Let's use algebra to solve this problem. Let "x" be the number of pens Karim originally bought, and "p" be the cost of each pen in taka. We are given two pieces of information: He bought some pens for 120 taka, which means: x * p = 120 If each pen cost him 2 taka less, he could buy 2 more pens, which means: (x + 2) * (p - 2) = 120 Now, we have a system of two equations: x * p = 120 (x + 2) * (p - 2) = 120 We can solve this system of equations simultaneously. Let's start with equation (1): x * p = 120 Now, let's express "p" in terms of "x" from equation (1): p = 120 / x Now, substitute this expression for "p" into equation (2): (x + 2) * ((120 / x) - 2) = 120 Let's simplify this equation: (x + 2) * ((120 - 2x) / x) = 120 Now, let's eliminate the fractions by multiplying both sides by "x": (x + 2) * (120 - 2x) = 120x Expand and simplify: 120x - 2x^2 + 240 - 2x = 120x Now, bring all the terms to one side: -2x^2 + 240 - 2x = 0 Divide the entire equation by -2 to simplify: x^2 - 120 + x = 0 Now, let's solve this quadratic equation for "x." We can use the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a) In this case, a = 1, b = -120, and c = 1. Plug these values into the formula: x = (-(-120) ± √((-120)² - 4(1)(1))) / (2(1)) Simplify: x = (120 ± √(14400 - 4)) / 2 x = (120 ± √14396) / 2 Now, calculate the square root: x = (120 ± 119.997) / 2 Now, consider both the positive and negative roots: x = (120 + 119.997) / 2 ≈ 119.998 / 2 ≈ 59.999 (approximately 60) x = (120 - 119.997) / 2 ≈ 0.003 / 2 ≈ 0.0015 (approximately 0) Since Karim cannot buy 0 pens, the only reasonable solution is that he originally bought approximately 60 pens. However, the problem states that if he could buy pens for 2 taka less, he could buy 2 more pens. This suggests that he bought 12 pens initially (60 / 5 = 12), and each pen cost 10 taka (120 / 12 = 10). So, Karim initially bought 12 pens.