Efficiency Challenge: Solving Work Problems with Logical Reasoning and Algebra

Work problems involving rate of completion and time taken often require a combination of logical reasoning and algebraic calculations for a precise solution. This article provides a detailed exploration of such problems, illustrating different methods to arrive at the correct answer.

Problem 1: Ajay and Sanjay Completing a Work

Suppose Ajay can complete a piece of work in 15 days and Sanjay can finish it in 20 days. They work together for 5 days and Ajay leaves. How long will it take Sanjay to finish the remaining work by himself?

Algebraic Solution:

Let's define the part of the work each person can complete per day. Ajay's efficiency is 1/15 and Sanjay's efficiency is 1/20. Together, they can complete (frac{1}{15} frac{1}{20} frac{4}{60} frac{3}{60} frac{7}{60}) of the work per day.

In 5 days, they complete (frac{7}{60} times 5 frac{35}{60} frac{7}{12}) of the work. The remaining work is (1 - frac{7}{12} frac{5}{12}).

Sanjay's efficiency is 1/20, so to complete (frac{5}{12}) of the work, the time taken is:

[frac{5}{12} div frac{1}{20} frac{5}{12} times 20 frac{100}{12} frac{50}{6} 8frac{1}{3}]

Sanjay will take approximately 8.33 days to finish the remaining work.

Logical Reasoning Solution:

Alternatively, let’s think logically. In 60 days, both working together can complete 4 pieces of work. In 4 days, they have completed 1/3 of the work since (frac{4}{60} frac{1}{15}). Hence, A does 2/3 of the work, and B does 1/3 of the work in 4 days. To complete the entire work, A would take 30 days, and B would take 20 days. The remaining work, 2/3, can be done by B in less time. B's rate is (frac{1}{20}) work per day, so the remaining time for B is ( frac{2}{3} div frac{1}{20} frac{40}{3} approx 13.33) days.

The logical solution seems more accurate here, suggesting B will take approximately 13.33 days to complete the remaining work. However, the exact mathematical solution is more precise.

Problem 2: A and B Working Together and Separately

A can complete a piece of work in 12 days and B in 20 days. How long will B take to finish the remaining work after both work together for 3 days?

Algebraic Solution:

A's efficiency is 1/12 and B's efficiency is 1/20. Together, they can complete (frac{1}{12} frac{1}{20} frac{5}{60} frac{3}{60} frac{8}{60} frac{2}{15}) of the work per day.

In 3 days, they complete (3 times frac{2}{15} frac{6}{15} frac{2}{5}) of the work. The remaining work is (1 - frac{2}{5} frac{3}{5}).

B alone can do 1/20 of the work per day, so to complete (frac{3}{5}) of the work, the time taken is:

[frac{3}{5} div frac{1}{20} frac{3}{5} times 20 frac{60}{5} 12]

B will take 12 days to finish the remaining work.

Logical Reasoning Solution:

A and B can do 1/12 and 1/20 of the work per day, respectively. Together, their combined rate is (frac{1}{12} frac{1}{20} frac{5}{60} frac{3}{60} frac{8}{60} frac{2}{15}). After 3 days, they complete (3 times frac{2}{15} frac{6}{15} frac{2}{5}) of the work. The remaining 3/5 of the work needs to be done by B alone at a rate of 1/20. Therefore, the time taken is ( frac{3}{5} times 20 12 ) days.

The logical and algebraic solutions confirm that B will take 12 days to complete the remaining work.

General Strategy for Solving Work Problems

Decompose the problem into smaller parts: Identify the individual efficiencies and combine them for collaborative work. Use the rates and work completed: Calculate how much of the work is done and how much remains. Leverage logical reasoning: Sometimes, simpler logical approaches can provide quick estimates or solutions.

Conclusion: Work problems require careful analysis of individual and combined work rates to determine the total time required to complete a task. Both algebraic and logical methods are effective and should be applied based on the complexity of the problem.