A shopkeeper marked an article 80% above the cost price. Sumit purchased the article at the discount of 50% and sold it to Ayush at a profit of 25%. If Ayush purchased the article in Rs. 4500, then find the cost price of the article.

The correct answer is Rs.4000.

Step-by-step Explanation:

Step 1: Define the initial Cost Price (CP)

Let the cost price of the article for the shopkeeper be $\mathrm{CP}$.

Step 2: Calculate the Marked Price (MP)

The shopkeeper marked the article 80% above the cost price.

$\mathrm{MP}=\mathrm{CP}+0.80\times \mathrm{CP}=1.80\times \mathrm{CP}$

Step 3: Calculate Sumit's Cost Price (Purchase Price)

Sumit purchased the article at a discount of 50% on the marked price.

$\text{Sumit's CP}=\mathrm{MP}\times (1-0.50)$

$\text{Sumit's CP}=1.80\times \mathrm{CP}\times 0.50=0.90\times \mathrm{CP}$

Step 4: Calculate Ayush's Purchase Price

Sumit sold the article to Ayush at a profit of 25% on his cost price.

$\text{Ayush's Purchase Price}=\text{Sumit's CP}\times (1+0.25)$

$\text{Ayush's Purchase Price}=0.90\times \mathrm{CP}\times 1.25=1.125\times \mathrm{CP}$

Step 5: Solve for the original Cost Price (CP)

We are given that Ayush purchased the article for Rs. 4500.

$1.125\times \mathrm{CP}=4500$

$\mathrm{CP}=\frac{4500}{1.125}$

$\mathrm{CP}=4000$

Therefore, the cost price of the article is Rs. 4000.