A shopkeeper earns a profit of 12% on selling a book at 10% discount on the printed price. The ratio of the cost price and the printed price of the book is

Options

A. `45:56`
B. `45:51`
C. `47:56`
D. `47:51`

Answer

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

- Cost Price (CP) = x
- Printed Price (PP) = y

### Step 1: Understand the Profit and Discount
The shopkeeper earns a profit of 12% on the cost price. This means:
\[ \text{Selling Price (SP)} = CP + 12\% \text{ of } CP = x + 0.12x = 1.12x \]

The shopkeeper sells the book at a 10% discount on the printed price. Therefore, the selling price can also be expressed as:
\[ SP = PP - 10\% \text{ of } PP = y - 0.10y = 0.90y \]

### Step 2: Equate the Two Expressions for Selling Price
Since both expressions represent the selling price, we can set them equal to each other:
\[ 1.12x = 0.90y \]

### Step 3: Rearranging the Equation
We can rearrange this equation to find the ratio of CP to PP:
\[ \frac{x}{y} = \frac{0.90}{1.12} \]

### Step 4: Simplifying the Ratio
To simplify \( \frac{0.90}{1.12} \):
1. Convert both numbers to fractions:
- \( 0.90 = \frac{90}{100} = \frac{9}{10} \)
- \( 1.12 = \frac{112}{100} = \frac{28}{25} \)

2. Now, substituting these fractions into the ratio:
\[ \frac{x}{y} = \frac{9/10}{28/25} = \frac{9}{10} \times \frac{25}{28} = \frac{9 \times 25}{10 \times 28} = \frac{225}{280} \]

### Step 5: Simplifying Further
Now simplify \( \frac{225}{280} \):
- The greatest common divisor (GCD) of 225 and 280 is 5.
- Dividing both the numerator and denominator by 5:
\[ \frac{225 \div 5}{280 \div 5} = \frac{45}{56} \]

### Final Ratio
Thus, the ratio of the cost price to the printed price is:
\[ \frac{CP}{PP} = \frac{45}{56} \]

### Conclusion
The ratio of the cost price and the printed price of the book is 45:56.

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