Printed price vs cost price ratio: A book is sold at a 10% discount on its printed (marked) price, yet the shopkeeper earns a profit of 12% on cost. Find the ratio of the cost price to the printed price.

Introduction / Context:
Relating cost, marked, and selling prices requires mapping discounts and profits as multiplicative factors. Here, discount is on the printed price, while profit is relative to cost. Express both with symbols and eliminate the selling price to directly get the ratio of cost to printed price.

Given Data / Assumptions:

• Printed price = P.
• Discount = 10% ⇒ Selling price S = 0.9P.
• Profit = 12% on cost ⇒ S = 1.12C.

Concept / Approach:
Set 0.9P = 1.12C. Rearranging yields C : P = 0.9 : 1.12. Reduce the fraction to a clean ratio by multiplying numerator and denominator by 100 and then simplifying.

Step-by-Step Solution:

Verification / Alternative check:
If P = 56 units, S = 0.9 * 56 = 50.4; with C = 45, profit% = (50.4 − 45)/45 = 12%, consistent.

Why Other Options Are Wrong:
8:11, 47:56, 3:4, 9:11 do not satisfy both the 10% discount and the 12% profit simultaneously.

Common Pitfalls:
Taking profit on printed price or discount on cost price; always apply each percentage to its proper base.

Final Answer:
45 : 56