If a shopkeeper sold a book with 20% profit after giving a discount of 10% on marked price. The ratio of cost price and marked price of the book is

Options

A. `6:5`
B. `5:6`
C. `3:4`
D. `2:3`

Answer

To solve the problem step by step, let's denote the following:

- Let the marked price (MP) of the book be \( X \).
- The discount given on the marked price is 10%.

### Step 1: Calculate the Selling Price (SP)
The selling price after a 10% discount on the marked price can be calculated as follows:

\[ SP = MP - \text{Discount} \]
\[ \text{Discount} = 10\% \text{ of } X = \frac{10}{100} \times X = \frac{X}{10} \]
\[ SP = X - \frac{X}{10} = \frac{10X}{10} - \frac{X}{10} = \frac{9X}{10} \]

### Step 2: Relate Selling Price to Cost Price (CP)
We know that the shopkeeper sold the book at a 20% profit. The relationship between selling price, cost price, and profit percentage is given by:

\[ SP = CP + \text{Profit} \]
\[ \text{Profit} = 20\% \text{ of } CP = \frac{20}{100} \times CP = \frac{1}{5} CP \]
Thus, we can express the selling price in terms of cost price:

\[ SP = CP + \frac{1}{5} CP = \frac{6}{5} CP \]

### Step 3: Set the two expressions for Selling Price equal to each other
Now we have two expressions for the selling price:

\[ \frac{9X}{10} = \frac{6}{5} CP \]

### Step 4: Solve for Cost Price (CP)
To find the cost price, we can rearrange the equation:

\[ CP = \frac{9X}{10} \times \frac{5}{6} = \frac{9 \times 5X}{10 \times 6} = \frac{45X}{60} = \frac{3X}{4} \]

### Step 5: Find the ratio of Cost Price to Marked Price
We need to find the ratio of cost price to marked price:

\[ \text{Ratio of CP to MP} = \frac{CP}{MP} = \frac{\frac{3X}{4}}{X} = \frac{3}{4} \]

### Final Answer
The ratio of the cost price to the marked price of the book is:

\[ \text{Ratio} = 3 : 4 \]

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