A shopkeeper sells a book at a discount of 20 percent on its printed price but still makes a profit of 15 percent on the cost price. What is the ratio of the cost price to the printed price of the book?

Introduction / Context:
This question combines the concepts of marked price (printed price), cost price and selling price with both discount and profit present. The shopkeeper offers a 20 percent discount on the printed price to customers but still earns a 15 percent profit on the cost price. We are asked to find the ratio of cost price to printed price. This tests understanding of how discounts and profits relate to each other through the selling price.

Given Data / Assumptions:
- Discount offered on the printed price is 20 percent.
- Profit made on the cost price is 15 percent.
- Selling price is therefore both 80 percent of the printed price and 115 percent of the cost price.
- We need the ratio cost price : printed price.

Concept / Approach:
Let the printed price be M and the cost price be C. The selling price S is 80 percent of M because of the 20 percent discount, so S = 0.80M. The same selling price is 115 percent of C because of the 15 percent profit, so S = 1.15C. By equating these two expressions for S, we can express C in terms of M and then write the ratio C : M. Finally, we simplify the ratio into integers by clearing decimals and reducing to lowest terms.

Step-by-Step Solution:
Step 1: Let printed price = M and cost price = C. Step 2: Discount 20 percent means selling price S = 80 percent of printed price = 0.80M. Step 3: Profit 15 percent on cost means S = 115 percent of cost price = 1.15C. Step 4: Since both expressions equal the selling price, set 0.80M = 1.15C. Step 5: Rearrange to express C in terms of M: C = (0.80 / 1.15) * M. Step 6: Write 0.80 as 80/100 and 1.15 as 115/100, so C = (80/100) / (115/100) * M = (80/115) * M. Step 7: Therefore, the ratio C : M = 80/115 : 1 which simplifies to 80 : 115. Step 8: Divide both numbers by 5 to simplify: 80 : 115 becomes 16 : 23.

Verification / Alternative check:
Pick actual numbers using the ratio 16 : 23. Suppose cost price C = Rs 16 and printed price M = Rs 23. Selling price after discount is 80 percent of 23 which is 0.80 * 23 = Rs 18.40. Profit on cost is then 18.40 minus 16 = Rs 2.40 which is 15 percent of 16. Both the discount and profit conditions are satisfied, confirming that the ratio 16 : 23 is consistent with the given information.

Why Other Options Are Wrong:
20 : 23 and 23 : 20 correspond to different relationships between cost and printed price that do not maintain both a 20 percent discount and a 15 percent profit when tested with actual values. 23 : 16 would imply cost price greater than printed price, which is unrealistic in this context. Only 16 : 23 gives a selling price that is exactly 80 percent of printed price and 115 percent of cost price at the same time.

Common Pitfalls:
A common mistake is to confuse discount and profit bases, for instance treating the 20 percent as a reduction on cost price or the 15 percent as based on printed price. Another error is to add or subtract the percentages directly without writing proper equations. Ensuring that discount is applied to printed price and profit to cost price, and equating both expressions for selling price, keeps the reasoning correct.

Final Answer:
The ratio of cost price to printed price of the book is 16 : 23.