A shopkeeper buys a certain number of books for RS.960 would be 4 more. Find the original cost of each book.

Answer

To solve the problem step by step, we will first define the variables and then set up the equation based on the information provided.

### Step 1: Define the variable
Let the original cost of each book be \( X \) rupees.

### Step 2: Determine the number of books bought
The number of books the shopkeeper can buy for Rs. 960 at the cost of \( X \) rupees each is given by:
\[ \text{Number of books} = \frac{960}{X} \]

### Step 3: Determine the new cost of the book
If the cost of each book was Rs. 8 less, the new cost would be:
\[ \text{New cost} = X - 8 \]

### Step 4: Determine the number of books that could be bought at the new cost
At the new cost, the number of books the shopkeeper could buy is:
\[ \text{Number of books at new cost} = \frac{960}{X - 8} \]

### Step 5: Set up the equation
According to the problem, the shopkeeper could have bought 4 more books at the new cost than at the original cost. Therefore, we can set up the equation:
\[ \frac{960}{X - 8} = \frac{960}{X} + 4 \]

### Step 6: Cross-multiply to eliminate the fractions
Cross-multiplying gives us:
\[ 960 \cdot X = 960 \cdot (X - 8) + 4X(X - 8) \]

### Step 7: Expand and simplify the equation
Expanding both sides:
\[ 960X = 960X - 7680 + 4X^2 - 32X \]
Now, cancelling \( 960X \) from both sides:
\[ 0 = 4X^2 - 32X - 7680 \]

### Step 8: Rearranging the equation
Rearranging gives us:
\[ 4X^2 - 32X - 7680 = 0 \]
Dividing the entire equation by 4 simplifies it to:
\[ X^2 - 8X - 1920 = 0 \]

### Step 9: Factor the quadratic equation
We need to factor the quadratic equation \( X^2 - 8X - 1920 = 0 \). We look for two numbers that multiply to \(-1920\) and add to \(-8\). The numbers are \( 40 \) and \( -48 \):
\[ (X - 48)(X + 40) = 0 \]

### Step 10: Solve for \( X \)
Setting each factor to zero gives:
\[ X - 48 = 0 \quad \text{or} \quad X + 40 = 0 \]
Thus, we have:
\[ X = 48 \quad \text{or} \quad X = -40 \]

### Step 11: Determine the valid solution
Since the cost of a book cannot be negative, we discard \( X = -40 \). Therefore, the original cost of each book is:
\[ X = 48 \text{ rupees} \]

### Final Answer
The original cost of each book is **Rs. 48**.

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