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If a seller gives a discount of $15\%$ on retail price, she still makes a profit of $2\%$. Which of the following ensures that she makes a profit of $20\%$?

- Give a discount of $5\%$ on retail price
- Give a discount of $2\%$ on retail price
- Increase the retail price by $2\%$
- Sell at retail price

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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 $:\text{D}$