Karim takes 2 days longer than Rahim to produce x units of products. At these rates, Karim & Rahim together take 2 days to produce \(5/6^{th}\) of x units. How many days would Karim take to produce 3x units of this product?

Explanation: Let, Rahim takes y days to produce x units. So, in 1 day Rahim produces \(x/y\) units. So, Karim takes \(y+2\) days to produce x units. So, in 1 day Karim produces \(x/(y+2)\) units. Together, in 1 day they produce \((x/y) + \{x/(y+2)\}\) units. So, in 2 days they produce \(2[(x/y) + \{x/(y+2)]\) units = \(x(4y+4)/y(y+2)\) units. According to question, \(\frac{x(4y+4)}{y(y+2)} = \frac{5x}{6}\). \(\Rightarrow 24y + 24 = 5y^2 + 10y \Rightarrow 5y^2 - 14y - 24 = 0\). \(\Rightarrow (5y+6)(y-4)=0\). So \(y=4\) or \(y=-6/5\). Since y can't be negative, so \(y=4\). So, Karim takes \(y+2 = 4+2 = 6\) days to produce x units. So, to produce 3x units, Karim need \(3\times6 = 18\) days.