**Question:
**

A and B can do a piece of work in 12 days, B and C in 15 days, and C and A in 20 days. How much time will A alone take to finish the job?

**Solution:**

$(\mathrm{A}+\mathrm{B})$ can complete the work in 12 days.

$(\mathrm{B}+\mathrm{C})$ can complete the work in 15 days.

$(\mathrm{C}+\mathrm{A})$ can complete the work in 20 days.

$(\mathrm{A}+\mathrm{B})$ 's 1 day work $=\frac{1}{12}$

$(\mathrm{~B}+\mathrm{C})$ 's 1 day work $=\frac{1}{15}$

$(\mathrm{C}+\mathrm{A})$ 's 1 day work $=\frac{1}{20}$

$2(\mathrm{~A}+\mathrm{B}+\mathrm{C})$ 's 1 day work $=\frac{1}{12}+\frac{1}{15}+\frac{1}{20}=\frac{5+4+3}{60}=\frac{12}{60}=\frac{1}{5}$

$(\mathrm{~A}+\mathrm{B}+\mathrm{C})$ 's 1 day work $=\frac{1}{10}$

$\mathrm{~A}$ 's 1 day work $=\{(\mathrm{A}+\mathrm{B}+\mathrm{C})$ 's 1 day work $\}-\{(\mathrm{B}+\mathrm{C})$ 's 1 day work $\}=\frac{1}{10}-\frac{1}{15}=\frac{3-2}{30}=\frac{1}{30}$

$\mathrm{~A}$ will take 30 days to complete the work, if he works alone.