A shopkeeper bought an article and sold it at a gain of 10%. If he had bought it at 20% less and sold it 30 rupees more, he would have made a profit of 45%

The problem doesn't specify a question, but I assume you want to find the cost price of the article. Here's how you can do it:

Step 1: Let's assume the cost price of the article is x rupees.

Step 2: According to the problem, the shopkeeper sold it at a gain of 10%. So, the selling price of the article is 110% of x, which is 1.1x rupees.

Step 3: The problem also states that if the shopkeeper had bought it at 20% less, the cost price would have been 80% of x, which is 0.8x rupees.

Step 4: If he sold it for 30 rupees more than the previous selling price, the new selling price would be 1.1x + 30 rupees.

Step 5: According to the problem, this new selling price is a 45% gain over the reduced cost price. So, 1.1x + 30 = 145% of 0.8x, which is 1.45 * 0.8x.

Step 6: Solving the equation 1.1x + 30 = 1.45 * 0.8x gives us the cost price x.

Step 7: Simplify the equation to find the value of x. This will give you the cost price of the article.