‘A’ can do a piece of work in 20 days and ‘B’ can do the same work in 15 days. How long will they take to finish the work, if both work together?

The correct answer is $8\frac{4}{7}$ days.

To find the total time taken by both 'A' and 'B' working together to complete the work, we can use the concept of work rates per day.

Step 1: Calculate the work done by each person in 1 day.

Since 'A' can complete the entire piece of work in 20 days, the fraction of work done by 'A' in 1 day is:

$\text{Work done by A in 1 day}=\frac{1}{20}$

Similarly, since 'B' can complete the same work in 15 days, the fraction of work done by 'B' in 1 day is:

$\text{Work done by B in 1 day}=\frac{1}{15}$

Step 2: Calculate the combined work done by both in 1 day.

When 'A' and 'B' work together, their combined 1-day work is the sum of their individual 1-day work rates:

$\text{Combined 1-day work}=\frac{1}{20}+\frac{1}{15}$

To add these two fractions, find the Least Common Multiple (LCM) of the denominators 20 and 15, which is 60:

$\text{Combined 1-day work}=\frac{3+4}{60}=\frac{7}{60}$

Step 3: Calculate the total time required to finish the work together.

The total time required to complete the work working together is the reciprocal of their combined 1-day work rate:

$\text{Total time}=\frac{60}{7}\text{days}$

Converting this improper fraction into a mixed fraction:

$\frac{60}{7}=8\frac{4}{7}$

Thus, working together, 'A' and 'B' will take $8\frac{4}{7}$ days to finish the work.