Establish Selling Price via Cost Price

Given a profit of $20\%$, the selling price ($SP$) can be expressed in terms of the cost price ($CP$) as: $SP = CP \times (1 + \frac{20}{100}) = 1.2 \times CP$.

$SP = 1.2 \times CP$

Establish Selling Price via Marked Price

Given a discount of $15\%$ on the marked price ($MP$), the selling price is: $SP = MP \times (1 - \frac{15}{100}) = 0.85 \times MP$.

$SP = 0.85 \times MP$

Equate the Selling Prices

Since the selling price is the same in both cases, equate the two expressions: $0.85 \times MP = 1.2 \times CP$.

$0.85 \times MP = 1.2 \times CP$

Calculate the Ratio

Rearrange the equation to find the ratio of marked price to cost price: $\frac{MP}{CP} = \frac{1.2}{0.85} = \frac{120}{85} = \frac{24}{17}$.

$\frac{MP}{CP} = \frac{24}{17}$