An article is sold at $20\%$ profit. If its cost price is increased by Rs. $50$ and at the same time if its selling price is also increased by Rs. $30$, the percentage of profit decreases by $3\frac{1}{3}\%$, then the cost price is

1. Let the original cost price be $CP=x$.Since the profit is $20\%$, the original selling price is:
$$SP=1.2x$$
According to the new condition:
$$\text{New }CP=x+50$$
$$\text{New }SP=1.2x+30$$
The new profit percentage drops by $3\frac{1}{3}\%=\frac{10}{3}\%$. Thus, the new profit percentage is:
$$\text{New Profit }\%=20\%-\frac{10}{3}\%=\frac{50}{3}\%=\frac{50}{3\times 100}=\frac{1}{6}$$
We set up the profit equation:
$$\text{New }SP=\text{New }CP\times \left(1+\text{New Profit }\%\right)$$
$$1.2x+30=(x+50)\times \left(1+\frac{1}{6}\right)$$
$$1.2x+30=(x+50)\times \frac{7}{6}$$
Multiplying the entire equation by 6 to clear the fraction:
$$6(1.2x+30)=7(x+50)$$
$$7.2x+180=7x+350$$
$$7.2x-7x=350-180$$
$$0.2x=170$$
$$x=\frac{170}{0.2}=850$$
Hence, the original cost price is Rs. $850$.