A shopkeeper sells a bicycle at a profit of 11.11%. If he had bought it at 16.66% less and sold it at Rs 400 less, he would have made a profit of 6.66% . What should be the list price of the bicycle in the case he gave a discount of 10% and thereby gained 20% on the transaction?

Let's break down the problem step by step:

Step 1: Let's assume the cost price of the bicycle for the shopkeeper is Rs. x.

Step 2: According to the problem, the shopkeeper sells the bicycle at a profit of 11.11%. So, the selling price of the bicycle is Rs. 1.111x.

Step 3: The problem also states that if the shopkeeper had bought the bicycle at 16.66% less, the cost price would have been Rs. 0.8334x.

Step 4: If he sold it at Rs 400 less, the selling price would have been Rs. 1.111x - 400.

Step 5: At this selling price, he would have made a profit of 6.66%. So, we can set up the equation 0.8334x * 1.0666 = 1.111x - 400. Solving this equation, we get x = Rs. 6000.

Step 6: Now, we need to find the list price of the bicycle in the case he gave a discount of 10% and thereby gained 20% on the transaction. If he gained 20%, the cost price is Rs. 6000 and the selling price is Rs. 1.2 * 6000 = Rs. 7200.

Step 7: If he gave a discount of 10%, it means the list price is 10% more than the selling price. So, the list price of the bicycle is Rs. 7200 / 0.9 = Rs. 8000.

So, the list price of the bicycle should be Rs. 8000.