Solving Work Rate Problems in Partnership: A Mathematical Journey

Imagine a scenario where two individuals, A and B, are tasked with completing a project. A can do the job in 16 hours, while B can do the same in 12 hours. Intriguingly, A joins B 4 hours before the project is completed, after B has been working alone. How many hours did B work on the project alone, and how does it impact their collaborative efforts?

Understanding the Work Rates

Let's start by expressing the work rates of A and B:

A can complete the work in 16 hours, so A's work rate is:

Work rate of A:1/16 work per hour

B can complete the work in 12 hours, so B's work rate is:

Work rate of B:1/12 work per hour

Evaluating B's Solo Work

Let t be the time (in hours) that B works alone.

The work done by B alone is:

Work done by B: t/12

Combining Efforts for the Final Push

A joins B 4 hours before the project is completed. This means both A and B collaborate for 4 hours, while B had already been working alone for t hours.

The total time B worked alone is t hours, and the total time spent on the project is t 4 hours.

Final Work Equation

The total work done by B alone and by A and B together is equal to 1 (the whole work):

Equation:(t/12 (1/12 1/16) * 41

Simplifying the Work Rate

We need to find a common denominator to solve the fractions 1/12 and 1/16:

1/12 4/48, and 1/16 3/48.

Therefore:

1/12 1/16 (4/48 3/48) 7/48

Substituting and Solving the Equation

Substituting back into the equation:

t/12 7/48 * 4 1

7/48 * 4 28/48, which simplifies to 7/12.

The equation now becomes:

t/12 - 7/12 1

Solving for T: B's Solo Work Time

Combining the terms:

t - 7 12

Multiplying both sides by 12:

t - 7 12

Subtracting 7 from both sides:

t 19

Conclusion: B worked alone for 5 hours before A joined. A works for 4 hours only, completing 1/4 of the work. B completes 3/4 of the work in 9 hours, resulting in a total of 5 hours of solo work for B.

Expressed in days, B's solo work time is:

5 hours 5/24 days

This detailed analysis highlights the importance of understanding individual work rates and teamwork in efficiently completing tasks. Solving such problems enhances our mathematical skills and helps in managing tasks more effectively.