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Let $100x$ be the cost price of the pen, and $100y$ be the cost price of the book.

- On selling a pen at $5\%$ loss.
- On Selling a book at $15\%$ gain.

He gains $7$ rupees. So, equation will be :

$\require{cancel} \begin{array} { c c } \text{Pen} & \text{ Book} \\ \cancel{100}x\times \frac{-5}{\cancel{100}} & \quad \cancel{100}y\times \frac{15}{\cancel{100}} \end{array}$

$\boxed{-5x+15y = \text{Rs.}\:7} \quad \longrightarrow (1)$

- On selling a pen at $5\%$ gain.
- On selling a book at $10\%$ gain.

He gains $13$ rupees. So, equation will be :

$\require{cancel} \begin{array} { c c } \text{Pen} & \text{ Book} \\ \cancel{100}x\times \frac{5}{\cancel{100}} & \quad\cancel{100}y\times \frac{10}{\cancel{100}} \end{array}$

$\boxed{5x+10y = \text{Rs.}\:13}\quad \longrightarrow (2)$

From equation $(1),$ and $(2),$ we get

$\begin{matrix} {\cancel{-5x}} +15y =7 \quad \longrightarrow (1) \\ \underline{{\cancel{-5x}} +10y =13} \quad \longrightarrow (2)\\ 25y=20 \end{matrix}\\ \Rightarrow\boxed{y=\frac{20}{25} = \frac{4}{5}}$

Now, the cost price of book $ = 100y = 100\times\frac{4}{5} = 80$

$\therefore$ The cost price of book is $\text{Rs.}\; 80.$

Correct Answer $: \text{A}$