Understanding the Problem: Combined Work Time

The question asks for the total time required to complete a job when two individuals, A and B, work together. We are given the individual times each person takes to complete the job alone.

• A completes the job in 10 days.
• B completes the job in 15 days.

We need to find the time it takes when they collaborate.

Calculating Individual Work Rates

To solve this, we first determine the rate at which each person works. The rate is the fraction of the job completed per day.

• Rate of A: If A completes the entire job (1 job) in 10 days, their rate is \(\frac{1 \text{ job}}{10 \text{ days}}\).
• Rate of B: If B completes the entire job (1 job) in 15 days, their rate is \(\frac{1 \text{ job}}{15 \text{ days}}\).

Calculating Combined Work Rate

When A and B work together, their individual rates add up to give the combined rate of work.

Combined Rate = Rate of A + Rate of B

Combined Rate = \(\frac{1}{10} + \frac{1}{15}\)

To add these fractions, we find a common denominator, which is 30.

Combined Rate = \(\frac{3}{30} + \frac{2}{30}\)

Combined Rate = \(\frac{3+2}{30}\)

Combined Rate = \(\frac{5}{30}\)

Simplifying the fraction, we get:

Combined Rate = \(\frac{1}{6}\) job per day.

This means that working together, A and B complete \(\frac{1}{6}\) of the job each day.

Calculating Total Time Together

The total time taken to complete the job when working together is the reciprocal of their combined work rate.

Time Together = \(\frac{1}{\text{Combined Rate}}\)

Time Together = \(\frac{1}{\frac{1}{6}}\)

Time Together = 6 days.

Conclusion

Therefore, if A and B work together, they will take 6 days to complete the job.