A person bought a certain number of pens for 800. If he had bought 4 pens more for the same money, he would have paid 10 less for each pen. How many pens did he buy?

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Define the variables
Let \( x \) be the number of pens bought for 800 rupees.

### Step 2: Calculate the cost per pen
The cost per pen when \( x \) pens are bought is given by:
\[ \text{Cost per pen} = \frac{800}{x} \]

### Step 3: Set up the equation for buying 4 more pens
If the person had bought 4 more pens, he would have bought \( x + 4 \) pens for the same amount of money (800 rupees). The new cost per pen would then be:
\[ \text{New cost per pen} = \frac{800}{x + 4} \]

### Step 4: Set up the equation based on the problem statement
According to the problem, the new cost per pen is 10 rupees less than the original cost per pen. Therefore, we can set up the equation:
\[ \frac{800}{x + 4} = \frac{800}{x} - 10 \]

### Step 5: Clear the fractions
To eliminate the fractions, we can multiply through by \( x(x + 4) \):
\[ 800x = 800(x + 4) - 10x(x + 4) \]

### Step 6: Expand and simplify the equation
Expanding both sides gives:
\[ 800x = 800x + 3200 - 10x^2 - 40x \]
Now, simplify the equation:
\[ 0 = 3200 - 10x^2 - 40x \]
Rearranging gives:
\[ 10x^2 + 40x - 3200 = 0 \]

### Step 7: Divide through by 10
To simplify, divide the entire equation by 10:
\[ x^2 + 4x - 320 = 0 \]

### Step 8: Apply the quadratic formula
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = 4, c = -320 \):
\[ x = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 1 \cdot (-320)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ x = \frac{-4 \pm \sqrt{16 + 1280}}{2} \]
\[ x = \frac{-4 \pm \sqrt{1296}}{2} \]
\[ x = \frac{-4 \pm 36}{2} \]

### Step 9: Solve for \( x \)
Calculating the two possible values:
1. \( x = \frac{32}{2} = 16 \)
2. \( x = \frac{-40}{2} = -20 \) (not valid since the number of pens cannot be negative)

Thus, the number of pens bought is:
\[ \boxed{16} \]