the cost of $n$ items is Rs. $m$ then the cost of $r$ items will be Rs. $\dfrac{{r \times m}}{n}$. Here given that the cost of a dozen soaps ($12$ soaps) is Rs. $153.80$. Therefore, the cost of $15$such soaps will be Rs. $\dfrac{{153.80 \times 15}}{{12}} = 192.25$.

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