A alone can do a piece of work in 20 days and B alone can do the same work in 30 days. In how many days can both A and B together complete the same work?

Steps to Solve

1. Calculate A's Work Rate

A can complete the work in 20 days. Thus, A's work rate is: $$ \text{Rate of A} = \frac{1}{20} \text{ (work per day)} $$

2. Calculate B's Work Rate

B can complete the work in 30 days. Thus, B's work rate is: $$ \text{Rate of B} = \frac{1}{30} \text{ (work per day)} $$

3. Calculate Combined Work Rate

To find the combined work rate of A and B together, add their individual rates: $$ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} = \frac{1}{20} + \frac{1}{30} $$

4. Find a Common Denominator

The least common multiple (LCM) of 20 and 30 is 60. Convert the fractions: $$ \frac{1}{20} = \frac{3}{60} \quad \text{and} \quad \frac{1}{30} = \frac{2}{60} $$

5. Add the Work Rates

Now, add the converted work rates: $$ \text{Combined Rate} = \frac{3}{60} + \frac{2}{60} = \frac{5}{60} $$ This simplifies to: $$ \frac{5}{60} = \frac{1}{12} $$

6. Calculate Time Taken to Complete the Work Together

To find the number of days (time) it takes for A and B to complete the work together, take the reciprocal of the combined rate: $$ \text{Time} = \frac{1}{\text{Combined Rate}} = 12 \text{ days} $$