Hemant purchased some book and by selling 40% of total books he will get cost price of 80% books. If he sells 70% of remaining books at half of its initial profit percent and rest of the books remain unsold, find his overall profit %.

Options

A. `40%`
B. `45%`
C. `75%`
D. `43%`

Answer

To solve the problem step by step, let's break it down:

### Step 1: Define the total number of books and their cost price.
Let the total number of books purchased by Hemant be 100.

### Step 2: Calculate the selling price of the first 40% of the books.
Hemant sells 40% of the total books:
- Number of books sold = 40% of 100 = 40 books.

According to the problem, by selling these 40 books, he gets the cost price of 80 books. Therefore, we can set up the following equation:
- Selling Price (SP) of 40 books = Cost Price (CP) of 80 books.

### Step 3: Establish the ratio of Selling Price to Cost Price.
Let the cost price of each book be 1 (for simplicity). Thus, the cost price of 80 books = 80.
- Therefore, SP of 40 books = 80.

Now, we can find the SP to CP ratio:
- SP of 40 books = 80
- CP of 40 books = 40 (since CP of each book is 1)

Thus, the ratio of SP to CP is:
\[ \text{SP} : \text{CP} = 80 : 40 = 2 : 1 \]

### Step 4: Calculate the initial profit percentage.
Using the profit formula:
\[ \text{Profit} = \text{SP} - \text{CP} \]
\[ \text{Profit} = 80 - 40 = 40 \]

Now, the profit percentage is calculated as:
\[ \text{Profit Percentage} = \left( \frac{\text{Profit}}{\text{CP}} \right) \times 100 \]
\[ \text{Profit Percentage} = \left( \frac{40}{40} \right) \times 100 = 100\% \]

### Step 5: Calculate the remaining books after the first sale.
After selling 40 books, the number of books remaining is:
- Remaining books = 100 - 40 = 60 books.

### Step 6: Calculate the number of books sold at half the initial profit percentage.
Hemant sells 70% of the remaining books:
- Number of books sold = 70% of 60 = 42 books.

### Step 7: Calculate the selling price of these 42 books at half the initial profit percentage.
The initial profit percentage was 100%, so half of that is 50%.

The new selling price for these books can be calculated as follows:
- New SP = CP + (50% of CP) = CP + 0.5 * CP = 1.5 * CP.

Since the CP of each book is 1, the SP for 42 books is:
\[ \text{SP} = 42 \times 1.5 = 63 \]

### Step 8: Calculate the total selling price.
Now, we can find the total selling price:
- Total SP = SP from first sale + SP from second sale
- Total SP = 80 + 63 = 143.

### Step 9: Calculate the total cost price.
The total cost price for 100 books is:
- Total CP = 100 (since each book costs 1).

### Step 10: Calculate the overall profit percentage.
Now, we can calculate the overall profit:
\[ \text{Overall Profit} = \text{Total SP} - \text{Total CP} = 143 - 100 = 43 \]

The overall profit percentage is:
\[ \text{Overall Profit Percentage} = \left( \frac{43}{100} \right) \times 100 = 43\% \]

### Conclusion
Thus, Hemant's overall profit percentage is **43%**.

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