A can do a piece of work in 10 days, B in 12 days and C in 15 days. All begin together but A leaves the work after 2 days and B leaves 3 days before the work is finished. How long did the work last?

A's 1 day work = $\dfrac{1}{10}$

B's 1 day work = $\dfrac{1}{12}$

C's 1 day work = $\dfrac{1}{15}$

Let the work get completed in x days.

∴ A's 2 days work + B's (x - 3) days work + C's x days work = 1

$\Rightarrow \dfrac{1}{10} \times 2 + \dfrac{1}{12} \times (x - 3) + \dfrac{1}{15} \times x = 1\\[1em] \Rightarrow \dfrac{1}{5} + \dfrac{(x - 3)}{12} + \dfrac{x}{15} = 1\\[1em] \Rightarrow \dfrac{[12 + 5(x - 3) + 4x]}{60} = 1\\[1em] \Rightarrow \dfrac{(12 + 5x - 15 + 4x)}{60} = 1\\[1em] \Rightarrow\dfrac{(9x - 3)}{60} = 1\\[1em] \Rightarrow 9x - 3 = 1 \times 60\\[1em] \Rightarrow 9x - 3 = 60\\[1em] \Rightarrow 9x = 60 + 3\\[1em] \Rightarrow 9x = 63\\[1em] \Rightarrow x = \dfrac{63}{9}\\[1em] \Rightarrow x = 7$

Hence, the work lasts for 7 days.