Collaborative Work Rate Calculation: How Long Will It Take for A and B to Finish the Work Together?

Let's assume that A can complete a piece of work in 30 days while B can do it in 40 days. The question at hand is to determine the number of days it would take for A and B to finish such a piece of work if they work together.

Understanding the Work Rate

The work rate can be described as the amount of work one can complete in a day. If A can finish the work in 30 days, A's work rate is expressed as:

A's work rate 1??/30 (work per day)

Similarly, if B can finish the work in 40 days, B's work rate is:

B's work rate 1??/40 (work per day)

Calculating the Combined Work Rate

The combined work rate can be calculated by adding A's and B's individual work rates. Therefore, the combined work rate is:

Combined work rate 1/30 1/40

To add these two fractions, let's find a common denominator, which is 120 in this case:

1/30 4/120

1/40 3/120

Combined work rate 4/120 3/120 7/120 (work per day)

Now, we know that A and B together can complete 7/120 of the work in one day. Therefore, the time taken to complete 1 whole work is:

Time 1 / (7/120) 120/7 ≈ 17.14 days

Hence, A and B together will take approximately 17.14 days to finish the work.

Solving the Problem Using Equations

Another approach to solving this problem involves setting up and solving simultaneous equations. Here's how to do it:

1 - 16/a - 44/b 0

1/a 1/b 1/30

Substitute the second equation into the first:

1 - 16(1/30 - 1/b) - 44/b 0

After solving the equation, you will find that:

b 60

This means that B can complete the work in 60 days alone, justifying the result obtained earlier from the work rates approach.

Conclusion

As shown, by understanding work rates and combining them, the time it takes for A and B to complete a task can be easily calculated. Using mathematical approaches, such as work rate calculations and solving equations, we have determined that A and B together will take approximately 17.14 days to finish the work.