Mohan takes 16 days less than Manoj to do a piece of work. If both working together can do it in 15 days, in how many days will Mohan alone complete the work?

Let no. of days in which Mohan can do work be x

No. of days in which Manoj can do work be x + 16

In one day, Mohan completes $\dfrac{1}{x}$ part of work

In one day, Manoj completes $\dfrac{1}{x + 16}$ part of work

Given, both can do work in 15 days,

$\therefore \dfrac{1}{x} + \dfrac{1}{x + 16} = \dfrac{1}{15} \\[1em] \Rightarrow \dfrac{x + 16 + x}{x(x + 16)} = \dfrac{1}{15} \\[1em] \Rightarrow \dfrac{2x + 16}{x^2 + 16x} = \dfrac{1}{15} \\[1em] \Rightarrow 15(2x + 16) = x^2 + 16x \\[1em] \Rightarrow 30x + 240 = x^2+ 16x \\[1em] \Rightarrow x^2 + 16x - 30x - 240 = 0 \\[1em] \Rightarrow x^2 - 14x - 240 = 0 \\[1em] \Rightarrow x^2 - 24x + 10x - 240 = 0 \\[1em] \Rightarrow x(x - 24) + 10(x - 24) = 0 \\[1em] \Rightarrow (x + 10)(x - 24) = 0 \\[1em] \Rightarrow x + 10 = 0 \text{ or } x - 24 = 0 \\[1em] \Rightarrow x = -10 \text{ or } x = 24.$

Since, no. of days cannot be negative,

∴ x ≠ -10.

Hence, Mohan alone completes the work in 24 days.