Understand the relationship between MP and CP

Let the Cost Price be $\text{CP}$ and Marked Price be $\text{MP}$. Given that the discount is $10\%$, the Selling Price is $\text{SP} = \text{MP} \times (1 - 0.10) = 0.9\text{MP}$.

$\text{SP} = 0.9 \times \text{MP}$

Express SP using Profit percentage

Given that there is a gain of $20\%$ on $\text{CP}$, the Selling Price is also given by $\text{SP} = \text{CP} \times (1 + 0.20) = 1.2\text{CP}$.

$\text{SP} = 1.2 \times \text{CP}$

Equate both expressions for SP

Equating the two expressions for Selling Price: $0.9 \times \text{MP} = 1.2 \times \text{CP}$.

$0.9 \times \text{MP} = 1.2 \times \text{CP}$

Calculate the ratio of MP to CP

Rearranging the terms to find the ratio $\frac{\text{MP}}{\text{CP}} = \frac{1.2}{0.9} = \frac{12}{9} = \frac{4}{3}$.

$\frac{\text{MP}}{\text{CP}} = \frac{4}{3}$