Given:
- Profit earned on selling a book = 12%
- Discount given on the printed price = 10%

To find:
- The ratio of the cost price and the published price of the book

Let's assume the cost price of the book is 'x' and the printed price is 'y'.

Calculating the Selling Price:
- As per the given information, the shopkeeper sells the book at a 10% discount on the printed price. So, the selling price (SP) can be calculated as:
SP = y - (10/100) * y = y - 0.1y = 0.9y

Calculating the Profit:
- The profit earned on selling the book is 12% of the selling price. So, the profit can be calculated as:
Profit = (12/100) * 0.9y = 0.108y

Calculating the Cost Price:
- The cost price is the difference between the selling price and the profit. So, the cost price can be calculated as:
Cost Price = Selling Price - Profit = 0.9y - 0.108y = 0.792y

Calculating the Published Price:
- The published price is the cost price plus the profit. So, the published price can be calculated as:
Published Price = Cost Price + Profit = 0.792y + 0.108y = 0.9y

Calculating the Ratio:
- The ratio of the cost price and the published price can be calculated as:
Cost Price : Published Price = 0.792y : 0.9y

- Simplifying the ratio by dividing both sides by 0.792y, we get:
Cost Price : Published Price = 0.792y/0.792y : 0.9y/0.792y
Cost Price : Published Price = 1 : 1.136

- Since the ratio should be in the form of integers, we can multiply both sides by a common factor to make the second value in the ratio an integer.
- Multiplying both sides by 44, we get:
Cost Price : Published Price = 44 : 50.024

- Rounding off the second value to the nearest integer, we get:
Cost Price : Published Price = 44 : 50

- Simplifying the ratio by dividing both sides by 4, we get:
Cost Price : Published Price = 11 : 12.5

- Multiplying both sides by 5, we get:
Cost Price : Published Price = 55 : 62.5

Therefore, the ratio of the cost price and the published price of the book is 55 : 62.5, which is not one of the given options.