A shopkeeper marks an article at 80\% above its cost price. Sumit buys the article at a 50\% discount on the marked price and then sells it to Ayush at a 25\% profit. If Ayush buys the article for Rs. 4500, what is the original cost price of the article?

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**Solution:**

1. Let the cost price of the article be $ x $.

2. The marked price (MP) is 80\% above the cost price:
   $$
 $  \text{MP} = x + 0.8x = 1.8x$
   $$

3. Sumit purchases the article at a 50\% discount on the marked price:
   $$
   $\text{Sumit's purchase price} = 1.8x \times (1 - 0.5) = 1.8x \times 0.5 = 0.9x$
   $$

4. Sumit sells the article to Ayush at a 25\% profit on his purchase price:
   $$
  4 $\text{Selling price to Ayush} = 0.9x \times (1 + 0.25) = 0.9x \times 1.25 = 1.125x$
   $$

5. Given Ayush buys the article for Rs. 4500, so:
   $$
   1.125x = 4500
   $$

6. Solve for $ x $:
   $$
  $ x = \frac{4500}{1.125} = 4000$
   $$

**Answer:** Rs. 4000

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