Let the retail price be $x$ and the cost price be $y$.
Now, $85 \%$ of $x = 102 \%$ of $y$
$\Rightarrow \frac{85}{100}\times x = \frac{102}{100}\times y$
$\Rightarrow 5x = 6y$
$\Rightarrow \boxed{ \frac{x}{y} = \frac{6}{5} } \quad \longrightarrow (1)$
Let, her give a discount of $d\%$ on retail price to make a profit of $20\%$.
Now, $120 \%$ of $y = d \%$ of $x$
$\Rightarrow \frac{120}{100}\times y = \left(\frac{100-d}{100}\right)\times x$
$\Rightarrow 120 y = 100 x – dx$
$\Rightarrow 120 y = (100-d)\left(\frac{6}{5}y\right)$
$\Rightarrow 600 = 600 – 6d$
$\Rightarrow 6d = 600-600$
$\Rightarrow \boxed {d = 0\% }$
$\therefore$ She will sell at retail price to get a $20\%$ discount.
$\textbf{Short Method:}$ Let ,
- Cost price $= 100$
- Selling price $= 102$
- Retail price $= x$
Now, $0.85 x = 102$
$\Rightarrow x = \frac{102}{85}\times 100 $
$\Rightarrow \boxed {x = 120}$
If the cost price is $100,$ to make a profit of $20\%,$ the selling price has to be $120.$ No discount will be given.
Correct Answer $:\text{D}$