A and B can finish a piece of work in 16 days and 12 days respectively.
Question:

A and B can finish a piece of work in 16 days and 12 days respectively. A started the work and worked at it for 2 days. He was then joined by B. Find the total time taken to finish the work.

Solution:

Time taken by $\mathrm{A}$ to complete the work $=16$ days

Work done per day by $\mathrm{A}=\frac{1}{16}$

Time taken by $\mathrm{B}$ to complete the work $=12$ days

Work done per day by $\mathrm{B}=\frac{1}{12}$

Work done per day by $\mathrm{A}$ and $\mathrm{B}=\frac{1}{12}+\frac{1}{16}=\frac{4+3}{48}=\frac{7}{48}$

Work done by $\mathrm{A}$ in two days $=\frac{2}{16}=\frac{1}{8}$

Work left $=1-\frac{1}{8}=\frac{7}{8}$

A and B together can complete $\frac{7}{48}$ of the work in 1 day.

Then, time taken to complete $\frac{7}{8}$ of the work $=\frac{7}{8} \div \frac{7}{48}=\frac{7}{8} \times \frac{48}{7}=6$ days

Time taken by $\mathrm{A}$ to complete the work $=16$ days

Work done per day by $\mathrm{A}=\frac{1}{16}$

Time taken by $\mathrm{B}$ to complete the work $=12$ days

Work done per day by $\mathrm{B}=\frac{1}{12}$

Work done per day by $\mathrm{A}$ and $\mathrm{B}=\frac{1}{12}+\frac{1}{16}=\frac{4+3}{48}=\frac{7}{48}$

Work done by $\mathrm{A}$ in two days $=\frac{2}{16}=\frac{1}{8}$

Work left $=1-\frac{1}{8}=\frac{7}{8}$

A and B together can complete $\frac{7}{48}$ of the work in 1 day.

Then, time taken to complete $\frac{7}{8}$ of the work $=\frac{7}{8} \div \frac{7}{48}=\frac{7}{8} \times \frac{48}{7}=6$ days

$\therefore$ Total time taken $=6+2=8$ days.