A publisher published 5000 books in 5 lakh rupees. If he gives 500 books in free, `(2)/(3)`rd of the rest he sell on 20% discount and remaining `(1)/(3)` rd on M.P. Find the profit% of the publisher if market price of each book is Rs 200.

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Calculate the Cost Price (CP) of Each Book
The total cost of publishing 5000 books is Rs 5,00,000.

\[ \text{Cost Price (CP) of each book} = \frac{\text{Total Cost}}{\text{Number of Books}} = \frac{5,00,000}{5000} = 100 \text{ Rs} \]

**Hint:** To find the cost price per item, divide the total cost by the number of items.

### Step 2: Determine the Number of Books Sold
The publisher gives away 500 books for free. Therefore, the number of books left for sale is:

\[ \text{Books left} = 5000 - 500 = 4500 \text{ books} \]

**Hint:** Subtract the number of free items from the total to find the remaining items.

### Step 3: Calculate the Number of Books Sold at a Discount
The publisher sells \( \frac{2}{3} \) of the remaining books at a 20% discount.

\[ \text{Books sold at discount} = \frac{2}{3} \times 4500 = 3000 \text{ books} \]

**Hint:** To find a fraction of a quantity, multiply the quantity by the fraction.

### Step 4: Calculate the Selling Price of Books Sold at Discount
The marked price (MP) of each book is Rs 200. The selling price (SP) after a 20% discount is:

\[ \text{Discount} = 20\% \text{ of } 200 = \frac{20}{100} \times 200 = 40 \text{ Rs} \]
\[ \text{Selling Price (SP) after discount} = 200 - 40 = 160 \text{ Rs} \]

The total selling price for the books sold at a discount is:

\[ \text{Total SP for discounted books} = 3000 \times 160 = 48,000 \text{ Rs} \]

**Hint:** To find the selling price after a discount, subtract the discount amount from the marked price.

### Step 5: Calculate the Number of Books Sold at Market Price
The remaining \( \frac{1}{3} \) of the books are sold at the market price.

\[ \text{Books sold at M.P.} = \frac{1}{3} \times 4500 = 1500 \text{ books} \]

**Hint:** Again, use multiplication to find the fraction of the remaining quantity.

### Step 6: Calculate the Selling Price of Books Sold at Market Price
The selling price for the books sold at market price is simply the marked price:

\[ \text{Total SP for M.P. books} = 1500 \times 200 = 3,00,000 \text{ Rs} \]

**Hint:** For items sold at the marked price, multiply the number of items by the marked price.

### Step 7: Calculate Total Revenue
Now, we can find the total revenue from selling all the books:

\[ \text{Total Revenue} = \text{Total SP for discounted books} + \text{Total SP for M.P. books} \]
\[ \text{Total Revenue} = 48,000 + 3,00,000 = 3,48,000 \text{ Rs} \]

**Hint:** Add the revenues from different selling methods to get the total revenue.

### Step 8: Calculate Profit
Profit is calculated as:

\[ \text{Profit} = \text{Total Revenue} - \text{Total Cost} \]
\[ \text{Profit} = 3,48,000 - 5,00,000 = -1,52,000 \text{ Rs} \]

**Hint:** Subtract the total cost from the total revenue to find profit or loss.

### Step 9: Calculate Profit Percentage
Profit percentage is calculated as:

\[ \text{Profit \%} = \left( \frac{\text{Profit}}{\text{Total Cost}} \right) \times 100 \]
Since the profit is negative, it indicates a loss:

\[ \text{Profit \%} = \left( \frac{-1,52,000}{5,00,000} \right) \times 100 = -30.4\% \]

**Hint:** To find the percentage, divide the profit by the total cost and multiply by 100.

### Final Answer
The profit percentage of the publisher is **-30.4%**, indicating a loss.

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