Step 1: Express reduced selling price in terms of Cost Price

Let the actual selling price of the article be $\text{SP}$. When sold at $\frac{2}{5}$ of its actual selling price, the selling price becomes $\frac{2}{5} \text{SP}$. Given there is a loss of $27\%$, the selling price is $73\%$ of the Cost Price ($\text{CP}$).

$\frac{2}{5} \text{SP} = 0.73 \times \text{CP}$

Step 2: Relate Cost Price to actual Selling Price

From the equation, express $\text{CP}$ in terms of $\text{SP}$:

$\text{CP} = \frac{2}{5 \times 0.73} \text{SP} = \frac{2}{3.65} \text{SP} = \frac{200}{365} \text{SP} = \frac{40}{73} \text{SP}$

Step 3: Calculate new Selling Price

The article is now sold at $84\%$ of its actual selling price, so the new selling price ($\text{SP}_2$) is:

$\text{SP}_2 = 0.84 \times \text{SP} = \frac{84}{100} \text{SP} = \frac{21}{25} \text{SP}$

Step 4: Determine profit percentage

Calculate the profit using the new selling price and cost price:

$\text{Profit} = \text{SP}_2 - \text{CP} = \frac{21}{25} \text{SP} - \frac{40}{73} \text{SP} = \left( \frac{1533 - 1000}{1825} \right) \text{SP} = \frac{533}{1825} \text{SP}$

Step 5: Compute final profit percentage

Now, compute the profit percentage as $\left( \frac{\text{Profit}}{\text{CP}} \right) \times 100$:

$\text{Profit } \% = \frac{\frac{533}{1825} \text{SP}}{\frac{40}{73} \text{SP}} \times 100 = \frac{533}{1825} \times \frac{73}{40} \times 100 = \frac{533 \times 73 \times 100}{1825 \times 40} = 53.3\%$