A shopkeeper buys a number of books for Rs 80. If the had bought 4 more books for the same amount, each book would have cost Rs 1 less. How many books did he buy?

Answer

To solve the problem step-by-step, let's define some variables and set up the equations based on the information provided.

### Step 1: Define Variables
Let the number of books the shopkeeper originally bought be \( x \).

### Step 2: Calculate the Cost per Book
The total cost for the books is Rs 80. Therefore, the cost per book when he buys \( x \) books is:
\[ \text{Cost per book} = \frac{80}{x} \]

### Step 3: Set Up the New Scenario
If the shopkeeper had bought 4 more books, he would have bought \( x + 4 \) books for the same amount of Rs 80. The cost per book in this case would be:
\[ \text{New cost per book} = \frac{80}{x + 4} \]

### Step 4: Set Up the Equation
According to the problem, if he bought 4 more books, each book would cost Rs 1 less. Therefore, we can set up the following equation:
\[ \frac{80}{x + 4} = \frac{80}{x} - 1 \]

### Step 5: Solve the Equation
Now, let's solve the equation step by step.

1. Start with the equation:
\[ \frac{80}{x + 4} = \frac{80}{x} - 1 \]

2. Clear the fractions by multiplying through by \( x(x + 4) \):
\[ 80x = 80(x + 4) - x(x + 4) \]

3. Distributing on the right side:
\[ 80x = 80x + 320 - (x^2 + 4x) \]

4. Simplifying the equation:
\[ 80x = 80x + 320 - x^2 - 4x \]
\[ 0 = 320 - x^2 - 4x \]

5. Rearranging gives us a quadratic equation:
\[ x^2 + 4x - 320 = 0 \]

### Step 6: Factor the Quadratic
Now we will factor the quadratic equation:
\[ (x + 20)(x - 16) = 0 \]

### Step 7: Find the Values of \( x \)
Setting each factor to zero gives us:
1. \( x + 20 = 0 \) → \( x = -20 \) (not a valid solution since the number of books cannot be negative)
2. \( x - 16 = 0 \) → \( x = 16 \)

### Conclusion
The shopkeeper originally bought **16 books**.

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