The cost price of 36 articles is equal to the selling price of $N$ articles. If the profit is $20\%$, then find the value of $N$.

Let the cost price of one article be $\text{CP}$ and the selling price of one article be $\text{SP}$.
According to the question:

$$36\times \text{CP}=N\times \text{SP}$$

$$\frac{\text{SP}}{\text{CP}}=\frac{36}{N}$$

Given that the profit is $20\%$, the ratio of $\text{SP}$ to $\text{CP}$ is:

$$\frac{\text{SP}}{\text{CP}}=1+\frac{20}{100}=\frac{120}{100}=\frac{6}{5}$$

Equating both fractional representations of $\frac{\text{SP}}{\text{CP}}$:

$$\frac{36}{N}=\frac{6}{5}$$

Cross-multiplying to solve for $N$:

$$6\times N=36\times 5$$

$$6N=180$$

$$N=\frac{180}{6}=30$$