Assume total quantity using LCM

Let the number of oranges bought of each type be equal to the LCM of 3 and 2, which is $6$ oranges of each type. Total oranges bought = $6 + 6 = 12$ oranges ($1$ dozen).

$\text{LCM}(3, 2) = 6$

Calculate Total Cost Price (CP)

Cost of $6$ oranges at the rate of $\text{₹}1$ for $3$ is $\frac{6}{3} = \text{₹}2$. Cost of $6$ oranges at the rate of $\text{₹}1$ for $2$ is $\frac{6}{2} = \text{₹}3$. Total $\text{CP} = 2 + 3 = \text{₹}5$.

$\text{Total CP} = \text{₹}2 + \text{₹}3 = \text{₹}5 \text{ for 12 oranges}$

Calculate Required Selling Price (SP) for 20% Profit

To gain a $20\%$ profit on the total cost price of $\text{₹}5$, calculate the required selling price using the profit formula: $\text{SP} = \text{CP} \times (1 + \text{Profit}\% / 100)$.

$\text{SP} = 5 \times \left(1 + \frac{20}{100}\right) = 5 \times \frac{6}{5} = \text{₹}6$