Both methods seem to approach the problem of determining how much work A and B can complete together, but they are structured slightly differently. Let's break down both approaches to evaluate their efficiency:

Method 1:

This method approaches the problem by:

1. Finding the LCM of the total days it takes for A and B to complete the work.
2. Determining the parts each of them can do in a day (A does 4 parts, B does 3 parts).
3. Calculating the total parts done in 4 days and the leftover work (28 parts completed out of 60 total parts).
4. Then computing the remaining work as a fraction of the total work.

Method 2:

This method uses the rate of work done by each person:

1. A completes the work in 15 days, so the work done by A in 1 day = $\frac{1}{15}$.
2. B completes the work in 20 days, so the work done by B in 1 day = $\frac{1}{20}$.
3. Adding their work rates together: $\frac{1}{15} + \frac{1}{20} = \frac{7}{60}$ of the total work completed in one day.
4. For 4 days, they complete $4 \times \frac{7}{60} = \frac{7}{15}$ of the total work.
5. Then, the remaining work is $1 - \frac{7}{15} = \frac{8}{15}$.

Comparison:

• Clarity and Ease: Method 2 directly uses work rates and gives a clearer picture of how much work is done daily by both A and B. It is easier to understand conceptually because it relies on the basic principle of work rates and fractions.

• Steps Involved: Method 2 has fewer steps and less manual computation than Method 1, which involves calculating parts, days, and the leftover work in terms of parts. In contrast, Method 1 may feel less straightforward because it converts the days into parts, which can add complexity.

• Efficiency: Method 2 is more efficient for this kind of problem because it directly calculates how much of the work is done by A and B together, avoids unnecessary intermediate steps, and leads quickly to the remaining fraction of work.

Conclusion:

Method 2 is more efficient because it uses the fundamental work-rate formula and involves fewer steps. It directly solves for how much work A and B complete together and what remains, making it easier to follow and quicker to compute.

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Do you want details on any part of this solution? Here are some related questions you might find interesting:

1. What is the concept of LCM in work problems, and how does it help?
2. How can you calculate the combined work rates of multiple workers?
3. How do you determine the remaining work when a portion is completed?
4. What is the significance of work rates like $\frac{1}{15}$ or $\frac{1}{20}$ in these problems?
5. Can this method be applied if more people are involved in the work?

Tip: When solving work problems, always focus on the rate of work (work done per day) for efficiency.