Understanding the Profit and Discount Scenario

This problem involves concepts of Cost Price (CP), Selling Price (SP), Printed Price (also known as Marked Price, MP), profit percentage, and discount percentage. We are given the profit percentage earned on the selling price and the discount percentage offered on the printed price, and we need to find the ratio of the Cost Price to the Selling Price.

Let's break down the information provided:

• Profit earned is 21%. Standard profit calculations are based on Cost Price. So, Selling Price (SP) = Cost Price (CP) + Profit. If the profit is 21% of CP, then Profit = 0.21 × CP. Thus, SP = CP + 0.21 × CP = 1.21 × CP.
• Discount offered is 21% on the Printed Price (MP). Selling Price (SP) = Printed Price (MP) − Discount. If the discount is 21% of MP, then Discount = 0.21 × MP. Thus, SP = MP − 0.21 × MP = 0.79 × MP.

We are asked to find the ratio of Cost Price and Selling Price, which is $\frac{\text{CP}}{\text{SP}}$ or CP : SP.

Step-by-Step Calculation of the Ratio

We have established the following relationships based on the given percentages:

1. Selling Price is 121% of Cost Price (due to 21% profit).
   $$ \text{SP} = \text{CP} + 21\% \text{ of CP} $$
   $$ \text{SP} = \text{CP} + \frac{21}{100} \times \text{CP} $$
   $$ \text{SP} = \text{CP} \left( 1 + \frac{21}{100} \right) $$
   $$ \text{SP} = \text{CP} \left( \frac{100 + 21}{100} \right) $$
   $$ \text{SP} = \frac{121}{100} \times \text{CP} $$
   $$ \text{SP} = 1.21 \times \text{CP} $$
2. Selling Price is 79% of Printed Price (due to 21% discount).
   $$ \text{SP} = \text{MP} - 21\% \text{ of MP} $$
   $$ \text{SP} = \text{MP} - \frac{21}{100} \times \text{MP} $$
   $$ \text{SP} = \text{MP} \left( 1 - \frac{21}{100} \right) $$
   $$ \text{SP} = \text{MP} \left( \frac{100 - 21}{100} \right) $$
   $$ \text{SP} = \frac{79}{100} \times \text{MP} $$
   $$ \text{SP} = 0.79 \times \text{MP} $$

We need the ratio CP : SP. From step 1, we have the direct relationship between SP and CP:

To find the ratio CP : SP, we can rearrange this equation:

Convert the decimal to a fraction:

So the ratio of Cost Price to Selling Price (CP : SP) is 100 : 121.

Note that the information about the 21% discount on the printed price is useful for finding the relationship between CP, SP, and MP, but the direct ratio of CP to SP can be found using only the profit percentage relative to the Cost Price, as profit is conventionally calculated on CP.

Summary of Findings

Based on the calculation, the ratio of the Cost Price and Selling Price of the book is 100 : 121.

Revision Table: Profit & Discount Concepts

Additional Information: Relating CP, SP, and MP

In problems involving both profit/loss and discount, there is a useful formula that relates CP and MP:

Let's verify this with the given problem:

Discount % = 21%

Profit % = 21%

So, CP : MP = 79 : 121.

We also know SP = 1.21 × CP and SP = 0.79 × MP.

Let's check if the ratio CP : SP = 100 : 121 holds true with CP : MP = 79 : 121.

From CP : MP = 79 : 121, we can say $\text{MP} = \frac{121}{79} \times \text{CP}$.

Substitute this into the SP equation involving MP:

This confirms our original relationship between SP and CP derived from the profit percentage. The ratio CP : SP is indeed 100 : 121.

This problem highlights the importance of understanding which value (CP or MP) the percentage is based on for profit, loss, and discount calculations.