A shopkeeper earns profit of `16(2)/(3)%` after selling a book at 12.5% discount on the printed price. Then, find the ratio of the cost price & printed price of the book?

Options

A. `1:2`
B. `2:3`
C. `3:4`
D. `4:5`

Answer

To solve the problem, we need to find the ratio of the cost price (CP) to the printed price (MP) of the book based on the given profit percentage and discount percentage.

### Step 1: Understand the Profit Percentage
The profit earned by the shopkeeper is given as \(16\frac{2}{3}\%\). We can convert this mixed fraction into an improper fraction:
\[ 16\frac{2}{3} = \frac{50}{3}\% \]

### Step 2: Understand the Discount Percentage
The discount given on the printed price is \(12.5\%\). We can also express this as a fraction:
\[ 12.5\% = \frac{12.5}{100} = \frac{1}{8} \]

### Step 3: Calculate Selling Price (SP) in terms of Printed Price (MP)
Let the printed price (MP) of the book be \(x\). The selling price (SP) after applying the discount is:
\[ SP = MP - \text{Discount} = x - \left(\frac{1}{8} \times x\right) = x \left(1 - \frac{1}{8}\right) = x \left(\frac{7}{8}\right) \]

### Step 4: Relate Selling Price to Cost Price
The profit percentage is given as \(\frac{50}{3}\%\). This means:
\[ SP = CP + \text{Profit} \]
The profit can be expressed in terms of CP:
\[ \text{Profit} = \frac{50}{3}\% \text{ of } CP = \frac{50}{3} \times \frac{CP}{100} = \frac{50 \times CP}{300} = \frac{CP}{6} \]
Thus, we can write:
\[ SP = CP + \frac{CP}{6} = \frac{6CP + CP}{6} = \frac{7CP}{6} \]

### Step 5: Set the Two Expressions for SP Equal
From Step 3, we have:
\[ SP = \frac{7x}{8} \]
From Step 4, we have:
\[ SP = \frac{7CP}{6} \]
Setting these two expressions equal gives:
\[ \frac{7x}{8} = \frac{7CP}{6} \]

### Step 6: Solve for the Ratio of CP to MP
To eliminate the 7 from both sides, we can simplify:
\[ \frac{x}{8} = \frac{CP}{6} \]
Cross-multiplying gives:
\[ 6x = 8CP \]
Rearranging gives:
\[ \frac{CP}{x} = \frac{6}{8} = \frac{3}{4} \]
Thus, the ratio of the cost price (CP) to the printed price (MP) is:
\[ CP : MP = 3 : 4 \]

### Final Answer
The ratio of the cost price to the printed price of the book is \(3 : 4\).

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